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Street numbers, money in bank accounts, points on number lines, quantum particles as contrasted with distinguishable manipulatives For our purposes, infinity is not a number Permanent link 2 Algebra Variables: At once arbitrary, yet specific and particular (a.s.a.p.) Functions, composition, and inverse f(x) is not a function. f(x) is not the function f. f(x) is the particular value associated, in the big picture of the function f, with x, a number that is at once arbitrary, yet particular and specific Inverse functions do not always exist First glimpse at the complex plane and i := √ -1 Permanent link 3 Quadratic formula Linear combination of terms in a polynomial Zeroes or "roots" of a function Completing the square Permanent link Combinatorics b: Combinations We obtain the famous (L + N)! / (L! N!) formula for counting the number of ways to arrange L indistinguishable objects and N indistinguishable objects together in a row. This is also the number of combinations of L objects that can be drawn from a container of L + N objects. 10 Combinatorics c: Binomial theorem We use the formula for combinations from the previous video to write an expression for the binomial quantity (x + y)p. In some applications, only a small number of terms in the resulting sum are necessary for approximate calculations. ε-δ definition of limit, notion of "arbitrarily close" Example of calculating a limit Limits do not always exist For an example of a strategy for writing ε-δ proofs useful for plots of functions that have curvature, please see Yosen Lin's examples (example # 4 on p. 3-4). Permanent link 12 Differentiation Differentiation a: Derivatives and differentials We define the derivative, caution against interpreting differentials as numbers, and remark that derivatives do not always exist. It is important to become familiar with derivatives because they provide a basic vocabulary for talking about dynamical systems in the natural sciences (including in biology). Permanent link 13 Differentiation b: Power rule We will later learn that many seemingly complicated functions can be approximated using sums of power law terms. To study the slopes of these terms, we use the power rule that we derive in this video, which is written d(xn)/dx = nxn-1. 14 Differentiation c: Chain rule (for composite functions) One way to combine functions is to nest functions within each other. The chain rule is used to study the slopes of "composite" functions. The rule is written d(g(f))/dx = dg/df df/dx. 15 Differentiation d: Products and quotients Another way to put basic functions together is to write their expressions next to each other as a product. In this video, we derive the product rule, which is used in such situations. The product rule is written d(fg)/dx = (df/dx)g + f(dg/dx). 16 Differentiation e: Sinusoidal functions The derivative of sine is cosine, and the derivative of cosine is negative sine. This back-and-forth relationship is a hallmark of dynamical systems that might support oscillations. Thus, this pattern, which you will derive in this video, is important to keep in mind when you later study biological oscillations. 17 Partial differentiation When a function depends on multiple independent variables, the curly-d symbol, ∂, denotes slopes calculated by jiggling only one independent variable at a time Permanent link 18 Power series representations Power series representations a: Second derivative and curvature Using a power series representation is like using decimal representation. Both techniques organize the description of the target object at levels of increasing refinement. In this first video, we show that the second derivative corresponds to the curvature of a plot. In this way, we strengthen intuition that higher-order derivatives can also have geometric interpretations. Permanent link 19 Power series representations b: Determining power series terms We imitate a function by combining the descriptions of its geometric properties as embodied in its value and the values of its higher derivatives at an expansion point. 20 Power series representations c: Power series for sine We obtain a power series representation for the sine function expanded about the point θ = 0. 21 Power series representations d: Decimal approximation for π Using the first three terms of the power series representation for sine we obtained in the previous video, we iteratively approximate π to four decimal places. 22 Integration Integration a: Area under a curve In these four videos, we develop a familiar with integration that will later be useful for deducing functions of time (e.g. number of copies of a molecule as a function of time) using rates of change (e.g. the first derivative of the number of copies of a molecule with respect to time). In this first video, we develop the concept of the definite integral in terms of the area under a curve. Permanent link 23 Integration b: First fundamental theorem of calculus In this video, we demonstrate that differentiation undoes integration. This is called the first fundamental theorem of calculus. 24 Integration c: Second fundamental theorem of calculus We demonstrate that integration undoes differentiation. This is called the second fundamental theorem of calculus. This theorem allows us to construct a table of integrals using differentiation rules we previously learned. 25 Integration d: Change of variables rule Sometimes, superficial differences can make it seem that a listing in an integration table does not match the integral we want to study. We develop a change of variables (also called a "u-substitution") rule that can sometimes help us to identify a match between an integral we want to study and a listing in a table. 26 Separation of variables Two wrongs make a right Tear two differentials apart as though they retained meaning in isolation Slap on the smooth S integral sign as though it were a unit of meaning itself, even without a differential You get the same integral expression you would obtain long-hand using u-substitution or "change of variables" in integrals Permanent link 27 Euler's number I Euler's number 1a: Compound interest Compounding interest with arbitrarily short compounding periods Power series representation of ex Permanent link This is a canonical worked problem from introductory systems biology. We will explain one way to fantasize about the classic protein dynamics equation dx/dt = β - αx and analytically demonstrate that protein "rise time" depends on degradation rate only. EGT 1a: Population dynamics with interactions Equations for collisional population dynamics using law of mass action An outcome of the prisoner's dilemma is simultaneous survival of the relatively most fit with decrease in overall fitness EGT 1b: Introduction to tabular game theory Tabular game theory An outcome of the prisoner's dilemma is simultaneous stability of D with, as a consequence, lower than maximum possible payoff for D Our first verbal suggestion (1) that payoffs from tabular game theory can be associated with rate coefficients from the population dynamics in part 1a, and (2) that part 1a should be referred to as evolutionary game theory 41 Evolutionary game theory II EGT 2a: Evolution resulting from repeated game play In the previous slide deck, we noted similarities between population dynamics and business transaction payoff pictures. In this and the next video, we provide deeper understanding of these connections. In this video, we derive the population dynamics equations in such a way that it is natural to say that cells being modeled repeatedly play games and are subject to game outcomes. Permanent link 42 EGT 2b: Relationship between time and sophisticated computation Repeated simple interactions in a population of robotic replicators can achieve results seemingly related to results obtained from sophisticated computations. The use of population dynamics and business transaction payoff matrix analyses from the previous slide deck to obtain this understanding is an example of quantitative reasoning. Uncertainty propagation a: Quadrature Quadrature formula is a result of Taylor expanding functions of multiple fluctuating variables, assuming that fluctuations are independent, and then applying the identity "variances of sums are sums of variances" Permanent link Uncertainty propagation c: Square-root of sample size (√n) factor Origin of the famous √n factor by which the standard deviation of the sample means is smaller than the standard deviation of the measurements (parent distribution) 52 Uncertainty propagation d: Comparing error bars visually Are error bars non-overlapping, barely touching, or tightly overlapping? What p-value do people associate with the situation in which error bars barely touch? 53 Uncertainty propagation e: Illusory sample size "I quantitated staining intensity for 1 million cells from 5 patients, everything I measure is statistically significant!" It is quite possible that you need to use n = 5, instead of 5 million, for the √n factor in the standard error. Poissonian copy numbers b: Stochastic synthesis and degradation Model: RNA polymerase makes many (usually unsuccessful) independent attempts to initiate transcription. Once a mRNA strand is produced, it begins to make independent (usually many unsuccessful) attempts to be degraded. Result: As in part a, mRNA copy numbers are Poisson distributed DEs IIIb: Eigenvector-eigenvalue analysis Determine the directions of "unbending" trajectories for a more precise hand sketch of the phase portrait 69 DEs IIIc: The cribsheet of linear stability analysis Use eigenvalue-eigenvector analysis to find analytic solutions for linear systems and describe the qualitative features of trajectories approaching, side-swiping, or departing from steady state. Additional activity: You may skim Ferrell, Jr., Tsai, and Yang, "Modeling the cell cycle: Why do certain circuits oscillate?" Cell, 144: 874-885 (2011)(online). Comment on how the positive-feedback term in Eqtn. 25 (pg. 882) contributes to the difference between the phase portraits in Fig. 4B (pg. 878) and Fig. 8B (pg. 883). The article describes the positive-feedback in terms of a time delay. Please describe the contribution of the positive-feedback term to stable oscillations instead in terms of "twisting nullclines" from the video tutorial. Seeing what computers can do In this activity, you will play against the computer in Blizzard's StarCraft for 2 hrs and in Sid Meier's Civilization for 2 hrs. WARNING: This activity might require rehabilitation and video game addiction treatment (PubMed). 74 Cellular automata Cellular automata a Deterministic cellular automata In this video, we see that limiting dispersal of seeds of annual plants can increase the proportion of the copper-colored subpopulation, whereas thorough mixing instead allows the denim plant subpopulation to dominate quickly. Additional activities: Refer to a similar model in Nowak and May, "Evolutionary games and spatial chaos," Nature359:826-829 (1992) (online). Watch Athena Aktipis talk about the walk-away model, which can contribute to the evolution of cooperation in highly-mobile populations (University of California, Los Angeles, Center for Behavior, Evolution, and Culture 2009, 1-hr video online) Permanent link Cellular automata b Stochastic cellular automata Toy agent-based model You will program a simple ABM For more extensive discussion, see Athena Aktipis's page on agent-based modeling (online). Statistical physics 101a: Fundamental postulate of statistical mechanics Systems have states and energy levels Energy can be exchanged between parts of a world If the Hamiltonian of the world is time-independent, the overall energy of the world is conserved Fundamental postulate of statistical mechanics: In an isolated system, all accessible microstates are accessed equally Permanent link 76 Statistical physics 101b: Notating configurations of a system with multiple parts Direct product 77 Statistical physics 101c: Distribution of energy between a small system and a large bath Bath: many parts Number of ways to find the bath configured exponentially decays with increasing system energy Boltzmann factor 78 Statistical physics 101d: Expressions for calculating average properties of systems connected to baths The system energy most typically observed is the one that corresponds to the greatest number, W, of configurations of the world Ways (W), entropy (sigma), free energy (F), probability (P), partition function (Z), taking derivative of Z Maximizing ways of the world Maximizing entropy of the world Minimizing free energy of the system 79 Ideal chain Ideal chain a: Introduction to model A series of links pointed up or down Permanent link Ideal chain c: Expectation of energy and elongation For heavy weights, the chain tends to be found extended fully. For lesser weights, the chain can be found partially crumpled, with the weight lifted, and with energy given to the bath. I take tonight's red eye to give a talk on quantitative biology in the morning. I haven't had time to learn this field. How can I learn to use buzzwords convincingly? Download the videos in this digest (follow the vimeo links in the video lightboxes). The indicated segments can be viewed in under 2 hours, and even an hour's sample should offer an informative taste of reasoning styles and topics commonly encountered in introductory quantitative biology. Buzzwords are like salt. Avoid unconvincing overuse. Observe that the videos in the digest never use the words "complexity" or "emergence" even though both terms could be used repeatedly. My institution already trains faculty in bioinformatics. How much of this website can I skip? This is not a bioinformatics course (there's only a little bit of rudimentary probability). Going in the other direction, much of the content from this course might be missing from your bioinformatics training. To determine whether you are already familiar with styles of reasoning in quantitative biology, respond to the quiz questions below and view the accompanying videos that follow. I just got funded to collaborate in quantitative biology. How do I find experts to coach me to understand the mathematical models I need to use? You could test possible instructors by asking them to help you work through some of the quiz questions below. See how they answer (it's probably best to watch the corresponding video sections ahead of time). Quiz 1: Protein level dynamics In a toy model of a cell, protein X is produced according to a translation rate coefficient and eliminated according to a degradation rate coefficient. The protein copy number at which the rates for these processes balance is called the steady-state level. The time it takes for a cell initially containing zero copies of protein X to accumulate half the steady-state level is called the "rise time." Which statement describes the rise time? Quiz 2: Law of mass action In this course, we use "law of mass action" to refer to an idea that chemical reaction kinetics can be modeled using rate formulas containing products of abundances of reactants raised to exponents. Which statement is best? Mass-action exponents and stoichiometric coefficients can be related using a probabilistic model of molecular collisions, so the stoichiometric coefficients of reactants in the proximate reaction that generates products under study can be deduced using the exponents in an equilibrium constant. Instantaneous collisions of more than two hard-spheres are exceedingly rare, so mass-action rate formulas cannot be used to model reactions involving more than two reactants. The warning against reading mass action exponents off of stoichiometric coefficients derives from the low likelihood of instantaneous collisions between more than two hard spheres. This warning can be relaxed when molecules have non-spherical geometries, and instantaneous collisions between three, or even four, molecules is common, as long as one of the reactants is not spherical. Taylor expansion of complicated functions of reactant concentrations can produce sums of mass-action terms. In other words, stringing together mass-action formulas can produce experimentally accurate mathematical descriptions while concealing insight in the same way that epicycles permit accurate description of celestial motion while obscuring Kepler's laws. Mass-action exponents must be empirically derived from time-course observations and cannot be deduced from stoichiometric coefficients. Quiz 3: Evolutionary game theory Evolutionary game theory is an example of a modeling framework used in the NCI Physical Sciences-Oncology Network to understand social aspects of biological systems. What statement about evolutionary game theory is best? Evolutionary game theory refers to a collection of mathematical models in which organisms (e.g. cells) are modeled as automated, robotic replicators. The propensity with which a replicator generates progeny is modified by the time-frequency with which it encounters other replicators (e.g. through pairwise interactions). In the prisoner's dilemma, defector population share and per capita fitness both increase. However, average fitness decreases because interaction between defectors and cooperators decreases the per capita fitness of cooperators more than enough to cancel out the fitness increase of defectors. In normal tissue microenvironments, individual cells typically express rational behaviors (i.e. protecting long-term survival of the host). Following carcinogenesis, however, cells can also display irrational behaviors, such as circumvention of stress-related barriers. Because cells can now choose between two classes of strategies, it becomes necessary to employ evolutionary game theory to determine how individual cells might make these choices and alter overall tissue system dynamics. John Maynard Smith understood the evolution of socially interacting biological individuals by describing the possible behaviors that individual agents could choose. The notion of choice in this context refers to the essential assumption that each individual organism expresses a strategy that maximizes its payoff, as computed using a payoff matrix. A startling result is the applicability of this kind of modeling, not just to organisms that perform advanced cognition, e.g. hawks and doves, but also, to individual cells, for which evolution had long been thought to proceed primarily in a Darwinian fashion. In traditional models of population dynamics, cell numbers vary as though cells individually solved for anti-derivatives and then adjusted their behavior to obey governing equations. In evolutionary game theory, individual cells, instead, perform sophisticated computations using payoff matrices to decide among strategies so as to realize Nash equilibria. Track Topic Slides Video Description D5 Evolutionary game theory I EGT 1a: Population dynamics with interactions Equations for collisional population dynamics using law of mass action An outcome of the prisoner's dilemma is simultaneous survival of the relatively most fit with decrease in overall fitness Quiz 4: Poissonian copy numbers Some quantitative biologists say that the copy numbers of mRNA in simple models of gene transcription are stochastic and Poisson distributed. Which of the following explains this claim? The copy number of a species of mRNA can be small (~1 copy per cell or less). When a system is composed of a small number of parts, intrinsic stochastic fluctuations are generated. The copy number counts are Poisson distributed because the transcription machinery "forgets" whether it has previously generated a strand of mRNA in a time interval much shorter than needed to produce, on average, one copy of mRNA. The copy number of a species of mRNA can be small (~1 copy per cell or less). This means that the molecule is "rare," so the limit of rare events, also known as the Poisson distribution, accurately models copy-number fluctuations. If the transcription machinery for a species of mRNA "forgets" whether it has previously generated a copy of mRNA over a time scale much shorter than the duration that produces, on average, one mRNA strand, then the generation of mRNA is a Poisson process across time. The integral over an interval is a function evaluated on the boundary. As a rough example, the number of copies of a species of mRNA at a time of observation equals the number of copies produced up until that time since a reference time in the past (minus the number of copies generated up until a lifetime earlier than the time of observation). In this example of the second fundamental theorem of calculus, the number of copies of mRNA counted at a time is Poisson distributed because it inherits the statistical properties of the Poisson process across time described above. A "Poisson distribution" refers to a distribution generated through natural random fluctuations. This is in contrast to fluctuations in engineered structures and circuits that use metabolic energy input to shape their distributions (i.e. the end products of biological signaling cascades can be log-normally distributed, rather than Poisson distributed). Hypothesizing a Poisson distribution for the copy numbers of a species of mRNA is equivalent to claiming that transcription occurs with minimal metabolic energy input. Quiz 5: Quasispecies Quasispecies models are used to understand the compositions of populations of cells in terms of mechanisms for genetic mutation. Choose the best statement: The genotype that dominates a population does not necessarily correspond to the genotype underlying those cells that most rapidly produce offspring. The dominant genotype is not necessarily the genotype with greatest fitness. Classical quasispecies models are sometimes referred to as mutation-selection models. A fundamental limitation of these models is an assumption of thorough mixture. The dynamics supported by cells undergoing mutation, selection, and spatial movement are examples of emergent phenomena because they are not easily predicted from models of mutation and selection alone. A mutational meltdown in cancer occurs when a sudden accumulation of dangerous mutations in cancer cells threatens the host. Dangerous mutations can include, for example, mutations that increase the likelihood of survival cancer cells (e.g. increased proliferation, insensitivity to growth control signals, etc.). Analyzing quasispecies models, mathematicians recommend decreasing the mutation rates of cancer cells to reduce the chances of a mutational meltdown in a tumor. Mutation leading away from the genotype of greatest fitness must be slow enough in order for this genotype to be dominant. Otherwise, this genotype will command only minority population share (in some models, no finite number will be small enough to represent population share). Condensation onto a dominant sequence at low mutation rates is often compared to the condensation of a population of bosons into the ground orbital at low temperatures because mutation is analogous with maximizing entropy of the thermal bath and selection is analogous with maximizing the entropy of the bosons. Quiz 6: Adaptation A gene-expression network composed precisely of nodes A, B, and C is said to exhibit adaptation. In the following, a step change in the level of node A is externally applied and maintained. Which one of the following is the best example for teaching how "adaptation" is achieved in systems biology? A step increase in node A is isolated. Nodes B and C are unchanged. A step increase in node A leads to a temporary increase (followed by a return to steady state) in node C. Node B is unchanged. A step increase in node A eventually leads a step increase in node C and a step decrease in node B. Both nodes C and B achieve new steady states, but with a delay so that no changes in node B and C are observed in the first moments after the increase in node A. A step decrease in node A initially causes a temporary decrease in node C. Node C rises and returns to its initial steady-state level owing to variation in node B. Track Topic Slides Video Description D8 Differential Eqtns IV DEs IVa: Adaptation Please see the excerpt from 19 min 14 sec to 21 min 27 sec. This video is inspired by Ma, Trusina, El-Samad, Lim, and Tang, "Defining network topologies that can achieve biochemical adaptation," Cell, 138: 760-773 (2009) (online). To view the short sequel to this video, jump to the Permanent link in main curriculum \| Permanent link in digest Quiz 7: Oscillations Choose the best statement regarding oscillations in mathematical biological models. Cyclic movement through phase space depends on deterministic, programmed behavior. Adding stochastic fluctuations to a deterministic system does not generate oscillatory trajectories. Instead, any oscillations originally present are damped out, if not completely eliminated. Oscillatory circuits can generate a variety of kinds of signals. These need not resemble sines and cosines in time plots or perfect circles and ellipses in phase planes. For example, time courses can look like momentary spikes, and phase-plane trajectories could resemble rounded squares and triangles. Oscillations correspond to loops in phase space. Such a structure is essentially 2-dimensional. Oscillations occur in high-dimensional phase spaces only when most of the degrees of freedom decouple from the pair that defines the phase portrait in which loops can be drawn. Even though spiral and closed-loop trajectories qualitatively resemble each other in phase portraits, they correspond to different biological network topologies. In other words, it is usually not possible to construct a protein-interaction network that can support both kinds of trajectories with merely adjustments to numerical parameters. This implies that therapies that only mildly suppress or activate a network component are unlikely to change the qualitative form of system oscillations. Oscillations are often visualized as loops in two dimensional slices of phase space. This kind of depiction is, in part, the consequence of arbitrary convention. Reversing one of the two axes by multiplying it by a negative sign reverses the orientation of its back-and-forth motion, and the motions along the two axes now synchronize in a way that traces cyclic motion along on a line, rather than a loop around a plane. Thus, oscillation can be represented in a 1-d "space." Quiz 8: Phenotypic stochasticity A cell can execute stochastic transitions between phenotypic states without need for gene sequence alterations or large-scale genomic rearrangement. This can lead to a type of cell individuality that has been called epigenetic or "non-genetic." Phenotypic stochasticity is actively studied in stem cell/developmental biology and cancer biology. Choose the best statement: Whereas a genotype rigidly corresponding to one phenotypic state might be suited to one environment, but stressed under another, a genotype underlying a plastic collection of interconvertable phenotypes is protected from selection. Darwinian evolution occurs through selection on underlying genetic variation. For this reason, genetic evolution does not proceed in the presence of stochastic phenotypic fluctuations and non-genetic network adaptations. Models based on phenotypic stochasticity and models based on cell-cycle phase specific therapeutic sensitivity both suggest adjusting dose timing according to the dynamics of non-genetic variation in the targeted cell population. However, models based on phenotypic stochasticity and cell-cycle specificity are distinct because stochastic transitions occur with fluctuating waiting times, whereas the cell-cycle proceeds in clockwork fashion through phases of well-defined duration. Therapeutic failure is inevitable any time cells with a phenotype impervious to a therapeutic modality are present at the onset of treatment. Even if such a subpopulation is initially small, it will be selected for and eventually dominate. In a population regenerated by tumor [re]-initiating cells (TICs, cancer stem cells, CSCs), for example, therapy will eventually fail if the treatment modality cannot directly kill the TIC phenotype. Drug-sensitive cells can enter relatively drug-resistant states through non-genetic mechanisms over time scales of days and weeks. Phenotypic stochasticity contributes to therapeutic failure. Owing to the "non-genetic" character of the cell-cell individuality that phenotypic stochasticity can generate, the population dynamics resulting from phenotypic fluctuations cannot be described using principles of Darwinian evolution. Quiz 9: Cellular automata and spatiality Two populations of annual plants, "cooperators" and "defectors," are sown on a field. Prisoner's dilemmas describe the chemical and mechanical contact that occurs repeatedly between pairs of neighboring plants. These interactions determine the numbers of seeds that the plants contribute to the next generation. Which statement is true? Increasing the spatial area over which offspring randomly disperse spreads the offspring of defectors too thin, making it more difficult for defectors to compete with dense pockets of cooperators. These pockets of cooperators survive and perpetuate heterogeneous co-existence. Increasing the spatial area over which offspring randomly disperse promotes heterogeneous co-existence because survival of cooperators relies on their ability to move away from defectors in an ongoing cat-and-mouse chase. Increasing the spatial area over which offspring randomly disperse makes it easier for defectors to take over the lattice and more difficult to realize heterogeneous co-existence. Increasing the spatial area over which offspring randomly disperse promotes heterogeneous co-existence because defectors and cooperators both have increased chances of bumping into other cooperators.
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"Math Connects: Concepts, Skills, and Problem Solving" was written by the authorship team with the end results in mind. They looked at the content needed to be successful in Geometry and Algebra and backmapped the development of mathematical content, concepts, and procedures to PreK to ensure a solid foundation and seamless transition from grade level to grade level. The series is organized around the new NCTM Focal Points and is designed to meet most state standards. "Math Connects" focuses on three key areas of vocabulary to build mathematical literacy, intervention options aligned to RtI, and a comprehensive assessment system of diagnostic, formative, and summative assessments. Description: This is the first systematic exposition of random sets theory since Matheron (1975), with full proofs, exhaustive bibliographies and literature notesInterdisciplinary connections and applications of random sets are emphasized throughout the bookAn extensive bibliography in the book is available ... Description: NoteablesTM Interactive Study Notebook with FoldablesTM is a complete note taking system with guided note taking for every lesson in a workbook format. Plus, vocabulary builder and a chapter test preparation section are included for every chapter.Book Format: Paperback. Number ... Description: Elementary Algebra, 6/e is part of the latest offerings in the successful Dugopolski series in mathematics. Given the importance of examples within a math book, the author has paid close attention to the most important details for solving the ...
The Importance of Mathematics to ConstructionPosted: Fri, 23 Dec 2011 17:13:00 +0000Imagine you building your first home or making your first set of cabinets without ever having to know how to read a tape measure. "Well, you've over slept and need to be wakened". Just the very thought of building or making anything requires that you completely understand how to properly read a standard tape measure which includes a combination of whole numbers, fractions,decimals and equivalents. Reviews: 5.0 out of 5 stars Very Useful, October 30, 2011 By E. Stole "Me" (LV usa) (REAL NAME) This review is from: Applying Mathematics to Construction: Carpentry Mathematics & Estimating (Paperback) I've read many construction books and math text books that do a pretty good job at enhancing my understanding of carpentry mathematics, but I must say this book tops them all. The methods used in the mathematical sense will blow your mind away. Lot's of times I found myself saying, "Wow I didn't even know it could be done this way" , or "That made it so much easier". This book will definitely help you in many areas, even if you think you know it all, you don't. It takes a gifted mind to be able to come up with these formulas that no one else really uses. I recommend this book to anyone seeking to better themselves in estimating, framing, exterior AND interior finish and trim work. instead of beating yourself up trying to figure out the math part of a project, take a moment and go through this book and you will be amazed at how much easier things can be done. "...
More About This Textbook Overview The Mathematical Olympiad examinations, covering the USA Mathematical Olympiad (USAMO) and the International Mathematical Olympiad (IMO), have been published annually by the MAA American Mathematics Competition since 1976. The IMO is the world mathematics championship for high school students. It takes place every year in a different country. The IMO competitions help to discover, challenge, and encourage mathematically gifted young people all over the world. The USAMO and the Team Selection Test (TST) are the last two stages of the selection process leading to selection of the US team in the IMO. The preceding examinations are the AMC 10 or AMC 12 and the American Invitational Mathematics Examination (AIME). Participation in the AIME, USAMO, and the TST is by invitation only, based on performance in the preceding exams of the sequence. In addition to presenting their carefully written solutions to the problems presented here, the editors have provided remarkable solutions developed by the examination committees, contestants, and experts, during or after the contests. They also provide a comprehensive guide to other materials on advanced problem solving. This collection of excellent problems and beautiful solutions is a valuable companion for students who wish to develop their interest in mathematics outside the school curriculum and to deepen their knowledge of mathematics. Preface This book is intended to help students preparing to participate in the USA Mathematical Olympiad (USAMO) in the hope of representing the United States at the International Mathematical Olympiad (IMO). The USAMO is the third stage of the selection process leading to participation in the IMO. The preceding examinations are the AMC 10 or the AMC 12 (which replaced the American High School Mathematics Examination) and the American Invitational Mathematics Examination (AIME). Participation in the AIME and the USAMO is by invitation only, based on performance in the preceding exams of the sequence. The top 12 USAMO students are invited to attend the Mathematical Olympiad Summer Program (MOSP) regardless of their grade in school. Additional MOSP invitations are extended to the most promising non-graduating USAMO students, as potential IMO participants in future years. During the first days of MOSP, IMO-type exams are given to the top 12 USAMO students with the goal of identifying the six members of the USA IMO Team. The Team Selection Test (TST) simulates an actual IMO, consisting of six problems to be solved over two 4 1/2 hour sessions. The 12 equally weighted problems (six on the USAMO and six on the TST) determine the USA Team. The Mathematical Olympiad booklets have been published since 1976. Copies for each year through 1999 can be ordered from the Mathematical Association of America's (MAA) American Mathematics Competitions (AMC). This publication, as well as Mathematical Olympiads 2000, Mathematical Olympiads 2001, Mathematical Olympiads 2002, and Mathematical Olympiads 2003 are published by the MAA. In addition, various other publications are useful in preparing for the AMC-AIME-USAMO-IMO sequence (see Chapter 6, Further Reading
Pre-Calculus Help In this section you'll find study materials for pre-calculus help. Use the links below to find the area of pre-calculus you're looking for help with. Each study guide comes complete with an explanation, example problems, and practice problems with solutions to help you learn pre-calculus. Study Guides Introduction to Systems of Inequalities The solution (if any) for a system of inequalities is usually a region in the plane. The solution to a polynomial inequality (the only kind in this book) is the region above or below the curve. We will begin with linear ... Introduction to Matrix Row Operations and Inverses We will use row operations to solve systems of equations and to find the multiplicative inverse of a matrix. These operations are similar to the elimination by addition method studied in Chapter 10. We ... Introduction to Matrices and Systems of Equations There are three ways we can use matrices to solve a system of linear equations. Two of them will be discussed in this Study Guide. Solving systems using these methods will be very much like finding inverses. We ... Matrix Determinant The last computation we will learn is finding a matrix's determinant . Although we will not use the determinant here, it is used in vector mathematics courses, some theoretical algebra courses, and in algebra ... Introduction to Ellipses Most ellipses look like flattened circles. Usually one diameter is longer than the other. In Figure 12.11, the horizontal diameter is longer than the vertical diameter. In Figure 12.12 the vertical diameter is longer than the horizontal ... Introduction to Hyperbolas The last conic section is the hyperbola. Hyperbolas are formed when a slice is made through both parts of a double cone. The graph of a hyperbola comes in two pieces called branches . Like ellipses, hyperbolas have a center, two ... Introduction to Coterminal and Reference Angles Two angles are coterminal if their terminal sides are the same. For example, the terminal sides of the angles 300° and −60° are the same. See Figure 13.3. Introduction to Trigonometric Functions There are six trigonometric functions, but four of them are written in terms of two of the main functions—sine and cosine. Although trigonometry was developed to solve problems involving triangles, there is a very ... Introduction to Graphing Sine and Cosine The graph of a trigonometric function is a record of each cycle around the unit circle. For the function f ( x ) = sin x, x is the angle and f ( x ) is the y ...
Publisher's Description From simple calculator operations to large-scale programming and interactive document preparation, Mathematica is the tool of choice at the frontiers of scientific research, in engineering analysis and modeling, in technical education from high school to graduate school, and wherever quantitative methods are used. - Powerful symbolic mathematics - Extensive built-in numerical routines - Easily handles day-to-day mathematical tasks for me. I work for NASA as an experimental cosmologist. - For me, this is a game changer. Had mathematica been available when I was a Ph.D. student in the 1980's, my career might have taken a different path. Cons - Mathematica is very good, but it is not as good as a good human. It does not "see" symmetries in systems of equations. Sometimes I have to go in by hand and solve things that Mathematica cannot. When this happens, the problem at hand is usually postgraduate level or higher. - Mathematica is clumsy with tensors. When doing tensor computations, I often have to trick it to get it to do what I want. I admit that more expert users might be able to get it to work straight away. Summary I use Mathematica every day in my work as an experimental cosmologist at NASA. I use it for designing scientific instruments and for solving many statistical problems. I tend to use IDL for numerical computation, but for symbolic work Mathematica is unparalleled in my experience. Yes, I still occasionally have to call on the standard post-gradualte level texts, but Mathematica is indispensable
More About This Textbook Overview This text for a second course in linear algebra, aimed at math majors and graduates, adopts a novel approach by banishing determinants to the end of the book and focusing on understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents - without having defined determinants - a clean proof that every linear operator on a finite-dimensional complex vector space has an eigenvalue. The book starts by discussing vector spaces, linear independence, span, basics, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite- dimensional spectral theorem. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. This second edition features new chapters on diagonal matrices, on linear functionals and adjoints, and on the spectral theorem; some sections, such as those on self-adjoint and normal operators, have been entirely rewritten; and hundreds of minor improvements have been made throughout the text. Editorial Reviews From the Publisher From the reviews: ZENTRALBLATT MATH "This second edition of an almost determinant-free, none the less remarkably far-reaching and didactically masterly undergraduate text on linear algebra has undergone some substantial improvements. First of all, the sections on selfadjoint operators, normal operators, and the spectral theorem have been rewritten, methodically rearranged, and thus evidently simplified. Secondly, the section on orthogonal projections on inner-product spaces has been extended by taking up the application to minimization problems in geometry and analysis. Furthermore, several proofs have been simplified, and incidentally made more general and elegant (e.g., the proof of the trigonalizability of operators on finite-dimensional complex vector spaces, or the proof of the existence of a Jordan normal form for a nilpotent operator). Finally, apart from many other minor improvements and corrections throughout the entire text, several new examples and new exercises have been worked in. However, no mitigation has been granted to determinants. Altogether, with the present second edition of his text, the author has succeeded to make this an even better book." AMERICAN MATHEMATICAL MONTHLY "The determinant-free proofs are elegant and intuitive." CHOICE "Every discipline of higher mathematics evinces the profound importance of linear algebra in some way, either for the power derived from its techniques or the inspiration offered by its concepts. Axler demotes determinants (usually quite a central technique in the finite dimensional setting, though marginal in infinite dimensions) to a minor role. To so consistently do without determinants constitutes a tour de forces in the service of simplicity and clarity; these are also well served by the general precision of Axler's prose. Students with a view towards applied mathematics, analysis, or operator theory will be well served. The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library." ZENTRALBLATT MATH "Altogether, the text is a didactic masterpiece." From the reviews of the second edition: S. Axler Linear Algebra Done Right "The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library."—CHOICE "A didactic masterpiece."—ZENTRALBLATT MATH "This book can be thought of as a very pure-math version of linear algebra … . it focuses on linear operators, primarily in finite-dimensional spaces … . Axler has come up with some very slick proofs of things that … makes the book interesting for mathematicians. The book is also very clearly written and fairly leisurely. … Axler concentrates on the properties of linear operators, and doesn't introduce other concepts unless they're really necessary." (Allen Stenger, The Mathematical Association of America, December 23, 2000 A Terrific Book for a Second Taste of Linear Algebra We used this as a textbook for the first quarter of a year long abstract algebra course. I can't recommend it enough. The proofs are clean, clear and beautiful. The text is uncluttered by unnecessary examples. Although many engineers interested primarily in computation will not likely enjoy this elegant work, anyone with good taste in mathematics will. Particularly beautiful and revealing is Axler's proof that every operator on a finite-dimensional, nonzero complex vector space has an eigenvalue--a brief proof completed without the use of determinants! This and so many other proofs throughout the book make linear algebra seem as simple and beautiful as it really is. 3 out of 3 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted April 30, 2003 Linear Algebra made Exciting This is a spellbinding book on linear algebra. Unlike the mechanistic thinking encouraged by the matrix-based approach, I found the direct view of vector spaces delightfully refreshing. Visualizing operators purely as mappings without the matrix-vector multipication 'crutch', provided an insightful understanding, that kept me enthralled. I read the entire book in a B&N store, often gasping alound at the simplicity and beauty of the proofs of several nontrivial results. By the way I bought the book to keep! Im an engineer and have used linear algebra matrix theory, but found this exposition deepened my understanding, while being a terrific read. Why not 5 stars? I did not want it to end!!Professor, plese write another book like this with more advanced topics. 2 out of 2 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted December 27, 2002 An elegant introduction to linear algebra from the right perspective As suggested by the wide spectrum of opinions on it, this is a book written from one perspective and one perspective only -- what many mathematicians consider the right one -- and therefore won't satisfy some engineering students who can't read through a math book that doesn't have concrete, carefully worked out examples, and nice little diagrams. But it seems awfully unfair to call it horrible just because you're used to linear algebra texts that all but spoon-feed you how to do a bunch of problems on matrices algebra. Most mathematics students, however, will agree that this indeed is the more elegant and intuitively-satisfying approach to writing a linear algebra textbook than the way the conventional lower-division linear algebra texts were written. The theory is built from a very intuitive and physically relevant concept of vector spaces, not some artificially concocted mathematical tools like matrices, which indeed arose from the need to describe linear maps on these vector spaces. If you're new to the subject or do not yet possess the mathematical maturity that most junior-level undergraduate math students have, you won't have nearly as much fun reading this book as the rest of us. (I recommend Howard Anton's Elementary Linear Algebra, which has all those nice examples and cute diagrams, if you're indeed not equipped well enough to appreciate the beauty of this textbook.) But if you care about understanding and appreciating linear algebra as a theory in itself, and not just as a tool, then by all means purchase this fine work by Professor Axler. Trust me, you will have more fun than you are allowed to have as an engineering student taking a math class. 1 out of 2 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted January 7, 2002 one of the best This a nice and straightforward book on linear algebra. If I was the author, I would not call it Linear Algebra Done Right (it does sound bad). Some people are obviously not happy with this book, but they need to realize that linear algebra is not a study of matrices; matrices only serve as a tool in some problems of linear algebra. Overall, highly recommended to math majors; not recommended to engineering majors! 1 out of 2 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted December 11, 2001 The title gives a false hype to the text After the semester was over I realized how poorly the text was put together. I ended up having to read out of two other books from the library and aided with Schaum's Outline for Linear Algebra, which I strongly recommend if you are stuck having to use this text for class. There were not enough problems worked out to gain a conceptual idea of the theory that is involved. Even those problems that are given have no answers so there is no way to judge your own answer to the questions. There is no beginning explanation of matrices and when they are introduced it is in a format that proves their validity but still gives no concrete examples of their use. This would definitely not be recommended if no class on matrix theory has been taken or possibly some previous work with the theory of linear algebra. The author understands the subject but he doesn't present it so that others can share his understanding. 0 out of 2 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted April 14, 2000 Loved It I think the one-star reviewer here is missing the point of the book. Matrices are indeed an important part of linear algebra, but what is much more important is a fundamental understanding of what a linear space is all about. This book communicates that essence in a clear and original way. I found it much easier to learn from than the books I had in college (that did things via determinants, matrices, etc). I'm a 3D game programmer so linear algebra is important to me. And this is a great book. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted March 29, 2000 It Should be Called 'Linear Algebra Done Wrong' In a word: horrible. The author takes a totally wrong approach to linear algebra. He introduces matrices towards the end of the book, and spends about a page on them. The book is written in an essay format, with very few examples and worked-out problems. A better choice would be Bernard Kolman's Elementary Linear Algebra. 0 out of 4 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
KeyMath-R KeyMath Revised: A Diagnostic Inventory of Essential Mathematics is an individually administered that can provide a comprehensive assessment of a student's understanding and application of important mathematics concepts and skills.
a free scientific calculator application for Windows, Mac, and Linux operating systems. Speed Crunch performs all of the functions necessary for high school Algebra and Geometry courses except graphing. In addition to performing all of these functions, Speed Crunch has a "math book" containing commonly used equations and formulas. One Speed Crunch feature that appealed to me from a design standpoint is the color coding of equations to differentiate between constants and variables A free scientific calculator download for Windows, Mac, and Linux operating systems. Speed Crunch performs all of the functions necessary for high school Algebra and Geometry courses except graphing. It also has a "math book" containing commonly used equa SpeedCrunch is a fast, high precision and powerful desktop calculator. * history and results on a scrollable display * up to 50 decimal precisions * unlimited variable storage * intelligent automatic completion * fully usable from the
A History of Mathematics Description For a junior or senior level course in the history of mathematics for math and math education majors. Blending relevant mathematics and history, this text immerses the student in the full, rich detail of mathematics. Students not only get a description of mathematics but they also learn how mathematics was actually practiced throughout the millennia by past civilizations and great mathematicians alike. As a result, students gain a better understanding of why mathematics developed the way it did. Features Detailed treatment of mathematics of the past-Not just a description of the mathematics, this is a math book. Allows for a better understanding of the evolution of mathematics. Reliance on original source material. Gives the student insight into how mathematicians of the past approached mathematics. Exercises, stated and solved, using period methods. Encourages students to think as the mathematicians of the past and gives instructors a wider variety of techniques to use in the classroom. Emphasizes the impact of secular history on mathematics. Shows students how mathematicians and the development of mathematics are affected by world events. The importance of history in teaching is emphasized. Gives potential teachers a wealth of alternative means of solving problems that they can use to create lesson plans and stimulate students who learn in "non-traditional" ways. Conjectures and propositions and demonstrations and proofs are distinguished. Shows students how the format of a mathematical argument has evolved throughout history.
Elementary Statistics, 11th Edition Author(s): Johnson/Kuby their own classrooms, through their popular texts, and in the conferences they lead, Robert Johnson and Patricia Kuby have inspired hundreds of thousands of students and their instructors to see the utility and practicality of statistics. Now in its Eleventh Edition, ELEMENTARY STATISTICS has been consistently praised by users and reviewers for its clear exposition and relevant examples, exercises, and applications. A focus on technology to help students succeed--including MINITAB®, Excel®, and TI-83/84 output and instructions throughout--is enhanced by a wealth of supplements that save instructors time and give students interactive guidance and support. All this and more have established this text's reputation for being remarkably accessible for students to learn from--and simple and straightforward for instructors to teach from.
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One of the problems in dealing with three equations and three unknowns is a lack of practice problems. This script provides a new problem set each time which has integer answers. Another difficulty fa... More: lessons, discussions, ratings, reviews,... This is a companion program to matrixpractice.htm [View Resource Connection below]. This program does not solve the 3 equations-3 unknowns system; rather it is a tool to help the student work t... More: lessons, discussions, ratings, reviews,... A TI-NspireTM file that students can use to reflect on the "Make a Mathematical Model" Activity from the Math Forum's Problem Solving and Communication Activity Series. This is designed to ... More: lessons, discussions, ratings, reviews,... A video that focuses on the TI-Nspire graphing calculator in the context of teaching algebra. In this program the TI-Nspire is used to explore the nature of linear functions. Examples ranging fromThis tool lets you plot functions, polar plots, and 3D with just a suitable web browser (within the IE, FireFox, or Opera web browsers), and find the roots and intersections of graphs. In addition, yo... More: lessons, discussions, ratings, reviews,... The Green family is planning a one-week vacation in Florida and needs to rent a car while there. They must decide which of four rental plans to choose. Students will enter data into a spreadsheet and ... More: lessons, discussions, ratings, reviews,... The Green family is planning a one-week vacation in Florida and needs to rent a car while there. They must decide which of four rental plans to choose. Students construct graphs to see which plan is b... More: lessons, discussions, ratings, reviews,... Highlight the rationale behind symbolic operations used to solve a linear equation with this tool that displays changes in the graphic and area models of functions as you change the value of each symb... More: lessons, discussions, ratings, reviews,... Using this virtual manipulative you may: graph a function; trace a point along the graph; dynamically vary function parameters; change the range of values displayed in the graph; graph multiple functi... More: lessons, discussions, ratings, reviews,... A free web-based function graphing tool. Graph up to three different functions on the same axes. Functions can refer to up to three independent variables controlled by sliders. As you move the... More: lessons, discussions, ratings, reviews,... An interactive applet that allows the user to graphically explore the properties of a linear functions. Specifically, it is designed to foster an intuitive understanding of the effects of chang
Summer Students are given a summer assignment in Precalculus from Chapter 1 in Calculus by Finney, De Mana, Waits, and Kennedy 1999 September Chapter 1 An intuitive study of limits, derivatives, integrals and definite integrals Chapter 2 Limits infinity. Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.) Continuity as a property of functions. An intuitive understanding of continuity. (Close values of domain lead to close values of the range.) Understanding continuity in terms of limits. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). Chapter 3 Concept of the derivative. Derivative presented geometrically, numerically, and analytical. Derivative interpreted as an instantaneous rate of change. Derivative defined as the limit of the difference quotient. Relationship betwee differentiability and continuity. Derivative at a point. Slope of a curve at a point. Examples are emphasized, including points at which are no tangents. Tangent line to a curve at a point and local linear approximations. Instantaneous rate of change as the limit of average rate of change. Approximate rate of change from graphs and tables of values. Applications of derivative with regart to displacement, velocity and acceleration, and speed October Chapter 4 Computation of derivatives. Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Basic rules for the derivative of sums, products, and quotients of functions. Chain rule and implicit differentiationChapter 5 Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values. Fundamental Theorem of Calculus. Use of the Fundamental Theorem to evaluate definite integrals. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Techniques of antidifferentiation. Antiderivatives following directly from derivatives of basic functions. Antiderivatives by substitution of variables (including change of limits for definite integrals). Applications of antidifferentiation. Finding specific antiderivatives using initial conditions, including applications to motion along a line. Solving separable differential equations and using them in modeling. In particular, studying the equation y ' = ky and exponential growth. November Chapter 5 cont. Derivative as a function. Corresponding characteristics of graphs of f and f' . Relationship between the increasing and decreasing behavior and the sign of f' . The Mean Value Theorem and its geometric consequences. Equations involving derivatives. Verbal descriptions are translated into equations concavity. Optimization, both absolute (global) and relative (local) extremes. Modeling rates of change, including related rates problems. Use of implicit differentiation to find the derivative of an inverse function. December Chapter 6 Derivatives and integrals of logarithmic and exponential functions. January Chapter 7 Applications of integrals dealing with growth and decayFebruary Chapter 8 Sections 8.1 to 8.6 Applications of integrals. Appropriate integrals are used in a and March Chapter 9 Sections 9.1 to 9.3 variety of applications to model physical, biological, or Chapter 10 Sections 10.1 to 10.6 economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. April Concentrated review of previous AP Calculus tests, both multiple choice and free response. May Continue test preparation. Assignment and completion of final project to be completed in class. Braintree High School Curriculum Guide Advanced Placement Calculus AB 414 Grade 12 Level 1 Expectations & Standards BHS 1. Come to school prepared and ready to learn. BHS 2. Communicate effectively in writing. BHS 3.Use appropriate research skills. BHS 4. Think critically. BHS 5. Work cooperatively in groups. BHS 6.Use technology appropriately. BHS 15. Use mathematics to solve problems. Advanced Placement Calculus AB 414 Grade 12 Level 1 Expectations & Standards Limits infinity. Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.) Continuity as a property of functions. An intuitive understanding of continuity. (Close values of domain lead to close values of the range.) Understanding continuity in terms of limits. Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem). Concept of the derivative. Derivative presented geometrically, numerically, and analytical. Derivative interpreted as an instantaneous rate of change. Derivative defined as the limit of the difference quotient. Relationship betwee differentiability and continuity. Derivative at a point. Slope of a curve at a point. Examples are emphasized, including points at which are no tangents. Tangent line to a curve at a point and local linear approximations. Instantaneous rate of change as the limit of average rate of change. Approximate rate of change from graphs and tables of values. Applications of derivative with regart to displacement, velocity and acceleration, and speed Computation of derivatives. Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions. Basic rules for the derivative of sums, products, and quotients of functions. Chain rule and implicit differentiation. Numerical approximations to definite integrals. Use of Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values. Fundamental Theorem of Calculus. Use of the Fundamental Theorem to evaluate definite integrals. Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Advanced Placement Calculus AB 414 Grade 12 Level 1 Expectations & Standards Techniques of antidifferentiation. Antiderivatives following directly from derivatives of basic functions. Antiderivatives by substitution of variables (including change of limits for definite integrals). Applications of antidifferentiation. Finding specific antiderivatives using initial conditions, including applications to motion along a line. Solving separable differential equations and using them in modeling. In particular, studying the equation y ' = ky and exponential growth. Derivative as a function. Corresponding characteristics of graphs of f and f' . Relationship between the increasing and decreasing behavior and the sign of f' . The Mean Value Theorem and its geometric consequences. Equations involving derivatives. Verbal descriptions are translated into equations concavity. Optimization, both absolute (global) and relative (local) extremes. Modeling rates of change, including related rates problems. Use of implicit differentiation to find the derivative of an inverse function. Derivatives and integrals of logarithmic and exponential functions. Applications of integrals dealing with growth and decay. Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, and the distance traveled by a particle along a line. Braintree High School Curriculum Guide Advanced Placement Calculus AB 414 Grade 12 Level 1B Outline Limits of functions Asymptotic and unbounded behavior. Continuity as a property of functions. Concept of the derivative. Derivative at a point. Applications of derivative with regart to displacement, velocity and acceleration, and speed Computation of derivatives. Numerical approximations to definite integrals. Fundamental Theorem of Calculus. Techniques of antidifferentiation. Applications of antidifferentiation. Derivative as a function. Second derivatives. Applications of derivatives. Derivatives and integrals of logarithmic and exponential functions. Applications of integrals dealing with growth and decay. Applications of integrals. Texts: Key Curriculum Calculus, Paul A. Foerster, 1998 and SFAW Calculus, Finney, De Mana, Waits, Kennedy
Well dont fret! After a course in calc your alg. skills will be given a makeover...that or you fail
To what extent do curriculum subjects, particularly highly abstract ones such as Mathematics, have their own unique special needs? This book celebrates the work done by subject specialists in mainstream classrooms to promote inclusive practice. It describes new and creative ways of developing mathematical thinking among pupils. Each chapter demonstrates... more... This book synthesizes research findings on patterns in the last twenty years or so in order to argue for a theory of graded representations in pattern generalization. While research results drawn from investigations conducted with different age-level groups have sufficiently demonstrated varying shifts in structural awareness and competence, which... more... Do your students have difficulty understanding math terms??Remembering the steps of algorithms in sequence? Interpreting data in displays? Our math study skills program will help you teach them math and how to learn! Essential to the program is the comprehensive Teacher's Guide. It provides a focus on the importance of study skills, directions for... more... During the past two decades, Chris Confer and Marco Ramirez have worked to deepen and improve mathematics instruction at schools around the country. Wherever they go, they find the raw ingredients for success already present: "The potential for positive change lies within each school. Abundance is present in the form of capable children, teachers,...This volume provides essential guidance for transforming mathematics learning in schools through the use of innovative technology, pedagogy, and curriculum. It presents clear, rigorous evidence of the impact technology can have in improving students learning of important yet complex mathematical concepts -- and goes beyond a focus on technology alone... more...
Mark McClure Mark McClure is an assistant professor at the University of North Carolina at Asheville researching real analysis, specifically fractal measure and dimension. Papers on these topics may be downloaded in gzipped PostScript format; a paper involving Julia ...more>> MasterGrapher - Pierre Vermaak Plot functions of one variable on-line. Up to ten functions may be displayed at the same time and features include zooming, editing etc. (Use Internet Explorer to see the MasterGrapher button needed to start the program.) ...more>> Mathematics of Cartography - Cynthia Lanius A map is a set of points, lines, and areas all defined both by position with reference to a coordinate system and by non-spatial attributes. These pages discuss how maps are used, give examples of different kinds of maps, and cover map history and mathGrapher A graphing tool for 2D and 3D functions and data, shaded surfaces, contour plots. It does linear and non-linear curve fitting, and you may integrate and analyse systems of up to 20 coupled differential equations. Analysis tools include power spectrum ...more>> MATHPLOTTER - Miguel Bayona A tool for graphing functions and mathematical equations, intended for teachers of mathematics. It will enhance the quality of printed mathematical documents such as tests, handouts, and quizzes. The programs can also be used for educational purposes. ...more>> maths online - University of Vienna, Austria A modern mathematics learning site on the Web: details of the project, persons and institutions participating, or project proposals (only in German). The Gallery consists of interactive multimedia learning units (Java applets), each of which includes ...more>> The Nature of Physical Science - Richard Brill A Honolulu Community College course, based around the Web and a series of television programs, that focuses on the history of science and its effects on our culture. Study guides are detailed outlines of various topics, including the foundations of arithmetic, ...more>> Poliplus Software Educational computer algebra, geometry, trigonometry, calculus software for Windows and Macintosh, and Java. Formulae 1 (F1) is a computer algebra system designed for the teaching and exploration of Mathematics. EqnViewer is a Java applet that allows ...more>> Polymath Love - Gary Smith Free and low-cost Macintosh software for middle school mathematics: over 200 programs written by a classroom teacher. Program descriptions and some free programs are available to download from the Web, or order the CD-ROM for $100 U.S., which includes ...more>> Pretty Functions - Ivars Peterson (MathTrek) About graphing calculators, and the software Graphing Calculator, a computer program for quickly visualizing two- and three-dimensional mathematical objects. Graphs illustrate plotting a given function, then seeing what happens when you modify the function ...more>> Quia! - The Learning Group Create your own Java-enabled quizzes, home pages, and learning games to play over the Internet: enter your material into a form, select a few options, and press a button. Games have been created and are available for browsing in mathematics, as well as: ...more>> Shodor Education Foundation, Inc. A non-profit research and education organization dedicated to the advancement of science and math education, specifically through the use of modeling and simulation technologies. Interactive middle school lessons aligned with curricula are available throughZap-a-Graph - Brain Waves Software A graphing program for use in the teaching and learning of high school mathematics and Calculus. Features include the ability to transform any relation, take the nth derivative of any function, and compose any two functions in nine different ways. The ...more>>
use of maths in other subjects wikipedia? Advertisement Possible Answer: How math is related to other subjects? Because in alot of subjects you use measurements and graphs/charts which is to do with maths. How is sequences in maths related to other subjects? - read morePlease vote if the answer you were given helped you or not, thats the best way to improve our algorithm. You can also submit an answer or check other resources.
Share to your social networks. Role of Mathematics in BusinessAvailability: Mathematics is used in most aspects of daily life. Many of thetop jobs such as business consultants, computer consultants,airline pilots, company directors and a host of others require asolid understanding of basic mathematics, and in some casesrequire a quite detailed knowledge of mathematics. It also playimportant role in business, like Business mathematics bycommercial enterprises to record and manage businessoperations. Mathematics typically used in commerce includeselementary arithmetic, such as fractions, decimals, and percentages, elementary algebra, statistics and probability.Business management can be made more effective in some cases by use of more advanced mathematics such as calculus, matrixalgebra and linear programming.Commercialorganizationsusemathematicsinaccounting,inventorymanagement,marketing,salesforecasting,andfinancial analysis.Inacademia,"BusinessMathematics"includesmathematicscoursestakenatanundergraduatelevelbybusinessstudents.Thesecoursesareslightlylessdifficultanddonotalwaysgointothesamedepthasothermathematicscoursesforpeoplemajoringinmathematicsorsciencefields.Thetwomost
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R. B. Manfrino, J. A. Gómez Ortega, and R. Valdez Delgado Based on fifteen years of preparing students for the Mexican Mathematical Olympiad and the International Mathematical Olympiad, this book presents a calculus-free introduction to inequalities and optimization problems with many interesting examples and exercises. The book starts with properties of order in the real numbers and the triangle inequality, and then continues through such mainstays of mathematical competitions as the arithmetic-geometric mean inequality and the inequalities of Cauchy-Schwarz, Hölder, Minkowski, Jensen, Nesbitt, and Schur. The careful treatment of convexity includes a discussion of Popoviciu's relatively recent (1965) inequality for convex functions, a very useful weapon in the problem solver's arsenal: If f is a real-valued convex function on an interval I, then for a, b, c in I, The first chapter concludes with Muirhead's theorem and a large number of its applications. Chapter 3 consists of 120 recent (1995-2008) national and international Olympiad problems, a mixture of numerical and geometric inequalities. In Chapter 4, the authors present solutions or hints to all exercises and problems appearing in the book. There is a twenty-one item bibliography (in which I was surprised to find a Spanish edition of the Courant and Robbins expository gem: ¿Qué son las Mathemáticas?). Most books on Olympiad-level competitions have sections on inequalities, but the book under review focuses on this genre of problems in a particularly attractive and effective way, providing good practice material. I recommend this softcover volume to anyone interested in mathematical competition preparation. I also suggest looking at the book Inequalities: An Approach Through Problemsby B. J. Venkatachala, a slightly more comprehensive treatment of inequalities, but still at the Olympiad level.
Course Goals. To strengthen and expand our understanding of fundamental mathematics, including arithmetic, algebra, and geometry, through comparative study of the mathematics of world cultures. To appreciate the contributions of all cultures to the development of mathematics. To explore the connections between mathematics, art, and music. The course will be particularly appropriate for majors in elementary education and fine arts and any student who is interested in world cultures. Text and Supplies. There is no textbook for this class. Course materials include class notes, handouts, and postings on the class web page. You will be required to purchase some supplies, such as scissors and a ruler. Projects. There will be two major projects, to be done in groups. An essential part of the project will be explaining your work to your classmates in a brief presentation. Part of your project grade will be based on peer evaluation. In-class work. This includes worksheets and group work. If you are absent on the day in-class work is assigned, you will receive a zero for that assignment. Homework. Homework assignments will be posted at You should start working on the homework problems for a section as soon as that section is covered in class. I will not collect the homework, but I strongly urge you to do it, as your quiz and test problems will be based on homework problems. I will also post solutions to homework problems on the web. Quizzes. We will have a quiz at the beginning of almost every class. At least one-third of your quiz grades will be dropped. No makeup quizzes will be given. Tests. There will be three midterms, scheduled for Thursday, September 26th, Thursday, October 31st, and Thursday, December 5th, and a final examination, given during the week of December 10th-16th. Makeup tests will only be given to students who contact me within 48 hours of missing a test. Students with a valid, verifiable reason for missing a test may take a makeup without penalty; those who have missed a test without a valid, verifiable reason may take a makeup with a 30% penalty. The C- Guarantee. I don't want anyone to fail this class. In fact, I'm willing to make you a promise: If you attend EVERY class, participate in in-class work, make an honest attempt on every quiz and exam, complete the projects, and hand them in ON TIME, then the lowest grade you will receive is a C-. GUARANTEED. Academic Honesty. Dishonesty includes cheating on a test, falsifying data, misrepresenting the work of others as your own (plagiarism), and helping another student cheat or plagiarize. Academic dishonesty will result in a grade of zero on that particular assignment; serious or repeated infractions of the Academic Honesty policy will result in failure of the course. For complete information about the University's policy on Academic Honesty, consult the Student Handbook 2002-2003. Attendance. Class attendance is mandatory. Although I do not have a rigid cut policy, anyone who has missed lots of classes and is doing poorly in the course should not expect much sympathy from me. If you do miss a class, it is your responsibility to make up the material. Schedule. Number systems. Finger counting. Number words. A comparative study of the earliest written number systems from around the world. Discussion of base and place value. History of the decimal system. Arithmetic. Algorithms for addition, subtraction, and multiplication of integers and why they work.
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Resources for the Math Community AIM is involved in several projects designed to address important issues for research mathematicians. Topics include the use of computers in research, introducing students to mathematics research, the career development of research mathematicians, maintaining a healthy research community, and several other issues. Preparing students to give talks If you have a student who is going to give a talk on their research, you may find our advice on preparing students to give a talk to be helpful. The advice is aimed at undergraduate students, but it should assist you in helping anyone who has not given too many research talks. AIM Research Experience for Undergraduates (REU) We have prepared a 6-page description of the AIM REU: individual projects with a common theme. Also included is general advice on selecting appropriate problems for undergraduate research, and some suggestions which may be useful in other REUs. The mentoring process The 17th annual Workshop on Automorphic Forms and Related Topics, held in Boulder Colorado, featured panel discussions on the mentoring process, research with undergraduates, and postdoctoral positions overseas. Applying for grants and other funding for professional development The 18th annual Workshop on Automorphic Forms and Related Topics, held at the University of California, Santa Barbara, in March 2004, also featured a panel discussion on the funding process. A summary of the discussion is available. Computation in ANTC A group of 22 research mathematicians whose core areas are in Algebra, Number Theory, and Combinatorics(ANTC) met at the NSF headquarters in Arlington, VA. The purpose of the meeting was to discuss various issues relating to computation in ANTC.
Web Codes Prentice Hall Connected Mathematics 2 (CMP2) Features and Benefits NSF funded. CMP2 has been classroom tested for five years as part of a new NSF grant prior to publication to ensure student success with the materials. Problem-centered, research-based approach. The same problem-centered, research-based approach proven successful with students as the original; content is developmentally appropriate for middle-school students. Embeds important mathematical concepts in interesting problems. Students learn important mathematical ideas in the context of interesting, interconnected problems. This exploration leads to understanding and the development of higher-order thinking skills and problem-solving strategies. Accessible to all levels of students. CMP2 is an effective combination of content and methodology designed to foster more "a-ha!" moments, regardless of a student's skill level or learning style. New technology to support learning! CMP2 now comes with updated technology to support teachers and students. Support is provided for digital presentations and StudentEXPRESS™ provides an interactive version of the textbook, with built-in homework help!
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Other Categories Algebra Help This section contains explanations of key algebra topics with interactive examples and quizzes. The topics are arranged in a cumulative way starting with some basic ideas and building up to more complicated aspects of algebra. You should be familiar with these basic algebraic properties as you start your lessons in algebra. These properties apply to all real numbers and include the cumulative property, the commutative property, and more. Before you try to solve algebra problems, learn these! Algebra equations can look complicated to new students. Learn how to decipher all those symbols with this primer on the Order of Operations. How do you evaluate an equation containing parentheses and exponents? Read this lesson to find out. The distributive property is a helpful algebra property that makes multiplying numbers easier. This lesson will explain how to multiply numbers inside parentheses. You'll also learn tips for quickly multiplying large numbers. Factorization is the process of breaking down an expression into products called factors. This concept will help you simplify large, complicated numbers into something you can actually work with. This lesson covers the different methods of factorization, such as factorization by grouping or factorization by taking the difference of two squares. Once you're familiar with the order of operations and different algebraic properties, you can finally get down to the business of solving equations. This lesson describes the terms "expression" and "equation" and walks you through solving a one variable equation. Graphs provide a visual representation of the relationship between two variables. In this lesson, learn how to graph and solve two variable equations, and become comfortable with coordinate planes, ordered pairs, and more. Inequalities, such as the "greater than" ( > ) and "less than" ( < ) relationships, can be visualized and solved just like normal equations. This lesson introduces inequalities and explains how to solve inequalities with variables, and how to show inequalities on a number line. After solving basic inequalities, you're ready to move onto solving and graphing two variable inequalities. In this lesson, you will learn how to solve a linear inequality, represent a linear inequality on a graph, and (most importantly!) check your work to make sure you've mastered the concept. The slope of a line is an essential concept in many areas of mathematics, algebra included. How do you calculate slope? What are some common mistakes to avoid? Learn all about "rise over run" in this lesson. Functions express the relationship between two variables. OK, now what does that mean? Read on for a simple definition and explanation of functions. Confused about the vertical line test? Not sure what the difference is between an even function and an odd function? You'll find the answers here. First Outside Inside Last. The FOIL method defines how two binomials are multiplied. Algebra students need to understand what FOIL stands for and means. Read on for an explanation and plenty of examples. A polynomial is an expression of finite length, including variables with positive whole number exponents. This lesson describes polynomials, polynomial roots, and includes an introduction to quadratic polynomials. What happens when you combine real numbers and imaginary numbers? You get a complex number. Learn how to solve equations involving complex numbers in this lesson. Need more information on imaginary numbers? This lesson covers that too. When you have a polynomial function of degree two, you have a quadratic function. When a quadratic function is equated to zero, you have what is called a quadratic equation. This lesson covers quadratic equations in depth. How are they formed, how do you graph them, and how do you solve them? The remainder theorem can be used to quickly factorize a polynomial of any degree. You can tackle difficult problems with this helpful theorem. Read this lesson to learn where the remainder theorem comes from and how to use it, with detailed examples. Also known as "powers of" numbers, exponents are operators used to multiply a number by itself a certain number of times. Exponents can be positive numbers, negative numbers, or many more special numbers. Learn about the different kinds of exponents and their properties in this lesson. A square root is a number which, when multiplied by itself, gives a square. Did you know every square has two square roots? How do you define a cube root? This lesson answers these questions and explains many concepts related to square roots and radicals. Rationalization is the process of making a fraction rational. When do you need to make a fraction rational? When it's irrational, of course. Read this lesson for examples of rationalization and a practice quiz. Also known as rational functions, a rational expression includes polynomials in its numerator and denominator. Can you find the domain of a rational expression? Do you know how to simplify a rational expression? This lesson will walk you through the process. Conic sections are formed by slicing a 3-D circular cone. The four kinds of conic sections are circles, ellipses, parabolas, and hyperbolas. In this lesson, learn how to represent all four conic sections with equations and graphs. Sign up for free to access more algebra 1 resources like . WyzAnt Resources features blogs, videos, lessons, and more about algebra 1 and over 250 other subjects. Stop struggling and start learning today with thousands of free resources!
Product Description Created for the independent, homeschooling student, Teaching Textbooks has helped thousands of high schoolers gain a firm foundation in upper-level math without constant parental or teacher involvement. Extraordinarily clear illustrations, examples, and graphs have a non-threatening, hand-drawn look, and engaging real life questions make learning pre-algebra practical and applicable. Textbook examples are clear while the audiovisual support includes lecture, practice and solution CDs for every chapter, homework, and test problem. The review-method structure helps students build problem solving skills as they practice core concepts and rote techniques. Teaching Textbooks' new Pre-Algebra Version 2.0 edition now includes automated grading! Students watch the lesson on the computer, work a problem in the consumable workbook, and type their answer into the computer; the computer will then grade the problem. If students choose to view the solution, they can see a step-by-step audiovisual solution. Teaching Textbooks Pre Algebra 2.0 includes the following new features: Automated grading A digital gradebook that can manage multiple student accounts and be easily edited by a parent. Over a dozen more lessons and hundreds of new problems and solutions Interactive lectures Hints and second chance options for many problems Animated buddies to cheer the student on Reference numbers for each problem so students and parents can see where a problem was first introduced Product Reviews Teaching Textbooks Pre-Algebra Kit, Version 2.0 4.8 5 10 10 I absolutely LOVE teaching textbooks. Teaching textbooks is fabulous. I recommend it to everyone. September 25, 2013 Teaching Textbooks Pre-Algebra My daughter, hates math with a passion. From the very first lesson of TT she has slowly begun to change her mind. She really enjoys what TT has to offer her. She's an audio/visual leaner with slight dyslexia so this has been a wonderful way for her to learn. I love the fact that version 2.0 does the grading for you, one less thing I need to grade but I do go in and check her progress. I can say, she's excited enough about this product that I get daily updates from her on how her lesson went and what her grade was. :) I hope that the rest of the year goes as smoothly and that she continues to enjoy TT. If so, we will continue with this for the remainder of her highschool years. August 27, 2013 What a blessing! This is our first year of Teaching Textbooks (pre-alg)... my 8th grade daughter has gone from crying, hating math last school year...to 'Math is my favorite subject!'...all because of Teaching Textbooks! Thank you so much for this curriculum, it makes it so much more understandable to a child that has always struggled in math, always cried about math because she didn't understand even the basics. August 27, 2012 Good foundation with slow pacing I am a home school mom, a college professor and an ACT tutor. We just finished our first year with this curriculum. There are several good things to consider. The book sets a good foundation for high school math. The information is presented clearly if your student can handle the tone of voice of the speakers. My daughter decided the textbook was better as she felt the voice droned too much. The text and the CD's present the exact same information in the same way. I appreciated the automatic grading and being able to go in and change grades or erase certain problems for her to redo. I appreciate that multiple students can use the curriculum at the same time. I appreciate that it presents many of the skills needed later on high school standardized tests. For someone who is not confident in teaching upper level math to their students this would be a good choice. But listen to your kids and take their concerns seriously, they may need some slight modifications to make this work for them. The negatives. The voice and tone will be tough for some students. The graphics are childish and distracting for some. The sound can be turned off though and the text can be used instead of listening to the lectures for those students this is difficult for. My sixth grade daughter liked this better when we turned off sound completely. Data can not be saved to the cloud. We had a computer crash and lost her work, despite backups. I would advise printing out grade sheets periodically if you need complete records of grades. The most disappointing part of the curriculum is the pacing. It is glacial. Lessons are presented broken up over several days and we found the best way to make the curriculum work for us was to combine 2-3 lessons a day as only small parts of the information were presented and it was more clear when several lessons were combined for my daughter. In addition the daily review in the lessons is extensive and becomes tedious. Every lesson has extensive review and it does become too repetitive and unnecessary for some students. I suppose some students need the continual daily review and they will appreciate this feature. It was driving my daughter to hate the curriculum until we modified it for her. To make the book work for us we combined several lessons per day and I only required review work on the unit tests rather than daily. If she got the review questions wrong then the next unit she had to do the daily review on the topic. The software will grade based on what you choose to do rather than all the problems listed so that worked well for us. We finished the curriculum in 6 months following these strategies and are beginning the Algebra 1 curriculum now. Overall I still recommend the curriculum especially as an ACT tutor. Teaching textbooks presents the math skills needed for standardized tests in a clear foundational way. For this reason I have chosen to continue with the curriculum for Algebra despite the downsides. Listen to your kids though and consider making modifications if needed. March 21, 2012
Partial Differential Equations: an Introduction (2ND 08 Edition) by Walter A. Strauss Publisher Comments Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the... (read more) Essentials of College Algebra, Alternate Edition by Lial Publisher Comments Essentials of College Algebra , Updated Edition, 1/e, has been specifically designed to provide a more compact and less expensive alternative to better meet the needs of colleges whose algebra courses do not include the more advanced topics. The authors... (read more) Understanding Basic Statistics (5TH 10 - Old Edition) by Charles Henry Brase Publisher Comments A condensed and more streamlined version of the very popular and widely used UNDERSTANDABLE STATISTICS, Ninth Edition, this book offers instructors an effective way to teach the essentials of statistics, including early coverage of Regression, within a... (read more) Algebraic Topology (02 Edition) by Allen Hatcher Publisher Comments An introductory textbook suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises.... (read more) Calculus of Variations by Robert Weinstock Publisher Comments This book by Robert Weinstock was written to fill the need for a basic introduction to the calculus of variations. Simply and easily written, with an emphasis on the applications of this calculus, it has long been a standard reference of physicists... (read more) Numerical Methods for Scientists & Engineers. 2nd Edition by R W Hamming Publisher Comments Numerical analysis is a subject of extreme interest to mathematicians and computer scientists, who will welcome this first inexpensive paperback edition of a groundbreaking classic text on the subject. In an introductory chapter on numerical methods and... (read more) Non-euclidean Geometry (55 Edition) by Roberto Bonola Publisher Comments This is an excellent historical and mathematical view by a renowned Italian geometer of the geometries that have risen from a rejection of Euclid's parallel postulate. Students, teachers and mathematicians will find here a ready reference source and... (read more) Cartoon Guide to Statistics by Larry Gonick Publisher Comments If you have ever looked for P-values by shopping at P mart, tried to watch the Bernoulli Trails on "People's Court," or think that the standard deviation is a criminal offense in six states, then you need The Cartoon Guide to Statistics to put you on the... (read more) Even You Can Learn Statistics (2ND 10 Edition) by David M. Levine Publisher Comments Even You Can Learn Statistics: A Guide for Everyone Who Has Ever Been Afraid of Statisticsis a practical, up-to-date introduction to statistics–for everyone! Thought you couldn't learn statistics? You can–and you will! One easy step at... (read more) Professor Stewart's Cabinet of Mathematical Curiosities by Ian Stewart Publisher Comments Knowing that the most exciting math is not taught in school, Professor Ian Stewart has spent years filling his cabinet with intriguing mathematical games, puzzles, stories, and factoids intended for the adventurous mind. This book reveals the most... (read more) Measure, Topology, and Fractal Geometry (2ND 08 Edition) by Gerald Edgar Publisher Comments For the Second Edition of this highly regarded textbook, Gerald Edgar has made numerous additions and changes, in an attempt to provide a clearer and more focused exposition. The most important addition is an increased emphasis on the packing measure, so... (read more) Analysis With Introduction To Proof (4TH 05 - Old Edition) by Steven Lay Publisher Comments "Let me begin by saying that I really like this book, and I do not say that of very many books. What impresses me most is the level of motivation and explanation given for the basic logic, the construction of proofs, and the ways of thinking about... (read more) Numerical Analysis (2ND 88 Edition) by Schaum Publisher Comments If you want top grades and thorough understanding of numerical analysis, this powerful study tool is the best tutor you can have! It takes you step-by-step through the subject and gives you accompanying related problems with fully worked solutions. You... (read more) Nature of Mathematics (12TH 12 Edition) by Karl J. Smith Publisher Comments Experience mathematics and hone your problem-solving skills with THE NATURE OF MATHEMATICS and its accompanying online learning tools. The author introduces you to Polya's problem-solving techniques and then shows you how to use these techniques to solve... (read more) Understanding Regression Analysis (86 Edition) by Larry D. Schroeder Synopsis The authors have provided beginners with a background to the frequently-used technique of linear regression. It is not intended to be a substitute for a course or textbook in statistics, but rather a stop-gap for students who encounter empirical
This is the first part of the sequence Math 2070, 2071 and is introduction _to numerical methods and the fundamentals of the analysis (stability, consistency, _convergence, efficiency and accuracy) of numerical methods. Course materials: The main course material will be lecture notes from the lectures in class. Attend and take good notes! You might find some material presented simpler than in the text, some differently than the text, some small extensions and omissions from the text. The topics are standard so variations will be minor rather than major. My lecture notes have evolved through the search for the simplest mathematically rigorous way to explain WHY and HOW and WHEN and WHEN NOT algorithms that are important for practical computing work. Textbooks are only useful when they give the same topics from a different point of view than the lectures. (This will not be a course where the teacher puts the book on the board.) The official text is: A Ralston and P Rabinowitz, A first course in Numerical Analysis, Dover reprint, 2001, an excellent and inexpensive book. Many years Atkinson's book is used here, K. Atkinson, An Introduction to Numerical Analysis, 2nd Ed, also an excellent book but more expensive. Misc. General Policies: Make Up Exams will NOT be given. HW will be collected and graded. You are not graded on what you can prove. You are graded on what your explanation of a proof conveys in a clear and direct manner to others. Write in complete and correctly punctuated sentences. Avoid intoxication with the symbols of symbolic logic. (Write for people not for automatons!) Negative partial credit is possible for extraneous and irrelevant nonsensical information. No quibbling over "points"; Focus on learning! Dr. Sussman has graciously provided instructions for accessing Matlab remotely. The first answer is: Yes, Matlab can be accessed remotely using your usual network access. I am not familiar with "PittNet" -- is it a dialup connection (slow) or some sort of fast connection? In either case, directions follow. ASSUMING YOU ARE USING LINUX or another type of Unix: Use ssh to get to unixs1 or unixs2, type "matlab": example ssh youraccountname@unixs2.cis.pitt.edu password matlab Wait a long while (1-2 minutes for a fast connection, a looooong time for dialup) and Matlab will come up. WARNING: if you have a (slow) dialup connection, you can turn off the graphical stuff and run matlab in command mode: (you get no graphics and no plotting) ssh YoUrAcCoUnTnAmE@unixs2.cis.pitt.edu password DISPLAY= matlab If you have a dialup and you want plotting but no gui, do NOT reset DISPLAY to nothing. Just do: matlab -nosplash -nojvm If you turn off the gui, you will need some sort of editor. Either open another ssh session with your favorite editor in it (pico, vi, emacs, etc.) or use the exclamation point to start an editor from the Matlab command line (!pico ...) ASSUMING YOU ARE USING WINDOWS You either have an "X-server" program or you don't. I do not know whether one is included in the programs they supply to students. ASSUMING YOU HAVE A DIALUP CONNECTION OR YOU DO NOT HAVE AN X-server Use the Start->Run menu and type telnet unixs2.cis.pitt.edu (or unixs1 or unixs) It will ask you to log in. Then just type matlab to get the LINE version of Matlab (no graphics, no plotting). There is no way to do plotting without an X-server. You also will not have the Matlab editor, but you can edit a file FROM THE MATLAB COMMAND LINE with the command "!pico filename.m" where you don't type the quotation marks and you put your own name where I have "filename.m". When you exit pico, you will return to Matlab. ASSUMING YOU HAVE BOTH AN X-server AND A FAST CONNECTION Start the X-server first. You will need to know the "display" name It will be something like 192.160.1.1:0 (your IP address first, then colon zero). Use the Start->Run menu and type telnet unixs2.cis.pitt.edu username, password, etc. DISPLAY=192.160.1.1:0 EXPORT DISPLAY matlab There might be a telnet or rlogin or ssh command that comes with the X-server. If so, the DISPLAY variable is automatically set. REMARK: There is a free Matlab clone called "octave" that runs on Windows (If you run linux, install it from your CDs). See It is not so pretty as Matlab, and it is mostly command-line driven, but it will do plotting and most simple stuff. Migration from FORTRAN or C to MATLAB If you have experience programming in FORTRAN or C, the least painful way to migrate to MATLAB is to begin by using MATLAB as a low level programming language and then learn the higher level features as you need them. These notes below were prepared by Dr. M. Sussman for exactly that purpose. For that reason, they parallel "A Synopsis of 5 statement FORTRAN". 1. INTRODUCTION: Matlab Statements. You can use "%" anywhere on a line to make Matlab skip everything you write past the "%". This is called "commenting it out." Ending a line without a semicolon (";") means its result will be printed. Ending a line with a semicolon means it will not be printed. Types of Variables. All simple variables in Matlab are actually complex matrices. Matlab tries to "treat them the way you meant them." Variable names are case sensitive, so iT is different from It is different from IT is different from it. Arithmetic Operations. These are denoted by the usual symbols +, -, , / and * Exponentiation is denoted by ^ The arithmetic equals (=) has a different meaning than in arithmetic. For example, the statement X=X+1.0; makes perfect sense in Matlab. It means, look at the left hand side, find the storage location set aside to store the variable called X and find the value of X. Add one to that value then put the result back in the X storage location. This requires the computer to do something; hence, it is called an executable statement. (It's described informally by: "look in the box marked X. Get the number there; add one to it and put it back in the X box.") CONTROL STATEMENTS Simply typing a variable name on a line by itself without a semicolon after it will cause it to be printed. A sequence of variables separated by commas will all be printed. Anything put in quotation marks is printed directly as written. Examples of this are given later. 2. The Heart of the Program. A computed basically can only do 5 things: * Input data, * perform arithmetic calculations, * decisions based on logical or arithmetic tests, * output data and * loop (meaning perform al the above iteratively). We've seen how to perform basic arithmetic calculations. Let's now review some of the remainder. The real power of the computer is unleashed with loops (doing a lot of calculations) and matrices and vectors (handling a lot of data). INPUT, OUTPUT and CONTROL A single variable can be read from the keyboard using the command r=input('What is the value?') and the string 'What is the value?' will be printed and then the user will type a number and the value of that number will become the value of r. DECISIONS. Decisions in Matlab are performed by "IF tests." There are many kinds. One basic kind takes the form if (condition is true) executable statement; else executable statement; end For example, the following adds 1 to x if x = 0 and subtracts 1 otherwise: if x>=0 x=x+1.0; else x=x-1.0; end Equality is denoted "==" to distinguish it from assignment. LOOPS. These are "DO-Loops" in FORTRAN and "FOR-Loops" in C and Matlab. for i = beginning: stepsize : end executable statements; end where "beginning", "end" and "stepsize" are all integers.For example, this computes the sum 1+1/2+1/4+1/8++ 1/(2*1000). sum=0.0 for I=0,1000,1 addend=1.0/(2.0^i); sum=sum+addend; end Suppose we want to exit a loop early when a condition holds. Use a "break" statement. For example, to exit the above when SUM> 1.5 we add an if-test sum=0.0 for k=0:1:1000 addend=1.0/(2.0^k); sum=sum+addend; if sum>1.5) break; end end k sum Matlab's BUILT IN FUNCTIONS Essentially all basic mathematical functitons are built-in. For example, exp(x) computes the exponential of x, sin(x) computes its sine, inv(M) computes the inverse of a matrix, eig(M) computes the eigenvalues of a matrix, etc. Inline functions Simple functions can be put directly in a Matlab statement: area=inline('pir^2','r'); defines a function area(r) that computes the area of a circle given its radius. 4. Programming "Rules of Thumb". These days one is more often modifying an existing code or linking pieces of codes than programming a new application from scratch. Nevertheless, the following "rules of thumb" will, if followed, save you hours of headaches in debugging: 1. Be careful and correct: every hour you spend in checking, planning and double checking will save four hours of debugging. 2. Use pseudo code to organize and plan your program. 3. Check it twice before running it! Then run it on several test cases with known true-solutions. 4. Use modules. Break the program into clear chunks (one way is to use FUNCTION m-files). Check and debug each module independently of the whole program before testing the whole program. 5. Use the right amount of generalization! 6. Print intermediate results. 7. Include warning messages. 8. Use variable names that are easy to understand and declare all variables. Some people might choose to use FORTRAN (or be forced to because they deal with legacy programms). I've inclluded some information next for those lucky ones. HOW TO RUN A FORTRAN PROGRAM Here are some basic , basic instructions on how to (a) write a fortran program [ie, create a file]. (b) how to run it. (a) To create a file you need to use an editor. In unix, the pico editor is probably easiest because it's menu driven. Dont forget though that unix is case sensitive: large and small letters have different meanings , so get the capitalization exactly right! At the prompt type: pico progname.for (the "progname " you may choose but be sure to make it .for at the end.) Next paste in or type in your program. Editing it is easy-read the commands at the bottom of the screen. When you are done type 'control'O then type 'control'X Now you've created a file progname.for. (b) Here's how to run progname.for At the prompt type setup sunpro At the next prompt type: f77 -C progname.for This compiles your program. Now run it by typing at the next prompt ./ a.out (i.e., " dot slash a dot out" without any spaces!) This tells the computer to dump your output to the screen. Try writing a simple program to add 2 numbers and output it. If it doesnt work, check carefully the cases [small letters or caps] and the spaces in your commands. Another comment: If you discover something interesting, email me and I'll send it to the whole class. More Remarks: These instructions are for our UNIXS system. There are a lot of free FORTRAN compilers available [FORTRAN 77] as well as a lot of free C compilers too. I'll email to you an addres on where to download one, if you are interested in working on a PC. I suggest though that you start on UNIXS if you've never programmed before. Have fun! FORTRAN Compilers- Here are some FORTRAN resources including free and/or cheap FORTRAN compilers for a PC. "The Fortran Library, is a comprehensive guide to online Fortran resources, including compiler vendors and resellers, benchmarks, programming tools, books and articles on Fortran and numerical methods, and commerical and public domain Fortran software. The Fortran Market, , also provides many links to Fortran resources and is a reseller for several Fortran 90 and 95 compilers, benchmarking and test suites, and books on Fortran." One former student installed one. Here below are his instruction on how to do this: Subject: Fortran Compiler for PC's First of all, the stuff has to be downloaded. It can be gotten via anonymous ftp (using Netscape, for instance) from ftp://ftp.bu.edu/pub/mirrors/simtelnet/gnu/djgpp/v2gnu/g770523b.zip and you also need ftp://ftp.bu.edu/pub/mirrors/simtelnet/gnu/djgpp/unzip386.exe There are other things like additional documentation, but this provides the bare essentials (I realized that it needed to be cut down for people to be able to download it). Then this file (g770523b.zip) needs to be unzipped. I recommend making the directory C:\DJGPP by typing mkdir \DJGPP at an MS-DOS prompt. Then, go to the directory where you downloaded g770523b.zip to, and type You'll also need to make sure you can access the compiler, so add C:\DJGPP\BIN to your PATH statement in your CONFIG.SYS. If you don't know how to do this, just type the following from an MS-DOS prompt. echo PATH=%PATH%;C:\DJGPP\BIN >> \CONFIG.SYS After that, you'll need to write the FORTRAN programs in something like Notepad (do not use a Word Processor for this!). To compile your program, named (for instance) progname.for, type the following. g77 -o progname progname.for and to run the new program, type progname A Synopsis of "5-Statement FORTRAN" ABSTRACT This is prepared solely for the benefit of my students. ItÕs intended to have 2 purposes. 1. To introduce math students whoÕve never programmed before but are used to the algorithmic thought in rigorous proofs to a language quickly and get them working, and 2. To do the same for students who have experience programming in another language like C, for example. 1. Introduction. FORTRAN is short for FORmula TRANslation language. It is a low level language designed for doing scientific calculations and lots of them. Although it is out of favor in CS departments, it is extremely handy to know a bit of FORTRAN because: (estimated) 95% of scientific software is in FORTRAN. Thus, if you must modify an existing program it will likely be a FORTRAN one. It is also useful for us because it is simple ( = easy to learn) and its logic parallels the logic of mathematical formulas. Generally, for these and other reasons, if you program in another language, it is a good idea to know how to call a FORTRAN subroutine from that language. 2. FORTRAN Statements. When you type a line of a FORTRAN program, different locations of characters (i.e., which column it is types in) have different meanings. Generally, try not to continue a statement from one line to another: Keep formulas short enough that you can glance at it and see what it is doing. Thus, try to leave column 6 blank. A character in column 1 tells the compiler to ignore the line so you use these lines to write notes to yourself. You can also use "C " in column 1 to de-activate a line. This is called "commenting it out." Types of Variables. Variable names can have no more than 7 characctters. You should declare variables but in FORTRAN you don't need to because there are defaults. INTEGERS: Integers are stored as such and added exactly. Any variable beginning with I, J, K, L, M, N is by default an integer. Thus, old-style FORTRAN would use the (undeclared) variable ICOUNT. Variables can also be declared as integers initially, meaning before the first executable statement. For example, _ _ _ _ _ _ _ _ _ INTEGER COUNT will make COUNT an integer even though the undeclared default of a variable COUNT is real. In integer arithmetic I =1/4 Returns I = 0 while in real arithmetic X = 1.0/4.0 Returns X = 0.25. Don't use mixed mode arithmetic! REAL NUMBERS: These can be declared as real before the first executable statement by, e.g., _ _ _ _ _ _ _ REAL X By default any variable beginning with A-H or O-Z is also real. Reals are numbers with decimal digits such as PI = 3.141592736 XINDEX = 1.0 They are stored in scientific notation and carry a certain number of significant digits. Carrying 7 digits base 10 is typical although not universal. DOUBLE PRECISION variables must always be declared n before the first executable statement, e.g., _ _ _ _ _ _ _ DOUBLE PRECISION X They work like reals but carry more than 2x the number of significant figures as reals (also called single precision). Double precision is the normal mode for many scientific calculations. If you add a real to a double precision variable, you immediately lose al the extra accuracy of the double precision variable so: Avoid mixed mode arithmetic. Arithmetic Operations. These are denoted by the usual symbols +, -, , / and * Exponentiation is denoted by with differences between real and integer exponents: (Integer) X2 means X * X (Real) X*2.0 means EXP(2.0 LOG( X ) ). The arithmetic equals (=) has a different meaning than in arithmetic. For example, the statement X=X+1.0 makes perfect sense in FORTRAN. It means, look at the left hand side, find the storage location set aside to store the variable called X and find the value of X. Add one to that value then put the result back in the X storage location. This requires the computer to do something; hence, it is called an executable statement. (It's described informally by: "look in the box marked X. Get the number there; add one to it and put it back in the X box.") CONTROL STATEMENTS Programs always end with two statements that terminate the program: STOP END Another useful statement is the unformatted print statement which just dumps the output to your computer screen. This is fine for moderate amounts of output. (For lots of it, read about formatted print statements in a FORTRAN book.) An unformatted print takes the form PRINT , variables to be printed separated by commas. Anything put in quotation marks is printed directly as written. Examples of this are given later. Generally, for moderate amounts of data, an unformatted print is fine. On your terminal screen, highlight the output. Next copy and paste it into your favorite text editor and format it and then print it. We've seen how to perform basic arithmetic calculations. Let's now review some of the remainder. The real power of the computer is unleashed with loops (doing a lot of calculations) and matrices and vectors (handling a lot of data). INPUT, OUTPUT and CONTROL Unformatted input is (usually from your terminal screen) READ, variables While unformatted output is PRINT, variables. You can label the variables by putting notes to be printed in quotes as in: PRINT, ÒNext we print the value of XÓ, X This results in a printted line like: Next we print the value of X .34567E2 when X has the value 34.567. DECISIONS. Decisions in FORTRAN are performed by "IF tests." There are many kinds. One basic kind takes the form Suppose we want to exit a loop early when a condition holds. In this case, in FORTRAN the only way is to use a "GO TO" statement. (This is the only valid use of GO TO's.) For example, to exit the above when SUM> 1.5 we add an if-test with a GO TO: FORTRAN's BUILT IN FUNCTIONS There are many built in functions in FORTRAN. If you put a "D" in front the function returns a double precision answer. Some common ones include: ABS(X)=\|x\| , TAN(X) = tangent(x), ATAN(X)= arctangent(x), COS(X), SIN(X),TAN(X),... , EXP(X) = exponential function of x= ex , LOG(X)= natural logarithm of x, LOG10(X)= logarithm base 10 of x, SQRT(X)=square root of x, FLOAT(I) and REAL(I) convert integer I into a real to avoid mixed mode arithmetic, DFLOAT(I) makes I a double precision real, and so on! FUNCTION STATEMENTS Function statements are very handy to code a simple formula. IMPORTANT: MUST BE BEFORE THE FIRST EXECUTABLE STATEMENT! Note that 4.0ATAN(1.0) is a standard way to code PI without typing in all its digits! If a function uses more calculations or a longer formula then use a function subroutine. Function subroutines are very useful!!! Here's an example: Notes: You must have "AREA = " somewhere in function subroutine AREA And you must always have "RETURN, END" at the end. 4. Arrays = Vectors and Matrices. Vectors and matrices are the way to store and manipulate lots of data. They must be set up with various dimensioning statements BEFORE THE FIRST EXECUTABLE STATEMENT!!! Examples of these dimensioning statements are the following: Remark: It's interesting to try running this changing "20" to, e.g., "50". You will soon see why FIBO(.) is stored as a real number rather than (at first and second consideration) the more natural integer. 5. Programming "Rules of Thumb". These days one is more often modifying an existing code or linking pieces of codes than programming a new application from scratch. Nevertheless, the following "rules of thumb" will, if followed, save you hours of headaches in debugging: 1. Be careful and correct: every hour you spend in checking, planning and double checking will save four hours of debugging. 2. Use pseudo code to organize and plan your program. 3. Check it twice before running it! Then run it on several test cases with known true-solutions. 4. Use modules. Break the program into clear chunks (one way is to use FUNCTION subroutines). Check and debug each module independently of the whole program before testing the whole program. 5. Use the right amount of generalization! 6. Print intermediate results. 7. Include warning messages. 8. Use variable names that are easy to understand and declare all variables.
Introductory Technical Mathematics - 5th edition ISBN13:978-1418015435 ISBN10: 1418015431 This edition has also been released as: ISBN13: 978-1418015459 ISBN10: 1418015458 Summary: Introdu sectio...show moren on basic statistics, new material on conversions from metric to customary systems of measure, and a section that supplements the basics of working with spreadsheets for graphing. Features: a new section on basic statistics features an all-new chapter on statistics and a chapter that consolidates all the statistical graphing techniques of bar, line, and circle graphs into one locationGood text book recycle ny malone, NY 2006-06-09 Paperback Good We ship everyday and offer PRIORITY SHIPPING. $6275 +$3.99 s/h Good CR Booksellers Punta Gorda, FL 1418015431 Will NOT arrive before Christmas. Used, in good condition. Book only. May have interior marginalia or previous owner's name. $85.87106
Basic Math and Pre-Algebra for Dummies '...more * Understand fractions, decimals, and percents * Unravel algebra word problems * Grasp prime numbers, factors, and multiples * Work with graphs and measures * Solve single and multiple variable equations(less) ebook, 384 pages Published January 29th 2008 by For Dummies (first published January 1st 2007) Community Reviews refr...more refresher. I find the basics are where college students screw up. Missing a sign, forgetting to carry the 1. Why not be different?(less)
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The Basics Overview An Introduction to Thinkwell's Calculus Page [1 of 1] Hi! And welcome to our studios. I'm Professor Edward Burger, professor of mathematics at Williams College, and I'd like to welcome you to this really interesting, new, and an innovative environment for learning. What we're going to do is we are going to actually turn things around and try to make calculus interesting, calculus understandable, and calculus fun. Now, you know, a lot of you, in fact, I'm hoping that you actually have a calculus book, it might look something like this, here's one, gee, it's a little heavy. Here it is. This one's called "Calculus, Math to Pump You Up". Well, the only pumping that I think this book is going to do is if you actually use it like this and bring it to the weight room because it's so heavy, it's so big. Maybe your book is just as big as this; maybe it's even bigger, I don't know. Have you read your book, by the way? Well, it would be great if you read it, seriously, but I'll let you in on a little secret. I think most people don't read this book, and, in fact, I'll let you in on a bigger secret. When I took calculus, I didn't read the book, but don't tell people. Anyway, the point is this is so cold and dry and filled with theorems and proofs and all sorts of stuff, and usually, what folks do is just run right to the back of each section and try the problems and when they can't do the problem, they look at the worked out problems. And it's hard to get a good understanding of what calculus is all about. We want to turn that around. What we want to do here is actually create an environment where we discuss the calculus, we explain it in simple easy to follow terms and really see that it's sort of an interesting endeavor, that there's a lot of sort of things that make a whole bunch of sense, and that it's actually sort of fun. And also empower you with the ability to understand it and to also understand other things. So, the difference is, what's the difference between what we are doing here and this. Well, first of all, this is on the information super highway. I'm coming to you courtesy of technology. I mean, I'm coming over the airwaves and you're seeing me -- ju, ju, ju. It's sort of fun, isn't it? Well, I'm sort of actually enjoying it, `cause it gives me a tingle right down my spine, but that's another story. Okay, now, what's going on here though, is we've created an environment where we are going to interact with each other. So, I'll talk about stuff and then we'll actually have an opportunity for you to try something, and for you to get your hands dirty and try a problem, or try a question, and then we'll talk about it and get some feed back with each other. Now you'll notice right over here there's another white right rectangle and that's the place where I'll actually sort of write things in if I want something to be there for you to see. I'll put it; for example, I'm not going to say, you know, calculus is cool. There you see. And I want to say, math isn't as awful as you may think. Hear me? So we use that a lot and that's basically what we're all about here. The actual set is going to be, when we actually lecture, there'll be a table here. I'll be writing on the table and you'll be able to see what I'm writing, and this is our set. Do you like our set back here, it's sort of blue, not very exciting, but there's a lot of possibilities, a lot of opportunities, and we'll have fun with it. For example, suppose I want to tell you the weather. Now you see, there it is, and you can see Massachusetts. It's beautiful, that's where I'm from. Maybe you want to actually go and to take a look at the moon. Well, there's a moon shot. Textbooks can't do that. It's going to be virtual. It's going to be now, and I hope you are going to really enjoy it. So, I want you to stay with us and I hope you have some fun. Thanks
This site provides a comprehensive summary of the Ontario Grade 12 Advanced Functions (MHF4U) and Calculus and Vectors (MCV4U) courses. Use the "Search" box to find information on specific topics. (copy in the search engine words you used to find this site) Keywords that match your search words will be displayed below each topic. The course tab headings will show you the number of matching topics. Click on the Examples tab for information about finding and working with examples. To navigate when not searching Select your course then click on the topics to expand or collapse them. When subtopics are displayed you can expand or collapse all details with a single click at the left side of the heading. a tool tip will tell you what will happen when you click - it changes depnding on what's displayed and how close you are to the left edge The recommended browsers for this site are IE8, Firefox, Chrome and Safari. Examples are attached to the topics displayed on the Course Summary page. To display an example position your cursor over the darker blue area that appears when the topic turns blue. When an example pops up, click on it to add it to this page. Every example has a three letter identifier displayed in the upper right corner. You can load examples by entering this identifier at the top of this page and clicking the [Load Examples] button. For example, enter aaa,aab at the top of this page and click the [Load Examples] button to load examples aaa and aab. WARNING: This text will disappear when examples are displayed. Click [Clear Examples] to get this text back. Write down the identifiers of any examples that you may wish to review again the next time you visit this site. You can also view the solutions to the examples directly on the Course Summary page by positioning your cursor over any blue or grey rectangle on any pop up example. Blue rectangles identify examples for the topic you are currently on. Grey rectangles identify examples for related topics. The blue/grey rectangle will turn red while the solution is displayed. Currently there are only a few examples loaded on this site. More will be added as they become available.
The course first covers sequences, series, their general properties, and using the Taylor series to approximate functions. Then it moves on to multivariable functions, vectors, and basic vector operations. Finally, it introduces multivariate derivatives and their applications, including partial derivatives, directional derivatives, and gradients. Student learning goals General method of instruction The class will be taught with a mixture of group activities, as well as interactive lectures. Recommended preparation BCUSP 125 (Calculus II) or equivalent. Class assignments and grading Homework will be assigned regularly. It will be completed using the Wiley Plus online system. You will need to purchase a registration code for this. Since that registration code includes a complete electronic version of the textbook, there is no need to buy hard copy of the text unless you prefer that and do not wish to print pages from the electronic version yourself. I will also distribute additional materials in class. Grades will be based on student's performance on assigned work, in-class worksheets and Bilin Z Stiber Date: 04/08/2012 Office of the Registrar For problems and questions about this web page contact icd@u.washington.edu, otherwise contact the instructor or department directly. Modified:November 27, 2013
The best way to study mathematics is to do as many problems as possible. However, it is often the case where a student doesn't completely understand the steps to solve a problem, or there are certain nuances which only someone who has been doing mathematics for years would pick up. That is what I am here for.
The Fourier Transform and its Applications Note: This course is being offered this summer by Stanford as an online course for credit. It can be taken individually, or as part of a master's degree or graduate certificate earned online through the Stanford Center for Professional Development. The goals for the course are to gain a facility with using the Fourier transform, both specific techniques and general principles, and learning to recognize when, why, and how it is used. Together with a great variety, the subject also has a great coherence, and the hope is students come to appreciate both. Topics include: The Fourier transform as a tool for solving physical problems. Fou Multidimensional Fourier transform and use in imaging. Further applications to optics, crystallography. Emphasis is on relating the theoretical principles to solving practical engineering and science problems. 29 Lectures Previous Knowledge Recommended (Matlab), The Fourier Series, Analysis V. Synthesis, Periodic Phenomena And The Fourier Series -Periodicity In Time And Space -Reciprocal Relationship Between Domains, The Reciprocal Relationship Between Frequency And Wavelength Periodicity; How Sine And Cosine Can Be Used To Model More Complex Functions, Example Of Periodizing A Signal, Discussion Of How To Model Signals With Sinusoids, "One Period, Many Frequencies" Idea In Modeling Signals, Modeling A Signal As The Sum Of Modified Sinusoids (Formula), Complex Exponential Notation, Symmetry Property Of The Complex Coefficients In The Fourier Series, Discussion Of The Generality Of The Fourier Series Representation For Modeling A Periodic Function Summary Of Previous Lecture (Analyzing General Periodic Phenomena As A Sum Of Simple Periodic Phenomena), Fourier Coefficients; Discussion Of How General The Fourier Series Can Be (Examples Of Discontinuous Signals), Discontinuity And Its Impact On The Generality Of The Fourier Series, Infinite Sums To Represent More General Periodic Signals, Summary Of Convergence Issues, Convergence: Continuous Case, Smooth Case (Fourier Series Converges To The Signal), Convergence: Jump Discontinuity, Convergence: General Case (Convergence On Average/ In Mean/ In Energy) Continued Discussion Of Fourier Series And The Heat Equation, Transition From Fourier Series To Fourier Transforms (Periodic To Nonperiodic Phenomena), Fourier Series Analysis And Synthesis; Relation To Fourier Transform And Inverse Fourier Transform, Fourier Series/ Coefficients With Period T, Spectrum Picture For Fourier Series With Period T, Effects Of A Change In T, The Complications Of Finding The Fourier Transform By Letting T Go To Infinity (Fourier Coefficients Go To 0) Review Of Fourier Transform (And Inverse) Definitions, Notation, Review Of Rect And Triangle Transforms, Example: Fourier Transform Of A Gaussian, The Duality Property Of The Fourier Transform, Example Of An Application Of The Duality Property Continuing Convolution: Review Of The Formula, Situiation In Which It Arose, Example Of Convolution: Filtering, The Ideas Behind Filtering, Terminology, Interpreting Convolution In The Time Domain, General Properties Of Convolution In The Time Domain, Derivative Theorem For Fourier Transforms, Heat Equation On An Infinite Rod Central Limit Theorem And Convolution; Main Idea, Introduction, Normalization Of The Gaussian, The Gaussian In Probability; Pictorial Demonstration With Convolution, The Setup For The CLT, Key Result: Distribution Of Sums And Convolution (With Proof), Other Assumptions Needed To Set Up CLT, Statement Of The Central Limit Theorem, Using The Fourier Transform To Prove The CLT Setting Up The Fourier Transform Of A Distribution, Example Of Delta As A Distribution, Distributions Induced By Functions (Includes Many Functions), The Fourier Transform Of A Distribution, The Class Of Tempered Distributions, FT Of A Tempered Distribution, Definition Of The Fourier Transform (By How It Operates On A Test Function), The Inverse Fourier Transform (Proof), Calculations Of Fourier Transforms Using This Definition (Distributions) Derivative Of A Distribution, Example: Derivative Of A Unit Step, Example: Derivative Of Sgn(X), Applications To The Fourier Transform (Using The Derivative Theorem), Caveat To Distributions: Multiplying Distributions, Distributions*Functions, Special Case: The Delta Function And Sampling, Convolution In Distributions, Special Case: Convolution When T = Delta, The Scaling Property Of Delta Application Of The Fourier Transform: Diffraction: Setup, Representation Of Electric Field, Approach Using Huyghens' Principle, Discussion Of The Phase Change Associated With Different Paths, Use Of The Fraunhofer Approximation, Aperture Function, Result; In General And For Single/ Double Slits More On Results From Last Lecture (Diffraction Patterns And The Fourier Transforms), Setup For Crystallography Discussion (History, Concepts), 1-Dimensional Version, The Fourier Transform Of The Shah Function, Trick: Poisson Summation Formula, Proof Of The Poisson Summation Formula, Fourier Transform Of The Shah Function: Result, Fourier Transform Of The Shah Function With Spacing P, Application To Crystals Review Of Main Properties Of The Shah Function, Setup For The Interpolation Problem, Bandwidth Assumption, Solving For Exact Interpolation For Bandlimited Signals, Periodizing The Signal By Convolution With The Shah Function, Solution Of The Interpolation Problem Aliasing Demonstration With Music, Transition To Discrete! The DFT, The Plan For Transitioning To Discrete Time, Creating A Discrete Signal From F(T) Creating A Discrete Version Of The Fourier Transform Of The Sampled Version Of F(T), Summary Of What We Just Did, Summary Of Results (Formulas), Moving From Continuous To Discrete Variables, Final Result: The DFT Review: Definition Of The DFT, Sample Points, Relationship Between N And Spacing In Time/Frequency, Complex Exponentials In The Discrete DFT, DFT Written With Discrete Complex Exponential Vector, Periodicity Of Inputs And Outputs In The DFT (More On This In Next Lecture), Orthogonality Of The Vector Of Discrete Complex Exponentials, Note On Orthonormality Of Discrete Complex Exponential Vector (Or Lack Thereof), Consequence Of Orthogonality: Inverse DFT Review Of Basic DFT Definitions, Special Case: Value Of The DFT At 0, Two Special Signals: One Vector, Delta Vector, DFT Of Deltas, Complex Exponentials, DFT As Nxn Matrix Multiplication, Periodicity Of Input/Output Signals In The DFT, Result Of Periodicity: Indexing, Result Of Periodicity: Duality FFT Algorithm: Setup: DFT Matrix Notation, One Intuition Behind FFT: Factoring Matrix, Our Approach: Split Order N Into Two Order N/2, Iterate, Notation (To Keep Track Of Powers Of Complex Exponentials), Plugging New Notation Into DFT; Split Into Even And Odd Indices, Result For Indices 0 To N/2-1, Result For Indices N/2 To N-1, Summary Of Results (DFT As Combination Of 2 Half Order Dfts) Review Of Last Lecture: Discrete V. Continuous Linear Systems, Cascading Linear Systems, Derivation Of The Impulse Response, Schwarz Kernel Theorem, Example: Impulse Response For Fourier Transform, Example: Switch, Special Case: Convolution, Time Invariance, Result: If A System Is Given By Convolution, It Is Time Invariant; Converse True As Well, Two Main Ideas Sumarized (Linear->Integration Against Kernel, Time Invariant If Given By Convolution) Review Of Last Lecture: LTI Systems And Convolution, Comment On Time Invariant Discrete Systems, The Fourier Transform For LTI Systems; Complex Exponentials As Eigenfunctions, Discussion Of Sine And Cosine V. Complex Exponentials As Eigenfunctions (Generally They Are Not), Discrete Version (Discrete Complex Exponentials Are Eigenvectors), Discrete Results From A Matrix Perspective Is our political ideology simply the result of a genetic coin toss? Mounting evidence suggests that biology may be a factor. In this video, Academic Earth explores some of the key research into the biology of politics.
More About This Textbook Overview Successful game programmers understand that in order to take their skills beyond the basics, they must have an understanding of central math topics; however, finding a guide that explains how these topics relate directly to games is not always easy. Beginning Math Concepts for Game Developers is the solution! It includes several hands-on activities in which basic math equations are used for the creation of graphs and, ultimately, animations. By the time you finish, you will have developed a complete application from the ground up that allows you to endlessly explore game development scenarios for 2D games. If you have a basic understanding of programming essentials and a desire to hone your math skills, then get ready to take a unique journey that examines what is possible when you combine game development with basic math concepts. Related Subjects Meet the Author John P. Flynt, Ph.D., works in the software development industry, has taught at colleges and universities, and has authored courses and curricula for several college-level game development programs. His academic background includes work in information technology, the social sciences, and the humanities. Among his works are In the Mind of a Game, Simulation and Event Modeling for Game Developers (with co-author Ben Vinson), and Software Engineering for Game Developers. John lives in the foothills near Boulder, Colorado. Boris Meltreger graduated from a top mathematics and physics high school in Russia. He went on to earn an advanced degree in optical engineering. After completing a dissertation on acoustics and optics, Boris took up work for the Russian government developing optical computers. He has been the recipient of engineering awards for his work and has owned his own engineering company. Boris has in recent years performed pioneering work in the development of optical technologies for medical applications and currently works as a software engineer. Boris lives in Aurora, Colorado Table of Contents 1. GETTING STARTED WITH C# AND THE MATH LIBRARY A. C# as a Game Development Language B. Setting up a Project C. Inspecting the Math Library D. Guess a Number 2. FUNCTIONS AND METHODS A. Understanding Functions as Patterns B. Creating Lab for Exploring Functions a.Generating Date for a Table b.Developing Classes c.Equations and Methods 3. CONCEPTS BEHIND FUNCTIONS A. Number Domains B. Restricted Values C. Handling Exceptions D. Making use of a List to Store Function Output E. Fields and Properties F. Division By Zero and Other Mysteries 4. EXTENDING THE LAB WITH A COORDINATE PLANE A. How to Graph B. Putting the List Values to Work C. Using a Flag D. Closing E. The True Value Game 5. LOCAL AND WORLD SPACE IN CARTESIAN TERMS A. Spawning a Cartesian Plane. B. Learning How to Do Things Twice C. Understanding Grids D. Axes and How to Make Them E. Making Your Point F. Finding the Curve 6. CHANGING THINGS: LINES, SLOPES, AND METHODS A. What Counts As a Valid Function B. Constant Functions C. How to Make a Profit D. Linear Things E. Making Things Visible F. Method Overloading H. The Factory Game 7. QUADRATICS AND OTHER FUNCTIONS A. Parabolas B. Minimum and Maximum C. Absolute Values D. Discontinuous Functions E. Stair Steps and Other Antics F. The Table Game 8. LIMITS AND METHODS FOR THEM A. Talk of Limits B. Bicycle Tires C. Different forms of Limits D. Continuity E. Infinity F. Creating Graphics That Merge 9. ANIMATING THE WORLD A. Threads and Timers B. Eliminating Flicker C. Derived Classes D. Working with Arrays of Continuous Values E. Queues and Coordinates F. Event Generation G. Event Detection 10. IN TO THE GAME A. Derivation and Acceleration B. Controlling Flight C. Multiplying Complexity D. A Target Game E. Extended Intelligence in Games
High School Math III Review (Review Smart) Book Description: If trudging through your textbook to study and complete homework assignments has become a frustrating grind, then get ready for a smooth ride to higher test scores and outstanding grades with The Princeton Review's High School Math III Review.We tell it to you straight, thoroughly explaining the important topics you'll need to understand to prepare for quizzes and tests, complete homework assignments effectively, and earn higher grades. We've carefully examined math textbooks just like yours to make sure that this book includes all the material essential to a thorough review. In this guide, we cover:Rational and Radical ExpressionsDegrees and RadiansTrigonometryComplex NumbersQuadratic EquationsTransformationsCircle RulesProbabilityStatisticsConic Sections*LogarithmsAnd since practicing your test-taking skills is just as important to getting good grades as knowing the material, we include two practice exams that feature the types of questions and problems that appear on in-class tests
Math Forum provides cadets an opportunity to explore math concepts and applications beyond the academic program. The intent is to foster interest in mathematics as an art and a science, and provide an arena for the exchange of ideas among individuals and organizations within and outside of the Academy. Club activities include monthly meetings at which guest speakers present topics ranging from math history to modern applications. Members have participated in national math modeling competitions and traveled to other service academies and universities to observe their math programs. Trips to research labs allows cadets to see the application of mathematics in defense and advanced technology
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books.google.com - A... Algebra and Trigonometry College Algebra and Trigonometry A with both instructor and student in mind, the text is easy to use, and each section can be covered in one class. Clearly marked subsections make it easy to omit more basic topics when necessary. The material is carefully organized and paced, offering thoughtful explanations through a combination of examples and theory. Contains an excellent review of basic algebra, with coverage of equations and inequalities, graphs and functions, complex numbers and more. This edition contains more exercises requiring the use of a calculator, new and numerous examples, and end-of-section exercises that provide a good test of the student's progress.
This is a concise introductory textbook for a one semester course in the history and philosophy of mathematics. It is written for mathematics majors, philosophy students, history of science students and secondary school mathematics teachers. The only prerequisite is a solid command of pre-calculus mathematics. It is shorter than the standard textbooks in that area and thus more accessible to students who have trouble coping with vast amounts of reading. Furthermore, there are many detailed (...) explanations of the important mathematical procedures actually used by famous mathematicians, giving more mathematically talented students a greater opportunity to learn the history and philosophy by way of problem solving. Several important philosophical topics are pursued throughout the text, giving the student an opportunity to come to a full and consistent knowledge of their development. These topics include infinity, the nature of motion, and Platonism. This book offers, in fewer pages, a deep penetration into the key mathematical and philosophical aspects of the history of mathematics. (shrink) This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research and how a number of historical accounts can be deepened by embracing philosophical questions. The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating (...) its interactions with the related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematical physics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink) Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges provide an opportunity to reconfigure particular philosophical problems – for example, the problem of individuation – and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction (...) of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. -/- In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seeming incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it. -/- Simon B. Duffy is a Senior Lecturer in Philosophy at Yale-NUS College, Singapore, and Honorary Research Associate in the Department of Philosophy at the University of Sydney, Australia. He is the author of The Logic of Expression: Quality, Quantity, and Intensity in Spinoza, Hegel and Deleuze (2006). (shrink) We have reached the peculiar situation where the advance of mainstream science has required us to dismiss as unreal our own existence as free, creative agents, the very condition of there being science at all. Efforts to free science from this dead-end and to give a place to creative becoming in the world have been hampered by unexamined assumptions about what science should be, assumptions which presuppose that if creative becoming is explained, it will be explained away as an illusion. (...) In this paper it is shown that this problem has permeated the whole of European civilization from the Ancient Greeks onwards, leading to a radical disjunction between cosmology which aims at a grasp of the universe through mathematics and history which aims to comprehend human action through stories. By going back to the Ancient Greeks and tracing the evolution of the denial of creative becoming, I trace the layers of assumptions that must in some way be transcended if we are to develop a truly post-Egyptian science consistent with the forms of understanding and explanation that have evolved within history. (shrink) Figures of Thought looks at how mathematical works can be read as texts and examines their textual strategies. David Reed offers the first sustained and critical attempt to find a consistent argument or narrative thread in mathematical texts. Reed selects mathematicians from a range of historical periods and compares their approaches to organizing and arguing texts, using an extended commentary on Euclid's Elements as a central structuring framework. He develops fascinating interpretations of mathematicians' work throughout history, from Descartes to (...) Hilbert, Kronecker, Dedekind, Weil and Grothendieck. Reed traces the implications of this approach to the understanding of the history and development of mathematics. (shrink) Social revolutions--that is critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. But can the idea of revolutionary upheaval be extended to the world of ideas and theoretical debate? The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences. A fascinating, but little known, off-shoot of this was a debate which began in the United States in the mid-1970's as to whether the concept of revolution could (...) be applied to mathematics as well as science. Michael Grove declared that revolutions never occur in mathematics, while Joseph Dauben argued that there have been mathematical revolutions and gave some examples. This book is the first comprehensive examination of the question. It reprints the original papers of Grove, Dauben, and Mehrtens, together with additional chapters giving their current views. To this are added new contributions from nine further experts in the history of mathematics, who each discuss an important episode and consider whether it was a revolution. The whole question of mathematical revolutions is thus examined comprehensively and from a variety of perspectives. This thought-provoking volume will interest mathematicians, philosophers, and historians alike. (shrink) Volume 9 of the Routledge History of Philosophy surveys ten key topics in the Philosophy of Science, Logic and Mathematics in the Twentieth Century. Each article is written by one of the world's leading experts in that field. The papers provide a comprehensive introduction to the subject in question, and are written in a way that is accessible to philosophy undergraduates and to those outside of philosophy who are interested in these subjects. Each chapter contains an extensive bibliography (...) of the major writings in the field. Among the topics covered are the philosophy of logic; Ludwig Wittgenstein's Tractatus; a survey of logical positivism; the philosophy of physics and of science; probability theory and cybernetics. (shrink) In this book, which is both a philosophical and historiographical study, the author investigates the fallibility and the rationality of mathematics by means of rational reconstructions of developments in mathematics. The initial chapters are devoted to a critical discussion of Lakatos' philosophy of mathematics. In the remaining chapters several episodes in the history of mathematics are discussed, such as the appearance of deduction in Greek mathematics and the transition from Eighteenth-Century to Nineteenth-Century analysis. The (...) author aims at developing a notion of mathematical rationality that agrees with the historical facts. A modified version of Lakatos' methodology is proposed. The resulting constructions show that mathematical knowledge is fallible, but that its fallibility is remarkably weak. (shrink) Diagrams are ubiquitous in mathematics. From the most elementary class to the most advanced seminar, in both introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase understanding, and prove results. They thus fulfill a variety of important roles in mathematical practice. Long overlooked by philosophers focused on foundational and ontological issues, these roles have come to receive attention in the past two decades, a trend in line with the growing philosophical interest in actual mathematical practice. This article canvasses five senses in which one might introduce an historical element into the philosophy of mathematics: 1. The temporal dimension of logic; 2. Explanatory Appeal to Context rather than to General Principles; 3. Heraclitean Flux; 4. All history is the History of Thought; and 5. History is Non-Judgmental. It concludes by adapting Bernard Williams' distinction between 'history of philosophy' and 'history of ideas' to argue that the philosophy of mathematics is unavoidably (...) historical, but need not and must not merge with historiography. (shrink) The thirty year long friendship between Imre Lakatos and the classic scholar and historian of mathematics Árpád Szabó had a considerable influence on the ideas, scholarly career and personal life of both scholars. After recalling some relevant facts from their lives, this paper will investigate Szabó's works about the history of pre-Euclidean mathematics and its philosophy. We can find many similarities with Lakatos' philosophy of mathematics and science, both in the self-interpretation of early axiomatic Greek (...) class='Hi'>mathematics as Szabó reconstructs it, and in the general overview Szabó provides us about the turn from the intuitive methods of Greek mathematicians to the strict axiomatic method of Euclid's Elements. As a conclusion, I will argue that the correct explanation of these similarities is that in their main works they developed ideas they had in common from the period of intimate intellectual contact in Hungarian academic life in the mid-twentieth century. In closing, I will recall some relevant features of this background that deserve further research. (shrink) Duhem's portrayal of the history of mathematics as manifesting calm and regular development is traced to his conception of mathematical rigor as an essentially static concept. This account is undermined by citing controversies over rigorous demonstration from the eighteenth and twentieth centuries. Choosing the history of statistics and operations research as a casestudy, several ways of setting the development of 20th century applied mathematics into a social context are discussed. It is shown that there is ample common ground between these contextualizations and several recent research programs in general contemporary history. It is argued that a closer cooperation between general historians and historians of mathematics might further the integration of the internalist and externalist approaches within the historiography of (...)mathematics. (shrink) This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of (...) programs for the training of preservice as well as inservice mathematics teachers. (shrink) A more and more important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application of mathematical methods. The history is studied more and more as a system of various processes, within which one can detect waves and cycles of different lengths – from a few years to several centuries, or even millennia. This issue is the third collective monograph in the (...) series of History & Mathematics almanacs and it is subtitled Processes and Models of Global Dynamics. The contributions to the almanac present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models. This issue of the almanac consists of two main sections: (I) Analyses of the World Systems and Global Processes, and (II) Models of Economic and Demographic Processes. We hope that this issue of the almanac will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences. (shrink) Geraldine Brady, "From Peirce to Skolem. A Neglected Chapter in the History of Logic", Studies in the History and Philosophy of Mathematics, vol. 4, Elsevier (imprint: North-Holland), 2000, ISBN-13: 978-0444503343, ISBN-10: 0-444-50334-X, 625~pp. The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting (...) with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink) Recent philosophical accounts of mathematics increasingly focus on the quasi-Empirical rather than the formal aspects of the field, The praxis of how mathematics is done rather than the idealized logical structure and foundations of the theory. The ultimate test of any philosophy of mathematics, However idealized, Is its ability to account adequately for the factual development of the subject in real time. As a text case, The works and views of felix klein (1849-1925) were studied. Major advances (...) in mathematics turn out to be most adequately understood as shifts to new conceptualizations at different levels of idealization and abstraction. The implications of the model are explored with special reference to modern philosophical views of the nature and the warrants of mathematical claims to knowledge, And the methodology of the development of this knowledge. (shrink) A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: 'omnipresent' and 'multipresent' theories, and 'ubiquitous' notions that form dependent parts, or moments, (...) of theories. The category of 'facets' is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. (shrink) From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors (...) and many others. The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink) The Pythagorean idea that numbers are the key to understanding reality inspired philosophers in late Antiquity (4th and 5th centuries A.D.) to develop theories in physics and metaphysics based on mathematical models. This book draws on some newly discovered evidence, including fragments of Iamblichus's On Pythagoreanism, to examine these early theories and trace their influence on later Neoplatonists (particularly Proclus and Syrianus) and on medieval and early modern philosophy. The first part of this paper consists of an exposition of the views expressed by Pierre Duhem in his Aim and Structure of Physical Theory concerning the philosophy and historiography of mathematics. The second part provides a critique of these views, pointing to the conclusion that they are in need of reformulation. In the concluding third part, it is suggested that a number of the most important claims made by Duhem concerning physical theory, e.g., those relating to the Newtonian (...) method, the limited falsifiability of theories, and the restricted role of logic, can be meaningfully applied to mathematics. (shrink) William Tait is one of the most distinguished philosophers of mathematics of the last fifty years. This volume collects his most important published philosophical papers from the 1980's to the present. The articles cover a wide range of issues in the foundations and philosophy of mathematics, including some on historical figures ranging from Plato to Gödel. Tait's main contributions were initially in proof theory and constructive mathematics, later moving on to more philosophical subjects including finitism and skepticism (...) about mathematics. This collection, presented as a whole, reveals the underlying unity of Tait's work. The volume includes an introduction in which Tait reflects more generally on the evolution of his point of view, as well as an appendix and added endnotes in which he gives some interesting background to the original essays. This is an important collection of the work of one of the most eminent philosophers of mathematics in this generation. (shrink) In this paper, I argue against Penelope Maddy's set-theoretic realism by arguing (1) that it is perfectly consistent with mathematical Platonism to deny that there is a fact of the matter concerning statements which are independent of the axioms of set theory, and that (2) denying this accords further that many contemporary Platonists assert that there is a fact of the matter because they are closet foundationalists, and that their brand of foundationalism is in radical conflict with actual mathematical practice. (...) of Peano's conception, but it is essential to understand the relations of Peano's logic with other philosophical traditions and some epistemological aspects of Peano's perspective, such as the search for a universal language. (shrink)
Suwanee StatisticsProblem solving skills are emphasized throughout, and time is devoted to advanced topics like telescoping sums and piecewise functions Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern ma...
frey Harold Hardy book reviews An introduction to the theory of numbers. An Older Approach to a Math book The book has a lot of good information, but the style is a bit dated. It's a lot of the basic ideas of number theory - prime numbers, etc - but the format makes it a little difficult to follow, since
SSAC General Collection Modules Following are short descriptions of the available modules. To access any of them, select from the list and you will connect to cover material including learning goals, context of use, and other information. Within the cover material (under "Teaching Materials") there is a link by which you can download the student version of the module. You can also request an instructor's version by clicking on the "request" link under Teaching Materials. There are a couple of ways of searching for modules. The box below left provides a full-text search of the cover material about the module, so you can search by author, subject, or keyword. The controlled vocabularies below right allow targeted searches by three different dimensions (click on the links to see the full, hierarchical vocabularies): Math content (Quantitative Concept)(Microsoft Word 41kB Jun17 10), Context (Subject)(Microsoft Word 29kB Jul13 07), and Excel Skill(Microsoft Word 32kB Jul13 07). Grade Calculationpart of Spreadsheets Across the Curriculum:General Collection:Examples Spreadsheets Across the Curriculum module. This activity introduces the student to the concept of weighted averages by asking them to calculate course grades and grade point averages. Driving Across Town for Cheaper Gas -- A Cost/Benefit Analysispart of Spreadsheets Across the Curriculum:General Collection:Examples Spreadsheets Across the Curriculum module. Students build a spreadsheet to explore the trade-offs between "bargain-priced" gas vs. the extra mileage needed to get it. A modeling problem. Calzones vs. Mini-Pizzas -- A Linear Programming Problempart of Spreadsheets Across the Curriculum:General Collection:Examples Spreadsheets Across the Curriculum module. Students build a spreadsheet to find the combination of mini-pizzas and calzones that maximizes revenue given constraints on labor time and baking time. Understanding Mortgage Paymentspart of Spreadsheets Across the Curriculum:General Collection:Examples Spreadsheets across the Curriculum module. Students build Excel spreadsheets to calculate monthly mortgage payments and evaluate how much of their payment is applied to the principle and interest.
In Math C257 students are expected to consistently write the derivative of expressions that contain the inverse trigonometric, logarithmic, exponential, hyperbolic, and inverse hyperbolic functions; evaluate integrals (definite and indefinite) by using fundamental integral formulas, partial fractions, integration by parts, and substitutions, trigonometric substitutions; expand skills with limits, including l'Hôpital's Rule and improper integrals; identify the conic section represented by a second degree equation and give the foci, vertices, and directricies; use polar coordinates to graph equations and to find area, arc length, and intersection of curves; use the tests for convergence and divergence of sequences and series; write infinite series representations of various functions; and use the fundamental concepts of vectors including sums, dot product, and projection. Students successfully demonstrating these Math C152 skills will be prepared for Math C257.
Boost your students understanding of Saxon Math with DIVE's easy-to-understand lectures! Each lesson concept in Saxon Math's textbook is taught step-by-step on a digital whiteboard, averaging about 10-15 minutes in length; and because each lesson is stored separately, you can easily move about from lesson-to-lesson as well as maneuver within the lesson you're watching. Taught from a Christian worldview, Dr. David Shormann also provides a weekly syllabus to help students stay on track with the lessons. DIVE teaches the same concepts as Saxon, but does not use the problems given in the text; it cannot be used as a solutions guide. Great supplement to Saxon's Algebra! We are very satisfied with our purchase. Share this review: 0points 0of0voted this as helpful. Review 3 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Excellent product, excellent seller! Date:June 27, 2011 Nicat Location:CA Age:55-65 Gender:male Quality: 5out of5 Value: 5out of5 Meets Expectations: 5out of5 The Saxon product is a useful adjunct to teaching math. Constant layering and call back of material through every year. Share this review: +1point 1of1voted this as helpful. Review 4 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 4out of5 Saxon Dive CD review Date:October 29, 2010 joyful91 Location:Alabama Age:35-44 Gender:female Quality: 5out of5 Value: 5out of5 Meets Expectations: 4out of5 This CD is excellent. The only thing that would make me love it more is if it COULD be used as a solutions manual. That's why we loved Teaching Textbooks because every problem in the text was also worked out on the CD for the student to review if they missed a problem. You can't do that with this CD BUT the curriculum is much stronger than TT, that's why we came back to Saxon. Their math curriculum is the best and this CD is a valuable assest to our daughter's daily instruction!! Share this review: +2points 2of2voted this as helpful. Review 5 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Date:September 11, 2010 Richard Hopp My oldest daughter loves this dvd program, she has asked for us to get the science and any other DIVE program for next year that is available. She really enjoys Dr. Shormanns style of teaching. This program also gets her used to taking notes which is something that sometimes is lacking in homeschooling. She wasn't too excited when I told her that she needed to watch the intro before she could begin school at first, but was extremely excited to know that he was a Christian. Now she looks forward to her math time. Share this review: +1point 1of1voted this as helpful. Review 6 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Date:September 9, 2010 Christine Roth This is the best money we have EVER spent on home school material! I love math but have five "students" all at different levels. Math is one of our most time consuming subjects so the DIVE CD has freed up this teacher to focus on the little students while the older children can now learn independently. It has not only blessed us in the area of time management but also in our attitudes. It is a much more enjoyable subject for all! CR Share this review: 0points 0of0voted this as helpful. Review 7 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Date:August 25, 2010 James Pruett This has been such a help for my son - math is not my strong suit, so I am thankful he can see the lesson taught and understand it! DIVE is a great addition to using Saxon's textbooks! Dr. David Shormann teaches Math in a way that makes it so easy to follow. He teaches these lessons on CD so you can go back anytime and recheck anything he has taught. He also makes himself available at his personal email in the event you have any questions regarding a lesson he has taught. Share this review: 0points 0of0voted this as helpful. Review 9 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 4out of5 Date:October 24, 2009 Purry We tried Algebra without this CD and our son was choking on it. His test scores have certainly improved. I'm glad we got it. Share this review: 0points 0of0voted this as helpful. Review 10 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Date:June 17, 2009 Sandra Soria It is better than the book. It builds confidence in the subject, visual step by step. Share this review: 0points 0of0voted this as helpful. Review 11 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Date:May 23, 2009 Debora Williams I was very satisfied with the Saxon DIVE 8/7 and know I will be satsified with the Algebra 1 DIVE. My son enjoys using the Saxon DIVE curriculum. Share this review: 0points 0of0voted this as helpful. Review 12 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Date:October 10, 2008 Teresa The Dive CD is wonderful!!! It's like having an Algebra teacher at your house! Dr. David Shormann is very informative and pleasant to listen to. I am so glad we purchased this product! Share this review: 0points 0of0voted this as helpful. Review 13 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Date:June 19, 2008 Cha This was my second time getting the DIVE CD with our Saxon Math. My son enjoyed using it so much he requested it again, he enjoys having a second explaination for solving problems. I enjoy the support and enthusiastic approach to keeping focus on a simple way to explain math. We love Saxon and the DIVE CD's. My son was left behind in the public school system when I started home schooling in sixth grade. When I tested him he tested out at a fourth grade level. Since we've used the Saxon math, he is now in Algebra 1. He moved up a full four grade levels in a years worth of work. Because my son was finding math so much fun, we continued on into the summer months until he was ahead of his class. His self esteem has sky rocketed. Saxon works for my son. Share this review: +1point 1of1voted this as helpful. Review 14 for Saxon Math Algebra 1 3rd Edition DIVE CD-Rom Overall Rating: 5out of5 Date:May 27, 2008 Kathy Gildez Great product. Very helpful! My daughter was having a lot of trouble with math, but this has helped her understanding concepts.
These suggestions are intended to make your life easier. Some of them may seem like extra work, but really they cause you less work in the long run. This is a preliminary document, i.e. a work in progress. 1. Be tidy and systematic. Whenever you have data in tables, especially when place-value is important, be sure to keep things lined up in neat columns and neat rows. See section 4 for an example of how this works in practice. 2. If your columns are wobbly, get a pad of graph paper and see if that helps. If you don't have a supply of store-bought graph paper, you can make graph paper on your computer printer. There are freeware programs that do a very nice job of this. If that's too much bother, start with a plain piece of paper and sketch in faint guidelines when necessary. 3. Paper is cheap. If you find that you are running out of room on this sheet of paper, get another sheet of paper, rather than squeezing the calculation into a smaller space. See section 4 for an example of using extra paper to achieve a better result. 4. The whole calculation should be structured as a succession of true statements. The first statement is true, the next statement is true, and the statement after that, et cetera. Each statement is a consequence of the previous statements (in conjunction with known theorems, and the "givens" of the problem). Finally, we get to the bottom line, and we know it is true. An example of this can be seen in reference 1. 5. Sometimes, especially for long and/or complex calculations, it helps to organize your calculation in two columns. An example of this can be seen in reference 1. In the left column, you write an equation. In the right column, make a note as to how you derived that equation. (Numbering all your equations makes this easier.) If there isn't enough width to do this easily, turn the paper 90 degrees, so it becomes wider and less tall. (Writing paper isn't suitable for this, so use graph paper ... or lacking that, plain white paper.) 6. Avoid writing down an un-named number like 17, or an un-named expression like a+17 ... because you might forget the meaning thereof. Instead, whenever possible, write down an equation such as b=a+17. That way you can point to every item on the page and say that's true, that's true, that's true... in accordance with the strategy described in item 4. If the meaning of variables such as a and b is not obvious, write down a legend somewhere (like the legend of a map), explaining in a sentence or two the meaning of each variable. That is, a name is not the same as an explanation. Do not expect the structure of a name or symbol to tell you everything you need to know. Most of what you need to know belongs in the legend. The name or symbol should allow you to look up the explanation in the legend. 7. It is important to be able to go back and check the correctness of the calculation you have done. See section 4 and/or reference 1 for examples of what this means in practice. 8. As a corollary of item 7: Don't perform surgery on your equations. That is, once you write down a correct equation, don't start crossing out terms (or, worse, erasing terms) and replacing them with other expressions. Such a substitution may be "mathematically" correct, if the replacement is equal to the thing being replaced ... but it is a bad strategy, because it makes it hard for you to check your work. Instead, write a new equation. Leave the old equation as it is. Paper is cheap. An example of this can be seen in reference 1. 9. Keep track of the units for each expression. For example, the statement "x=2.5 inches" means something rather different from "x=2.5 meters" ... and if you shorthand it as "x=2.5" you're just asking for trouble. Sometimes the penalty for getting this wrong is 328 million dollars, as in the case of the Mars Climate Orbiter (reference 2). Most computer languages do not automatically keep track of the units, so you will have to do it by hand, in the comments. If the calculation is nicely structured, it may suffice to have a legend somewhere, spelling out the units for each of the variables. If variables are re-used and/or converted from one set of units to another, you need more than just a legend; you will need comments (possibly quite a lot of comments) to indicate what units are being used at each point in the code. One policy that is sometimes helpful (but sometimes risky) is to convert everything to SI units as soon as it is read in, even in fields where SI units are not customary. Then you can do the calculation in SI units and convert back to conventional units (if necessary) immediately before writing out the results. (This is problematic when writing an "intermediate" file. Should it be SI or customary? How do you know the difference between an "intermediate" result and a final result?) It is certainly possible for computer programs to keep track of the units automatically. A nice example is reference 3. It is a shame that such features are not more widely available 10. The factor-label method is a convenient and powerful way of converting units when necessary. The correctness of this method is a direct consequence of the axioms of algebra, since it starts by multiplying by unity, which is allowed by the axioms. 11. Haste makes waste, especially with multi-step processes. If you work methodically, you'll get the right answer the first time, and that's all there is to the story. If you try to do it twice as fast, you'll get the wrong answer, and then you'll have to do it over again ... and again ... and again. Here's a classic example: The task is to add 198 plus 215. The easiest way to solve this problem in your head is to rearrange it as (215 + (200 − 2)) which is 415 − 2 which is 413. The small point is that by rearranging it, a lot of carrying can be avoided. One of the larger points is that it is important to have multiple methods of solution. This and about ten other important points are discussed in reference 4. The classic "textbook" diagram of an inequality uses shading to distinguish one half-plane from the other. This is nice and attractive, and is particularly powerful when diagramming the relationship between two or more inequalities, as shown in figure 2. Obviously the hatched depiction is not as beautiful as the shaded depiction, but it is good enough. It is vastly preferable on cost/benefit grounds, for most purposes. Some refinements: I recommend hatching the excluded half-plane, so that the solution-set remains unhatched rather than hatched. This is particularly helpful when constructing the conjunction (logical AND) of multiple inequalities. I recommend using solid lines versus dashed lines to distinguish "≥" inequalities from ">" inequalities. If you're going to hatch the excluded half-plane, use solid lines for the ">" inequalities, to show that the boundary itself is excluded. As suggested in figure 4, for linear inequalities you can do the hatching faster and more accurately with the help of a ruler or straightedge, which makes it easy to ensure that none of the hatches stray into the wrong region. Short multiplication refers to any multiplication problem where you just memorize the answer. You must memorize the multiplication table for everything from 0×0 through 9×9. You get the next step (up to 10×10) practically for free, and it is often worthwhile to keep going up to 12×12. Long multiplication refers to multiplying larger numbers. This works by breaking the numbers down into their individual digits, then multiplying on a digit-by-digit basis (using the short multiplication facts for zero through nine) and then combining all the results with due regard for place value. We now discuss a nice way to do long multiplication. The first steps are shown in figure 5. There are two parts to the figure, representing two successive stages of the work. Anything shown in black is something you actually write down, whereas anything shown in color is just commentary, put there to help us get through the explanation the first time. We start with the leftmost part of the figure. This is just the statement of the problem, namely 4567×321. The important point here is to line up the numbers as shown, so that the ones' place of the top number lines up with the ones' place of the bottom number, et cetera. Keeping things aligned in columns is crucial, since the colums represent place value. If you have trouble keeping things properly lined up, use grid-ruled paper. You can see such a grid in figure 5. If you don't have grid-ruled paper, you can always sketch in some guide lines. As mentioned in item 1, tidiness pays off. You may omit the multiplication sign (×) if it is obvious from context that this is a multiplication problem (as opposed to an addition problem). Tangential remark: Some people attempt to call one of these numbers the multiplier and the other the multiplicand, but since multiplication is commutative the distinction is meaningless. People often use the terms in ways inconsistent with the supposed definition. I call both of them multiplicands, which is more-or-less Latin for "thing being multiplied". In figure 6 we have two things being multiplied. Note that since multiplication is associative, you could easily have many things being multiplied, as in 12×32×65×99, in which case it again makes sense to call each of them multiplicands, and it is obviously hopeless to attempt to distinguish "the" multiplier from "the" multiplicand. As a related issue, there are also holy wars as to whether 4567×321 means 4567 "times" 321, or 4567 "multiplied by" 321. Again the distinction is meaningless. Don't worry about it. If one of the multiplicands is longer than the other, it will usually be more convenient to place the longer one on top. That's not mandatory, but it makes the calculation slightly more compact. We now proceed digit by digit, starting with the rightmost digit in the bottom multiplicand, which in this case is a 1. We multiply this digit into each digit of the upper multiplicand, working in order right-to-left, which makes sense because it is the direction of increasing place value (even though it is opposite to the direction of reading normal text). We place these results in order in row c, with due regard for place value. The 1×7 result goes in the ones' place, the 1×6 result goes in the tens' place, and the 1×5 result goes in the hundreds' place, and so forth. Actually, multiplication by 1 is so easy that you could just copy the whole number 4567 into row c without thinking about it very hard. You may wish to leave a little bit of space above the numbers in row c, for reasons that will become apparent later. That is all we need to do with the low-order digit of the bottom multiplicand. We now move on to the next digit, working right-to-left. In this case it is a 2. Again we multiply this digit into each digit of the upper multiplicand. The result of the first such multiplication is 2×7=14, which we place in row d. This is most clearly seen in the middle column of the figure 6. Alignment is crucial here. The 14 must be aligned under the 2 as shown. That's because it "inherits" the place value of the 2. Next we multiply 2×6=12. You might be tempted to write this in row d, but there is no room for it there, so it must go on row e, as shown in the middle column of the figure. Again alignment is critical. The 12 is shifted one place to the left of the 14, because it inherits additional place value. It inherits some from the 2 and some from the 6. Since we are working systematically right-to-left, you don't need to think about this too hard; just remember that each of these short-division products must be shifted one place to the left of the previous one. We have now more than half-way finished. We have reached the stage shown in figure 5. The next step is the 2×5=10 multiplication. There is room for this on row d, which is a good place to put it. Next we do the 2×4=8 multiplication. There is room for this on row e, which is a good place to put it. Note the pattern of placing successive short-division results on alternating rows. This is guaranteed to work, because the product of two one-digit numbers can never have more than two digits. At this point (or perhaps earlier), if you are not using grid-ruled paper, you should lightly sketch in some vertical guide lines, as shown by the dashed lines in figure 7. The tableau has become large enough that there is some risk of messing up the alignment, i.e. putting things into the wrong columns, if you don't put in guide lines. That's all for the "2" digit in the bottom multiplicand. We now progress to the "3" digit. The work proceeds in the same fashion. All the short-multiplication results are put in rows f and g. In the tableau, you can see where everything comes from. The color-coded background indicates which digit of the upper multiplicand was involved, and the row indicates which digit of the bottom multiplicand was involved. At this point you can draw a line under row g as shown in the figure. All that remains is a big addition problem, adding up rows c through g inclusive. You can use the space above row c to keep track of carries if you wish, but this is not mandatory. (There are never very many carries, so keeping track of them is easy, no matter how you do it. Some people just count them on their fingers.) The result of the addition is the result of the overall multiplication problem, as shown on row h. Let's do another example, as shown in figure 7, which illustrates one more wrinkle. This example calls attention to the situation where some of the short-multiplication products have one digit, while others have two. You can see this on row c of the figure, where we have 3×3=9 and 3×8=24. In most cases it is safer to pretend they all have two digits, which is what we have done in the figure, writing 9 as 09. Similarly on line d we write 6 as 06. This makes the work fall into a nice reliable pattern. It helps you keep things lined up, and makes the work easier to check. In some cases, such as 345×1 or 432×2, all the short-multiplication products have one digit, so you can write them all on a single line, if you wish. This saves a little bit of paper. On the other hand, remember the advice in item 3: paper is cheap. You may find it helpful to write the short-multiplication products as two digits even if you don't have to. Mathematically speaking, writing one-digit products as two digits is unconventional, but it is entirely correct. It has the advantage of making the algorithm more systematic, and therefore easier to check. In any case, the result of the addition is the result of the overall multiplication problem, as shown on row g of figure 7. That's all there is to it. This algorithm uses two rows of intermediate results for each digit in the bottom multiplicand (except when the digit is zero or one). This has two advantages: First of all, you don't need to do any adding or carrying as you go along; you just write down the short-multiplication results "as is". Secondly, it makes it easy to check your work. Each of the short-multiplication results is sitting there in an obvious place, almost begging to be checked. This differs from the old-fashioned "textbook" approach, which uses only one row per digit, as shown below. The old-fashioned approach supposedly uses less paper – but the advantage is slight at best, and if you allow room keeping track of "carries" throughout the tableau, the advantage becomes even more dubious. What's worse is that the old-fashioned approach is significantly more laborious. It may look more compact, but it involves more work. You have to do the same number of short multiplications, and a greater number of additions. It requires you to do additions (including carries) as you go along. Last but not least, it makes it much harder to check your work. 4 5 6 7 × 3 2 1 4 5 6 7 9 1 3 4 1 3 7 0 1 1 4 6 6 0 0 7 The old-fashioned approach. Not recommended. Remember, paper is cheap (item 3) and it is important to be able to check your work (item 7). The usual "textbook" instructions for how to do long division are both unnecessarily laborious and unnecessarily hard to understand. There's another way to organize the calculation that is much less mysterious and much less laborious (especially when long multi-digit numbers are involved). Note that as discussed above, keeping things lined up in columns is critical. It may help to use grid-ruled paper, or at least to sketch in some guidelines. After writing down the statement of the problem, and before doing any actual dividing, it helps to make a crib, as shown in the lower left of the figure. This is just a multiplication table, showing all multiples of 13 (or, more generally, all multiples of the assigned divisor). It is super-easy to construct such a table, since no multiplication is required. Successive addition will do the job. We need all the multiples from ×1 to ×9, but you might as well calculate the ×10 row, by adding 117+13, as a check on all the additions that have gone before. The first step is to consider the leading digit of the dividend (which in this case is a 7). Since this is less than the divisor (13), there is no hope of progress here, so we proceed to the next step. Now consider the first two digits together, namely the 7 and the 5. Look at the crib to find the largest entry less than or equal to 75. It is 65, as in 5×13=65. Write copy this entry to the division problem, on row b, directly under the 75. Since this came from row 5 of the crib, write a 5 on the answer line, aligned with the 75 and the 65, as shown. Check the work, to see that 5 (on the answer line) times 13 (the divisor, on line a), equals 65 (on line b). Now do the subtraction, namely 75−65=10, and write the result on line c as shown. We now shift attention from the first column to the middle column of figure 8. Bring down the 2 from the dividend, as shown by the red arrow in the middle column of the figure. So now the number we are working on is 102, on line c. The steps from now on are a repeat of earlier steps. Look at the crib to find the largest entry less than or equal to 102. It is 91, as in 7×13=91. Write copy this entry to the division problem, on row d, directly under the 102. Since this came from row 7 of the crib, write a 7 on the answer line, aligned with the 102 and the 91, as shown. Check the work, to see that 7 (on the answer line) times 13 (the divisor, on line a), equals 91 (on line d). Do the subtraction. As a check on the work, when doing this subtraction, the result should never be less than zero, and should never be greater than or equal to the divisor. Otherwise you've used the wrong line from the crib, or made an arithmetic error. We now shift attention to the rightmost column of figure 8. Bring down the 7, look in the crib to find the largest entry less than or equal to 117, which is in fact 117, as in 9×13=117. Since this came from line 9 of the crib, write a 9 on the answer line, properly aligned. The final subtraction yields the remainder on line g. The remainder is zero in this example, because 13 divides 7527 evenly. Perhaps the crib's most important advantage, especially when people are first learning long division, is that the crib removes the mystery and the guesswork from the long division process. This is in contrast to the "trial" method, where you have to guess a quotient digit, and you might guess wrong. Using the crib means we never need to do a short division or trial division; all we need to do is skim the table to find the desired row. We have replaced trial division by multiplication and table-lookup. Actually we didn't even need to do any multiplication, so it would be better to say we have replaced trial division by addition. Another advantage is efficiency, especially when the dividend has many digits. That's because you only need to construct the crib once (for any given divisor), but then you get to use it again and again, once for each digit if the dividend. For long dividends, this saves a tremendous amount of work. (This is not a good selling point when kids are just learning long division, because they are afraid of big multi-digit dividends.) Setting up the crib is so fast that you've got almost nothing to lose, even for small dividends. Another advantage is that it is easy to check the correctness of the crib. It's just sitting there begging to be checked. When bringing down a digit, you may optionally bring down all the digits. For instance, in the middle column of figure 8, if you bring down all the digits you get 1027 on row c. One advantage is that 1027 is a meaningful number, formed by the expression 7527−13×5×102. This shows how the steps of the algorithm (and the intermediate results) actually have mathematical meaning; we are not not blindly following some mystical mumbo-jumbo incantation. I recommend that if you are trying to understand the algorithm, you should bring down all the digits a few times, at least until you see how everything works. A small disadvantage is that bringing down all the digits requires more copying. The countervailing small advantage is that it may help keep the digits lined up in their proper columns. Whether the advantages outweigh the cost is open to question, and probably boils down to a question of personal preference. Another remark: Division is the "inverse function" of multiplication. In a profound sense, for any function that can be tabulated, you can construct the inverse function – if it exists – by switching columns in the table. That is, we interchange abscissa and ordinate: (x,y)↔(y,x). That's why we are able to perform division using what looks like a multiplication table; we just use the table backwards. The modern numeral system is based on place value. As we understand it today, each numeral can be considered a polynomial in powers of b, where b is the base of the numeral system. For decimal numerals, b=10. As an example: Given two expressions such as (a+b+c) and (x+y), each of which has one or more terms, the systematic way to multiply the expressions is to make a table, where the rows correspond to terms in the first expression, and the rows correspond to terms in the second expression: In the special case of multiplying a two-term expression by another two-term expression, the mnemonic FOIL applies. That stands for First, Outer, Inner, Last. As shown in figure 9, we start with the First contribution, i.e. we multiply the first term from in each of the factors. Then we add in the Outer contribution, i.e. the first term from the first factor times the last term from the last factor. Then we add in the Inner contribution, i.e. the last term from the first factor times the first term from the last factor. Finally we add in the Last contribution, i.e. we multiply the last terms from each of the factors.
MATH AND SCIENCE FOR YOUNG CHILDREN, 6th Edition is a unique reference that focuses on the integration of math and science along with the other important areas of child development during the crucial birth-through-eight age range. The purpose of this book is to provide an integrated development of modern analysis and topology through the integrating vehicle of uniform spaces. The reader should have taken an advanced calculus course and an introductory topology course. International agreements, such as those governing arms control and the environment, virtually always require some degree of verification so that compliance can be established. To ensure that the verification process is regarded as efficient, effective and impartial, it is important to model it mathematically.
Knowing my math This is a discussion on Knowing my math within the C Programming forums, part of the General Programming Boards category; I'm kind of ashamed to admit it but I never really had any school, and because of that I never ... Knowing my math I'm kind of ashamed to admit it but I never really had any school, and because of that I never learned any math. And I'm talking about none, zero, zip, dd if=/dev/null of=/proc/math_skills. I know enough about simple calculations and tables to get me by with a calculator in the IT world so far, but it's just not enough for C. I could, of course, just google "Learn basic math" but that's a bit too random, I wouldn't know if it where any good until I start failing when I try to apply it, or if later chapters would make any sense because the writer starts slow and then takes off like a rocket, that's why I'm here now. Does anyone know of a good book or website he or she can recommend me, something that that can take me from zero to knowing enough to functionally learn C? [PS.] I know this is not a question about C directly but it does apply to programming C / C++, especially since programming is the reason I am asking this. These seem to get updated and corrected like a few times a year, which is cool. Lots of exercises. I usually just pound away at wikipedia when I don't understand something. But IMHO unless there's something specific you need, don't get bogged down and distracted with this. Maybe you are using the wrong programming resources if that is the case. Programming is much more about logic and you do not learn logic from math. Algorithms can be described abstractly using equations but that is just a convenience for people who already understand the notation. Wokzombie, there's no reason to be ashamed! In fact, heck, I wish I worked with more people as eager as you are to learn. It's never too late to learn buddy so put all the shame or whatever it is you feel behind you and start working/studying. In my experience, at least, people that are so passionate about moving forward are actually the ones that get far. Now, if I may recommend a site that helps it's Math.com - World of Math Online. They have exercises for every part of math you should focus on to improve your math skills : algebra, geometry and calculus. However, I disagree slightly with MK27. While Programming may certainly not be about math, all the theory of computation that lies behind programming languages is certainly a very solid and easily identifiable part of math. Programming languages are nothing but a method of expressing algorithms, which in essence, belong more to the field of math than of software development. There are different degrees to which you can develop your skills, but overall it's up to you to determine how much you want to learn. The good news is that there is really no real limit to that, and no school or diploma can really limit what you can know! So good luck buddy! I'm cheering for you! Programming is much more about logic and you do not learn logic from math.As for linear algebra, calculus, and geometry? Well, you're honestly not going to need that unless you're into 3D graphics... and even then, there are numerous libraries to get you by with next to no knowledge. Computers are discrete machines, so discrete math is by far the easiest to work with as well as the most important (in my opinion) branch of mathematics for programming. Wikipedia - Discrete MathematicsRight, my point is: math is an application of logic and not vice versa. The people who invented mathematical notation were applying logic. Math is an expression of logical principles. I guess to be fair some of it may rub off in the process, but you may just as well study science or philosophy or programming or anything which trusts reason as it's foundation. Originally Posted by claudiu However, I disagree slightly with MK27. While Programming may certainly not be about math, all the theory of computation that lies behind programming languages is certainly a very solid and easily identifiable part of math. A fascinating topic but like I said, not required knowledge for most programming. Programming languages are nothing but a method of expressing algorithms, which in essence, belong more to the field of math than of software development. There are plenty of algorithms that actually cannot be expressed mathematically, at least not in a reasonable way -- and plenty more that are best expressed in a human* (or programming) language. Again, contrary to popular opinion, "algorithm" IS NOT simply a mathematical concept, although the term may be most commonly used by mathematicians, hence the etymology. I'm just saying this because W Math does have a very very broad range of applicability for description, so mathematicians who are also curious people will use it to describe a very very broad range of things. Which if you ain't a curious person you probably would never have taken an interest in math . If you are interested in it, then go, but it should not be holding you back much from learning C. No doubt, it will help you in many circumstances! * Like, with words. Pretty sure there are some that CANNOT be expressed using math notation but I won't claim that as irrefutable truth. Thanks for the pick-me-up, claudiu, I needed that =) I'll take a better look at the link tomorrow since it's pretty late over here, but it looks like what I was looking for. Originally Posted by MK27 ...W You are correct, I do not know the terminology. Whatever the right term for it is, my understanding of it is about the same as an 8 year old child. Although I'm pretty sure I'm not talking about algorithms, I am also dutch so there's a language barrier as well. Originally Posted by MK27 If you are interested in it, then go, but it should not be holding you back much from learning C. No doubt, it will help you in many circumstances! In this case you are correct again. The information I have on C just got to floating-point numbers, where the book gave me a value, the same value with an exponent and how it could/would be written in C. For example the number 0.0000000000000005 is 0.5 to the power of -15 (correct?) which can be written in C as 0.5E-15. I know a very tiny bit about how calculations like these are made, not even the logic. And until I understand the why and how I will always write numbers like that as 0.0000000000000005 instead of 0.5E-15. Okay. That is indeed math-ish. But you can figure it out -- you just did. Most of the math-ish things involved in programming are like that. You do not have to quit programming to study math first. I think you are "psychologically intimidated". BUT: perhaps studying some math will help you overcome that. Maybe you will like it. If not, don't worry. Let me give you an analogy (do they have "analogy" in math?): the only thing you have to know for this one is, "I like music". Music, even more than programming, can be very easily described in mathematical terms. If you can play a stringed instrument, it is very easy to understand how this is so. However, it is also easy to see how the people who invented, developed, and mastered the stringed instrument did not need formal math to do it, because everything about music which can be described and analyzed with math is intuitively obvious.* "Intuitively obvious" may seem like an abstraction, blah blah blah -- let me get to my point. The only TWO things I have heard described as "universal language" are music, and math. In fact, there are probably more, but I will spare you the philosophy. To complete the analogy: make a list of your favourite musicians and consider, how much time did they spend studying math? Programming is A SKILL. Practice it. * intuitive here might be like "empirical", a very important reality in learning music (where is the sound?) and learning programming (what happened?).I am writing a microcontroller (embedded) program to reproduce some instrument sounds using trig functions, including some harmonics at different amplitudes. That would at least require basic function transformation stuff (change frequency and amplitude), and to generate the actual spectrum would require Fourier transform. Math may not be "required" for basic programming, but many fields of programming sure require more than basic math. Graphics, audio, optimizations, statistics, physical and biological simulations, AI, etc. Programming is much more about logic and you do not learn logic from math. That's not true. Mathematics is formally a language used to represent logic, and perform logical operations. It's just that a lot of the logic of computer programs can be expressed without mathematical formalism. However, mathematics can be used to characterise anything that software can do. Originally Posted by cpjustJust because you've never seen it does not mean it doesn't happen. Try writing programs that realistically simulate real physical phenomena, and you will use plenty of mathematics. For example the number 0.0000000000000005 is 0.5 to the power of -15 (correct?) which can be written in C as 0.5E-15. The latter part is right, but not the former. 0.5^(-15), or "0.5 to the power of -15" is not the number you've shown (0.000...0005). This long number you've shown is 0.5 x 10^(-15). This is what "scientific notation" is, its basically multiplying some number (in this case 0.5) by a power of 10 (in this case to the power -15). The "0.5E-15" is a way of representing "scientific notation". So "0.5E-15" is a way of writing "0.5 x 10^(-15)". As for your concerns: once you get into it, math is actually an incredible and very powerful tool. However, and this is in accordance with what others are saying, "programming" doesn't require much math. All of the math I have studied I don't, and I know I probably won't in the near future, apply when "programming". What I have got out of studying math for practical purposes is a way to think more critically and abstractly. However, mathematics can be used to characterise anything that software can do. Excellent point. The modern concept of computers was based on highly theoretical mathematical research. Thanks to Alan Turing, the "Godfather of (Theoretical) Computer Science", are we able to even be having this conversation (I mean via computers).Absolutely, I don't know exactly why but math has always scared me. On more than one occasion I have tried to learn basic math and failed because at the start it all seems logical and I can follow it, then suddenly the writer/teacher speeds up and it's like they are talking ancient hebrew and tell me how logical it all is. Originally Posted by waterborneI have thought about this before, but I think you make a good point. The issue is that I was planning on saving up for a course in C.
0821803 the lectures presented at the Sixth International Conference on Formal Power Series and Algebraic Combinatorics held at DIMACS in May 1994. The conference attracted approximately 180 graduate students and junior and senior researchers from all over the world. Generally speaking, algebraic combinatorics involves the use of techniques from algebra, algebraic topology, and algebraic geometry in solving combinatorial problems; or it involves using combinatorial methods to attack problems in these areas. Combinatorial problems amenable to algebraic methods can arise in these or other areas of mathematics, or in areas such as computer science, operations research, physics, chemistry, and, more recently, biology. Because of this interplay among many fields of mathematics and science, algebraic combinatorics is an area in which a wide variety of ideas and methods come together. The papers in this volume reflect the interesting aspects of this rich
Welcome to Practice Quizzes for MA 110, Survey of College Mathematics.These quizzes were written by Professor Joyce Riseberg, Montgomery College, Rockville, MD, to coordinate with the text Finite Mathematics for Business, Economics, Life Science and Social Sciences, 10th edition, by Barnett, Ziegler, and Byleen.This book is currently being used in all sections of MA 110 at the Rockville Campus of Montgomery College. The purpose of these practice quizzes is to give you, the student, an opportunity to solve problems on your own and to receive immediate feedback.Each quiz has a multiple-choice format.You should work out the problem on your own, using pencil and paper and a calculator if appropriate, and then choose one of the answers.After you select an answer, you will be told if your answer is correct.If it is not, you will receive some hints as to what you might have done wrong.If you choose "None of the answers given; click to see the solution," a solution of the problem will be shown.After you finish the quiz, you can click "Return to Table of Contents" in order to return to the Table of Contents so that you can try another version of the same quiz or in order to go on to a quiz on a different section. For Chapter 3, the Mathematics of Finance, I have written two types of quizzes for sections 3-2, 3-3, and 3-4.The quizzes which are called "Alternate Quizzes" assume that you know how to use the TVM Solver on a TI-83, TI-83+ or TI-84 calculator.If you do not know how to use the TVM Solver, you should not try these quizzes. Occasionally, I have used notation which is a little different from the notation used in the textbook.When this seemed appropriate, I have used the same type of notation as is used on a graphing calculator. For exponents, I have usually used standard notation, but on occasion I have used the symbol used on the calculator, so that, for example, xn might be written as x^n. For multiplication, I have used the symbol , so that, for example, 12 times 4 would be written as 124. I hope that you find these quizzes useful and that they help you to do well in your math course.If you have any comments or suggestions about the quizzes, or of you find any errors, please contact me by e-mail at joyce.riseberg@montgomerycollege.edu
0321652797 9780321652799 Using and Understanding Mathematics: A Quantitative Reasoning Approach, Fifth Edition increases readers' mathematical literacy so that ...See more details below Details about this item Using and Understanding Mathematics: Using and Understanding Mathematics: A Quantitative Reasoning Approach, Fifth Edition increases readers' mathematical literacy so that they better understand the mathematics used in their daily lives, and can use math effectively to make better decisions every day. Contents are organized with that in mind, with engaging coverage in sections like Taking Control of Your Finances, Dividing the Political Pie, and a full chapter about Mathematics and the Arts. Note: This is the standalone book, if you want the book with the Access Card please order the ISBN below: 0321727746 / 9780321727749 Using and Understanding Mathematics: A Quantitative Reasoning Approach with MathXL (12-month access) * Package consists of 0201716305 / 9780201716306 MathXL -- Valuepack Access Card (12-month access) 0321652797 / 9780321652799 Using and Understanding Mathematics: A Quantitative Reasoning Approach Back to top Rent Using and Understanding Mathematics 5th edition today, or search our site for Jeffrey O. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson.
Summary: Susanna Epp's DISCRETE MATHEMATICS WITH APPLICATIONS, FOURTH EDITION, provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learnin...show moreg about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses. ...show less Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book $72.06 +$3.99 s/h New PaperbackshopUS Secaucus, NJ New Book. Shipped from US within 4 to 14 business days. Established seller since 2000 $73.13 +$3.99 s/h New indoo Avenel, NJ BRAND NEW $78.95 +$3.99 s/h New Textbook Barn Woodland Hills, CA PAPERBACK New 0495826138 Premium Books are Brand New books direct from the publisher sometimes at a discount. These books are NOT available for expedited shipping and may take up to 14 business day...show mores to receive. ...show less $80.01 +$3.99 s/h New PaperbackshopUS Secaucus, NJ New Book. Shipped from US within 4 to 14 business days. Established seller since 2000 $85.49 +$3.99 s/h New Bookeyez2 snellville, GA Brand New Title. We're a Power Distributor; Your satisfaction is our guarantee! $96.28 +$3.99 s/h New Russell Books Victoria, BC PAPERBACK New 0495826138
History of Mathematics, Volume 2 Volume II of an unusually clear and readable two-volume history — from Egyptian papyri and medieval maps to modern graphs and diagrams. Evolution of arithmetic, geometry, trigonometry, calculating devices, algebra, calculus, more. Includes problems, recreations, and applications. References from web pages History of Mathematics: Textbooks Elements of the history of mathematics. Springer-Verlag, New York, 1994. Translated from the French edition: Eléments d'histoire des mathématiques. ... aleph0.clarku.edu/ ~djoyce/ mathhist/ textbooks.html JSTOR: An Eleventh Lesson in the History of Mathematics MATHEMATICS MAGAZINE An Eleventh Lesson in the History of Mathematics by ga MILLER 20. Questionable historical statements, In 1893 a meeting was held in ... links.jstor.org/ sici?sici=0025-570X(194709%2F10)21%3A1%3C48%3AAELITH%3E2.0.CO%3B2-8 CALCULUS Here is a list of some of the best books and most reliable web sites containing information on the history of mathematics and, in particular, the history of ... media/ 1_inside_history.php BIBLIOGRAPHY Eves, Howard, An Introduction to the History of Mathematics, 6th edition, ... Dauben, Joseph, The History of Mathematics from Antiquity to the Present: A ... www66.homepage.villanova.edu/ thomas.bartlow/ mat2930/ biblio.html Bibliography An Introduction to the History of Mathematics. 1964. Rpt. ... Katz, Victor J. A History of Mathematics: An Introduction. New York:. harpercollins, 1993. ... press.princeton.edu/ books/ maor/ biblio.pdf
Matrix Theory and Applications with MATLAB This book is designed for an advanced course on linear algebra and covers the basics of the subject - from a review of matrix algebra through vector spaces to matrix calculus and unitary similarity. It integrates MATLAB throughout the text. Each chapter includes a MATLAB subsection that discusses the various commands used to do the computations and offers codes for the graphics and algorithms used in the text. All material is presented from a matrix point of view. The treatment includes optional subsections covering applications. The final chapters move beyond basic matrix theory to discuss more advanced topics, such as decompositions, positive definite matrices, graphs, and topology. Free Mathematical Modeling Technical Kit Learn how you can quickly build accurate mathematical models based on data or scientific principles.
Alabama General Studies Curriculum Prerequisites and Restrictions Credit 3 Semester Hours. Course Materials The current text is Precalculus, 3rd edition, by Lial, Hornesby, and Scneider, published by Addison-Wesley (2005). Grading System This course is graded A, B, C, D, F. The grade typically depends on a combination of class tests, homework, quizzes, and a comprehensive final exam. A grade of C or better is required for entry into MA 171, Calculus A.
Instructor Class Description Topics in Mathematics Across the Curriculum Examines mathematical theories and concepts within their historical and cultural contexts. Topics vary with instructor and may include mathematical symmetries, the organization and modeling of space, cryptology, mathematical models of social decision making, and/or theories of change and strategy. Class description SPRING 2007: The topic this quarter will be symmetry. We will look at the mathematical modeling of symmetry and how it reflects what is happening in nature. We will compare these mathemtical models to the models of symmetry generated by artists from across many cultures. Student learning goals - Understand the basic tools of mathematical modeling of symmetry. - Learn the vocabulary of mathematical symmetry. - Learn and appreciate commonalities and differnces in how various cultures use symmetry. - Generate symmetrical patterns using both traditional compass and ruler methods as well as computer software. General method of instruction This class will be a combination of lecture, small group work, and computer simulation. Recommended preparation Openess to problem solving and seeing mathematics in new ways. Class assignments and grading Assignments will reinforce the mathematical and cultural theories of symmetry developed throughout the quarter. It will include math problems, computer simulations, cultural investigations, and paying attention to the role of symmetry in your own life. Grades will be based on homework problems, computer labs, quizes, and final Cinnamon Hillyard Date: 02/07/2007 Office of the Registrar For problems and questions about this web page contact icd@u.washington.edu, otherwise contact the instructor or department directly. Modified:November 27, 2013
ALGEBRATOR ALGEBRATOR Description Detailed Features Algebrator step-by-step math problem solverAlgebrator software is your 24/7 math tutor. You can literally type in your homework assignment and see it solved step-by-step (just like your teacher would solve it on the board, only more patient!). When a particular step is not clear, Algebrator will explain it in an easy to understand way. It will not only tell you what rule is applied, but also how and why it is applied in your particular problem. Not only will your homework assignment be done in minutes, but you will learn the important math concepts while observing Algebrator at workWhat does Algebrator cover?ALgebrator covers every important math concept starting with pre-algebra, all the way to college algebra. Here are some feature highlights:simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic and long division ...), exponential expressions , fractions and roots (radicals) , absolute values)factoring and expanding expressionsfinding LCM, GCFoperations with complex numbers (simplifying, rationalizing complex denominators...)solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)solving a system of two and three linear equations (including Cramer's rule)graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)graphing general functionsoperations with functions (composition, inverse, range, domain...)simplifying logarithmsbasic geometry and trigonometry (similarity, calculating trig functions, right triangle...)arithmetic and other pre-algebra topics (ratios, proportions, measurements...)linear algebra (addition, subtraction and multiplication of matrices, matrix inverse, determinants)
This webinar offers educators a quick and easy way to learn some of the fundamental concepts of Maple. Learn a few simple techniques that will allow you to use Clickable Math™ features to compose, visualize, and solve a wide variety of mathematical problems without commands. This webinar will also provide an introduction to some of the technical documentation features in Maple, including the use of interactive components such as buttons and sliders. This webinar offers a quick and easy way to learn some of the fundamental concepts for using Maple. Learn the basic steps on how to compose, plot and solve various types of mathematical problems. This webinar will also demonstrate how to create professional looking documents using Maple, as well as the basic steps for using Maple packages.
The applet shows graphically and numerically consecutive terms of a sequence or consecutive partial sums of a series. The user enters a formula for a sequence or a series and the terms are plotted. Ma... More: lessons, discussions, ratings, reviews,... This activity is an introduction to the concept of convergent infinite series using an iterative geometric construction. This activity has been adapted from the following article: Choppin, J. M. (1994
״I intend this book to be, firstly, a introduction to calculus based on the hyperreal number system. In other words, I will... see more ״I aimed the text primarily at readers who already have some familiarity with calculus. Although the book does not explicitly assume any prerequisites beyond basic algebra and trigonometry, in practice the pace is too fast for most of those without some acquaintance with the basic notions of calculusIt is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The... see more It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The importance of algebra, in particular, cannot be overstated as it is the basis of all mathematical modeling used in applications found in all disciplines.Traditionally, the study of algebra is separated into a two parts, Elementary and Intermediate Algebra. This textbook by John Redden, Elementary Algebra, is the first part written in a clear and concise manner, making no assumption of prior algebra experience. It carefully guides students from the basics to the more advanced techniques required to be successful in the next course.John Redden's Elementary Algebra takes the best of the traditional, practice-driven algebra texts and combines it with modern amenities to influence learning, like online/inline video solutions, as well as, other media driven features that only a free online text can deliver. Using the online text in conjunction with a printed version of the text could promote greater understanding (at a lower cost than most algebra texts).From the traditional standpoint, John employs an early and often approach to real world applications, laying the foundation for students to translate problems described in words into mathematical equations. It also clearly lays out the steps required to build the skills needed to solve these equations and interpret the results. With robust and diverse exercise sets, students have the opportunity to solve plenty of practice problems. Elementary Algebra has applications incorporated into each and every exercise set. To do this John makes use of the classic "translating English sentences into mathematical statements" subsections in chapter 1 and as the text introduces new key terms.A more modernized element; embedded video examples, are present, but the importance of practice with pencil and paper is consistently stressed. This text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today.In addition, textual notation is introduced as a means to communicate solutions electronically throughout the text. While it is important to obtain the skills to solve problems correctly, it is just as important to communicate those solutions with others effectively in the modern era of instant communications.While algebra is one of the most diversely applied subjects, students often find it to be one of the more difficult hurdles in their education. With this in mind, John wrote Elementary Algebra from the ground up in an open and modular format, allowing the instructor to modify it and leverage their individual expertise as a means to maximize the student experience and success. According to OER Commons, 'These are the lecture notes of a one-semester undergraduate course which we taught at SUNY... see more According to OER Commons, 'These.'
More About This Textbook Overview This book introduces the student to numerous modern applications of mathematics in technology. The authors write with clarity and present the mathematics in a clear and straightforward way making it an interesting and easy book to read. Numerous exercises at the end of every section provide practice and reinforce the material in the chapter. An engaging quality of this book is that the authors also present the mathematical material in a historical context and not just the practical one. Mathematics and Technology is intended for undergraduate students in mathematics, instructors and high school teachers. Additionally, its lack of calculus centricity as well as a clear indication of the more difficult topics and relatively advanced references make it suitable for any curious individual with a decent command of high school math. Editorial Reviews From the Publisher From the reviews: "Christiane Rousseau and Yvan Saint-Aubin here present a valuable collection of diverse and detailed applied mathematics examples. … presented to work as a standalone guide to mathematics at work today, usable for self-study and enlightenment or as a text for coursework. … Chapters conclude with a rich collection of exercises followed by references for further study. … together with the clear signposts to help students get around and through the more difficult topics, make Mathematics and Technology suitable for any diligent reader … ." (Tom Schulte, MAA Online, February, 2009) "This book takes a more detailed view of mathematics in action, in several areas of technology … . This is an excellent book for a varied audience. … This book will also be attractive to college students … and to researchers in mathematics, computer science (CS), and technology, who want to acquire a more thorough understanding of the applications covered in the book. … the authors give several pointers and suggestions to instructors. … I like this book and I recommend it." (Edgar R. Chavez, ACM Computing Reviews, June, 2009) "The authors highlight how mathematical modeling, together with the power of mathematical tools, has been crucial for innovation in technology. … The text is written for students who have a familiarity with Euclidean geometry and have mastered multivariable calculus, linear algebra, and elementary probability theory. … undergraduates in their junior or senior years are the ideal audience for the course." (Tzvetan Semerdjiev, Zentralblatt MATH, Vol. 1211
Mathematics - Edexcel Who is this course for? Mathematics at AS and A Level is a course worth studying in its own right. It is challenging but interesting. It builds on work met at GCSE, but involves new ideas that some of the greatest minds of the millennium have produced. During AS, we offer options in Statistics and Mechanics, for those of you who have particular degree courses in mind. AS in Mathematics is very valuable as a supporting subject to many courses at A Level and degree level, especially in the sciences, Geography, Psychology, Sociology and medicine. A Level mathematics is a much sought after qualification for entry to a wide variety of full-time courses in Higher Education. There are also many areas of employment that see Mathematics A Level as an important or vital qualification. Formal Entry Requirements All students studying for A Level would be expected to have five GCSEs A*-C in academic subjects (of which two must be B grades) including GCSE English Language. We will count Level 2 Btec Diplomas towards this total, but only merits and distinctions will be counted and each diploma will count as one GCSE. Additionally, you will need GCSE Maths at B or above in the higher paper. What does the course involve? Mathematics at AS and A level is divided into three branches: Pure Mathematics (Modules C1, C2, C3, C4) Pure Mathematics at AS and A level extends your knowledge of topics such as algebra and trigonometry as well as introducing new ideas such as calculus. If you enjoyed the challenge of problem solving at GCSE then you should find this course very appealing. Mechanics (Modules M1, D1) Mechanics describes the motion of objects and how they respond to forces acting upon them, from cars in the street to satellites orbiting a planet. You will learn techniques of mathematical modelling by turning a complicated problem into a simpler one that can be analysed and solved using mathematics. Many of the ideas you will meet form an essential introduction to modern fields of study such as cybernetics, robotics, biomechanics and sports science, as well as the more traditional areas of engineering and physics. Statistics(Modules S1, D1) Statistics covers the analysis of numerical data in order to arrive at conclusions about it. Many of the ideas met have applications in a wide range of other fields: from assessing what car insurance costs to how likely the Earth is going to be hit by a comet. In order to get an AS Level, you will need to take three modules. For a full A Level, you will need to take three further modules. We offer two options: Pure and Mechanics: This option is usually taken by students studying Science (especially Physics), Engineering or Construction. Pure and Statistics: This option is usually taken by students studying Business (Economics, Accounting or Business Studies), Psychology or Biology. How will I be assessed? Both AS and A level Mathematics are assessed through a series of written examinations. If necessary, students can retake any module. There is no course work involved in assessing mathematics. Where can I go next? Most A level students go on to study at university. Some have used Mathematics to go directly into a career in accountancy. The study of mathematics opens the door to many varied professions. Obvious choices would be in the area of Science and Finance. To see the many varied careers that a student of mathematics and statistics may follow, from games programmer to weather forecasting, go to
Journal Article Detail Page This study investigates how students understand and apply the area under the curve concept and the integral-area relation in solving introductory physics problems. We interviewed 20 students in the first semester and 15 students from the same cohort in the second semester of a calculus-based physics course sequence on several problems involving the area under the curve concept. We found that only a few students could recognize that the concept of area under the curve was applicable in physics problems. Even when students could invoke the area under the curve concept, they did not necessarily understand the relationship between the process of accumulation and the area under a curve, so they failed to apply it to novel situations. We also found that when presented with several graphs, students had difficulty in selecting the graph such that the area under the graph corresponded to a given integral, although all of them could state that "the integral equaled the area under the curve." The findings in this study are consistent with those in previous mathematics education research and research in physics education on students' use of the area under the curve. Disclaimer: ComPADRE offers citation styles as a guide only. We cannot offer interpretations about citations as this is an automated procedure. Please refer to the style manuals in the Citation Source Information area for clarifications.
Precalculus - 10 edition Summary: Engineers looking for an accessible approach to calculus will appreciate Young's introduction. The book offers a clear writing style that helps reduce any math anxiety they may have while developing their problem-solving skills. It incorporates Parallel Words and Math boxes that provide detailed annotations which follow a multi-modal approach. Your Turn exercises reinforce concepts by allowing them to see the connection between the exercises and examples. A five-step problem solving ...show moremethod is also used to help engineers gain a stronger understanding of word Edition / ISBN-10: 0471756849 / Mint condition / Never been read / ships out in one business day with free tracking. $202.92 +$3.99 s/h New PaperbackshopUS Secaucus, NJ New Book. Shipped from US within 4 to 14 business days. Established seller since 200017568273.96 +$3.99 s/h New indoo Avenel, NJ BRAND NEW $330
+ By Trevor Doyle REAL TEACHER TAUGHT LESSONS Algebra Readiness: This course teaches whole numbers, operations on whole numbers, rational numbers, operations on rational numbers, symbolic notation, equations and functions, and the coordinate plane. This course offers 11 full chapters in three parts with 6-8 lessons each chapter that present short easy to follow algebra readiness videos. These 5 to 10 minutes videos take students through the lesson slowly and concisely. Algebra Readiness is taken by students after Pre-Algebra if they still need to review skills before algebra or instead of Pre-Algebra for students who only need a review class before Algebra or if students want to firm skills prior to Algebra.
ical Excursions Teaches students that mathematics is a system of knowing and understanding our surroundings. This title helps students learn those facets of ...Show synopsisTeaches students that mathematics is a system of knowing and understanding our surroundings. This title helps students learn those facets of mathematics that strengthen their quantitative understanding and expand the way they know, perceive, and comprehend their world.Hide synopsis Description:Fine in very good dust jacket. Sewn binding. Cloth over boards....Fine in very good dust jacket. Sewn binding. Cloth over boards. 1008 p. Audience: General/trade. INSTRUCTOR'S EDITION SAME AS STUDENT EDITIONNew. No dust jacket. Sewn binding. Cloth over boards. 877 p....New. No dust jacket. Sewn binding. Cloth over boards. 877 p. Audience: General/trade. Perfect Condition. Never Used Annotated Instructor's Edition. 2013 Same text as the standard student edition. Includes Answers to ALL Exercises. Does NOT includes CD"s or Access code
AQA GCSE Maths (Nelson Thornes) is used as the text book. Problem solving and reasoning skills are encouraged. Discussion about Mathematics is encouraged by students working in pairs or groups. Students are also encouraged to consolidate their learning with the use of mymaths.co.uk. Topic lessons are availbale and targeted revision can be done with the use of the "Booster Packs" at the appropriate level, all students have their own password and login details. Before each exam, students will sit a mock exam and have plenty of opportunities to practice papers and consolidate learning.
This booklet contains a number of case studies of lessons that emphasise working with what students already know and working from concrete ideas to the abstract. Each case study explains the task, contains examples of students' work and explains how the lesson was followed up. The function game is one way in which algebra game from the Contemporary School Mathematics collection published by Edward Arnold, was written in the belief that there is still value in the examination of Euclidean geometry as a logical system, whatever other approach may be used to increase the students' spatial perception. The topics covered are: What is…
Library of Congress Cataloging-in-Publication Data on File. Vice President and Editorial Director, ECS: Marcia J. Horton Vice President and Director of Production and ... Algebra 2 and Trigonometry is a new text for a course in intermediate algebra and trigonometry that continues the approach that has made Amsco a leader in presenting ... 2/Algebra 2 and Trigonometry.pdf 2 3 A New Perspective on Math Prentice Hall Algebra 1, Geometry, Algebra 2 2011 is changing the way South Carolina students see mathematics! By integrating digital ... Copyright 2010 PearsonEducation, Inc. or its affiliates. All rights reserved. 1 Program Components Introduction This guide describes the program components ... Preface Welcome to the exciting world of college algebra. In this course, you begin to learn the many ways that mathematics can be used to solve real-world problems ...
This site was built for the purpose of advancing math education and curiosity around the world. We hope you enjoy using the function grapher. We are constantly adding features and improving the user experience.
Course Description: It is an informal general education course for non-science majors designed to introduce students to the role and usefulness of mathematics in contemporary society. With only simple mathematical tools, students get a glimpse of the power of mathematical thinking. We will consider some of the greatest ideas of humankind. The great ideas we will explore are within the realm of mathematics - an artistic endeavor which requires both imagination and creativity. In this course, we will experience what mathematics is all about by investigating some beautiful and intriguing issues. Thus, the informal prerequisites for this course are an open and curious mind and the willingness to put aside any preconceived prejudices or dislikes for mathematics. Course Objectives: The students will be able to understand some rich mathematical ideas; demonstrate some new skills for analyzing life issues that transcend mathematics; appreciate mathematical perspective as a way to view the world. In particular, they will see many uses of mathematics in the world today such as using of mathematics in Risky Business, Prediction, Chaos theory, Logic and so on. We will work not only with the textbook, but also with additional sources. There will be many handouts, papers, projects with presentations. Material Covered: Chapters:1, 2, 4, 5, 6 (see tentative schedule). Attendance: Students are required to attend all class sessions. You will be counted absent if you are more than 5 minutes late. Bring special attendance problems to my attention immediately. Homework:Each set of homework assignments consists of reading from the text and doing several Mindscapes. (See HW assignments) Homework will be assigned each class day and collected every class day. Students are expected to work on homework and reading assignments as assigned. Daily consistent effort is essential. Clarity of exposition is important, and well written, polished solutions are the goal. Homework turned in MUST indicate (in the upper right hand corner of the first page): your name, section of Math class you are enrolled in, homework number, problem numbers, chapter, page in the textbook. Late homework without excusable reason will not be accepted. I will start each class by going over the homework, answering your questions regarding the problems assigned to you at the previous class meeting. You are expected to participate in the demonstration of your solutions. The problems on tests and on the final exam will be very similar to the ones from the homework assignment. So to be successful you have to complete all of the assigned problems. Tests and Exams: There will be 3 tests, 4 quizzes and one final exam (see tentative schedule). There will be no makeup tests and quizzes unless a valid, documented reason is presented. In exceptional circumstances, you may be allowed to take a test or exam early. Any such arrangement must be made in advance, and you must have a serious reason for doing so. Tutorial Center Hours: If your tests/quizzes' scores are less than 100%, you must do the error corrections for the tests/quizzes in the Tutorial Center. Your corrections should be sign by the tutor and the hours that you spend in the tutorial center should be recorded. The Tutorial center attendance is the part of your overall grade. (See below.) If you will accumulate more than 10 hours in the tutorial center per semester, you can accumulate up to 2 extra percent toward your Tutorial center grade. Portfolio: You will be expected to keep your work and handouts from the class organized in a binder. The binder should be separated into four sections: 1) Course information, 2) Homework, 3) Class notes and handouts, 4) Tests. 5) Papers. 6) Project. 7) Error corrections. I will check your portfolio before the final exam. Final Exam: a comprehensive cumulative final exam will be held on: Wednesday, May 6 from 11 to 1 :30. Cell phone policy: All cell phones are strictly prohibited on the tests. All other time they must be put on vibration. How Papers Will Be Graded Each writing assignment I give will be in the form of a reaction or reflection paper: at least 2 pages in length, 10 points possible credit. Basically, I want to know that you gave the topic some thought and made the paper easy to read. My grading method is: Format: 3 points Your paper should be typed on a computer . Your name, the due date for assignment, and the course number on each page. The title of the assignment should appear at the top of the first page. Content: 4 points Here I am looking engagement and/or originality. I want to know that you understood the topic and thought about it. Reciting or reporting facts may be useful, but that is not the point of these assignments. I want you to comment on the facts or ideas that you are writing about. Flow: 3 points I expect that you will connect your ideas into a readable whole. Often you will be able to do this without rewriting. Please reread what you wrote and edit it, if necessary. Research Project: The only way to really understand mathematics is to learn and discover it on one's own. Thus, students will select a mathematical topic not covered in our class, read and teach themselves any necessary background to understand it, and then investigate the topic. A student will write a paper on his/her findings and present a poster/power-point talk at the end of the semester.
Algebra and Trigonometry With Analytic Geometry - 12th edition ISBN13:978-0495108269 ISBN10: 049510826X This edition has also been released as: ISBN13: 978-0495383420 ISBN10: 0495383422 Summary: Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this text popular among students year after year. This latest edition of Swokowski and Cole's ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY retains these features. The problems have been consistently praised for being at just the right level for precalculus students like you. The book also provides calculator examples, includin...show moreg specific keystrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematics27.7781.05 +$3.99 s/h Acceptable SellBackYourBook Aurora, IL 049510826
View Classes: Descriptions Math Our philosophy for math can be summed up in two words: real life. We believe that math is best learned when it is made meaningful and relevant to a student's life. Therefore, our classes are designed to incorporate real world applications and hands-on activities. We offer a comprehensive math program for students ages 3-18 -from beginning math concepts through calculus and beyond. Please note: Many of the math classes have prerequisites. We set these to ensure, as much as possible, that students will have the necessary skills to be successful in each class. Please make sure your child meets all requirements listed for the class. For more information, contact the specific teacher. In addition, while it is anticipated that each class will cover the topics listed in the class description, this may not always be possible. Here's our philosophy: We believe it is important for students to work at their own level. Depending on the make-up of the class, each year's class may move at a different pace. We consider depth of knowledge to be more important than breadth. Given a choice between "finishing the book" and delving as deeply into each topic as the students need in order to grasp it fully, we will choose the latter every time. Therefore,if a class is unable to "finish the book" by the end of the year, families will be responsible to make sure their child accomplishes any additional material they need to move on to the next level. Teachers will let families know if there is anything that they need to work on over the summer. Most classes each year are able to cover the necessary topics for each level. Students work in small groups to collect data, develop and analyze mathematical models, explore patterns and relevant questions, make and defend predictions, discuss and present their findings, and write about mathematics. The text also presents algebraic problems in such a manner as to encourage students to approach problems from multiple perspectives. Tests are used as a tool for students to assess their own mastery. Purchase of a Texas Instrument 83+ graphing calculator is required. Outside-of-class work is mandatory for this class. This course is presented as a study of four fundamental concepts: limits, derivatives, and two kinds of integrals. Review of pre-calculus topics is done as it is needed, rather than all at once at the beginning of the course. From day one students work on learning concepts using a four-pronged approach: numerical, graphical, algebraic, and verbal-whichever is appropriate. With these approaches, realistic applications are possible, such as fitting a logistic equation to actual census data. As a result, students with a wider range of abilities can be successful in calculus, not just those students strong in algebra. Developed specifically with the high school student in mind, but with all the content of a college-level course, this class takes full advantage of graphing calculator technology. Tests are used as a tool to assess their own mastery. Purchase of a Texas Instrument 83+ graphing calculator is required. Outside-of-class work is mandatory for this class. A class based on the book Housebuilding for Children by Lester Walker. The students will design and build sheds and/or playhouses using real world house construction techniques. We will use mostly hand tools, except for electric drills, to cut down on the possibility of injuries. Each student is required to have: Hammer Crosscut Saw Keyhole Saw Square Measuring tape Adjustable Wrench Screwdriver Pliers Sandpaper Drill and bits Paintbrush(es) Carpenters Apron The first activities would be to build a toolbox and work tables. The students will work in groups to design their shed/playhouse and to build it. The students will learn how to use tools, design and draft structures, understand the building process and the proper way things should be built. We will discuss foundations in class (since we can't build them) and the sustainability movement and its effect on construction techniques, materials and design (we will include passive solar techniques in our design). The students will also gain an understanding of the importance of keeping moisture out of a building. Depending on the speed and abilities of the children the class may explore plumbing by installing a sink that could attach to a hose, build a solar collector, or hook up a solar panel connected to lighting, etc. The benefits of the class include: team work, physical activity, a working knowledge of construction that could lead to a job or at least the ability to make home repairs, plus, three of the top 10 up and coming professions are in sustainability. Buildings are currently the number one supplier of greenhouse gases and while the architecture community is dedicated to bringing that number down to zero, it won't happen until consumers demand something different and better. Houses are rarely designed by architects. In fact you don't need any qualifications to design a residence. Contractors often learn their trade through experience and internship which limits their knowledge of new construction techniques. Change won't occur until the consumer demands something different. Geofinity Rm #25 Teacher: To be determined Outside Work: Minimal Ages: 7 - 9 1 Hour Class; 1 day/week Yearly Materials Fee : $40.00 WELCOME TO A GEOMETRIC ADVENTURE! You have been selected to take a trip to a new planet called Geo. Learn Geolanguage, build a spaceship, meet aliens, and much more, during this OUT-OF-THIS WORLD hands-on geometry space simulation. Also, in a mysterious galaxy far, far, away, there are planets shaped like polygons and aliens with bodies made of angles. This galaxy is called INFINITY LANDING. Are you ready for a visit? This is a comprehensive, high school level, hands-on, cooperative approach to learning geometry. Students will explore geometric relationships using a wide variety of tools-from compasses to computers, from patty papers to graphing calculators. This class guides students to discover and master concepts and relationships before they are introduced to formal proofs, drawing students in with real life examples and applications from many cultures and disciplines. Tests are used as a tool to assess their own mastery. Texas Instrument 83+ graphing calculator required, but no previous experience with the calculator is necessary. Outside-of-class work is mandatory for this class. This class uses the Discovering Algebra book by Key Curriculum Press which integrates the traditional algebra curriculum with statistics, data analysis, functions, discrete mathematics, geometry, and probability. Students work with data from real-world situations and applications in a curriculum that places algebra in an applications-based context and where investigation precedes introductions of formulas and expressions. By focusing on exploration, rather than simply naming concepts and completing computation exercises, students gain a deeper understanding and build skills that last. Learning new material through their own investigations enables students to make meaningful personal connections to the numbers, patterns, arguments, and mathematics they discover. This class makes the transformation from rule-dominated, symbol-oriented manipulation-based algebra to an activity and experiment-based algebra. Word problems relate algebra to everyday situations, which help students understand abstract concepts. Tests are used as a tool for students to assess their own mastery. Texas Instrument 83+ graphing calculator required, but no previous experience with the calculator is necessary. Outside-of-class work is mandatory for this class. This class gives students the opportunity to work on the specific math skills needed by each student as an individual. Students take a preliminary test to determine where they are and continue from there. This class can accommodate students from Pre-Algebra through Algebra II. Each student works at their own speed and level. Regular daily homework required. This is a class designed to give students the opportunity to work on the specific math skills needed by each as an individual.Students take a preliminary test to determine where they are and continue from there. This class can accommodate students from General Math Skills through Algebra I. Each student will work at his/her own speed and level. Regular daily homework required. This is a comprehensive math class that assumes students have a basic understanding of the concepts and computation skills necessary to work competently with the four basic operations. Students will sharpen these basic skills through practice. They will also study fractions, decimals, percents, measurement, simple geometry, and have experiences with probability. This class will be a balance of skills-driven strategies & real-world practice to effectively help students understand math & internalize math concepts. The comprehension of mathematical concepts, operations, & relations, emphasizing reasoning & critical thinking skills, will help develop math proficiency in all areas. The curriculum is designed to include continual practice & review of skills, depth of understanding, & differentiated instruction. (Outside work: moderate) Book purchase required. This is a comprehensive math class that covers the topics necessary to prepare students for Pre-Algebra. Students intuitively explore concepts for a basic understanding so that they know when to apply what they have learned. Several hours of homework will need to be completed each week for success in this class. This is a comprehensive math class that will cover the topics necessary to prepare students for Pre-Algebra. Students will intuitively explore concepts for a basic understanding so that they will know when to apply what they have learned. Several hours of homework will need to be completed each week for success in this class. This class is designed to help young students explore economics and personal finance in ways that are exciting and meaningful to them. We'll explore these concepts using children's literature and lots and lots of hands-on projects and activities. We'll find out how money works, design our own currency, learn how to be a smart consumer and much, much more. Students will learn what is involved in making a product, and the class will run its own small business.This is a hands-on class covering operations on integers, variable terms and expressions, solving equations, graphing, basic geometry, formulas, measurement, statistics and probability. For the most part, in-class time is spent on interactive activities where students experience mathematics in a concrete way. The at-home work will be a paper and pencil reinforcement of the in-class topic. Tests are used as a tool for students to assess their own mastery. Outside-of-class work is mandatory for this class. Prerequisites: Students must be competent in the four basic operations (+, -, x, /) on whole numbers, decimals, fractions (including multi digit) and percents and their applications; Measurement: choosing appropriate units, converting from one unit to another, operations (+-x/) on measurements - in both standard and metric system. In addition, students must have been exposed to data gathering; data display; basic geometry; and probability. Analytic geometry and advanced topics in algebra are presented in this course. Students study algebraic, logarithmic, exponential, and matrix functions. A thorough treatment of trigonometric concepts and applications through the study of identity proofs, solutions of right and oblique triangles, solutions of trigonometric equations, logarithms, and vector applications and complex numbers. Tests are used as a tool to assess their own mastery. Purchase of a Texas Instrument 83+ graphing calculator is required. Outside-of-class work is mandatory for this class.
Math 521 ``Graduate Algebra I'' Fall 2011 Introduction: Abstract Algebra is the branch of pure mathematics that tries to reduce mathematical questions to symbolic manipulation. Here is an example. Consider a cube, an object most people feel they understand. 6 congruent square faces, 12 congruent edges, 8 vertices. How much symmetry does it have? This question seems to make sense already, but it is hard to quantify until we define ``symmetry''. In algebra, a symmetry of a geometric object is a bijection of the object to itself that preserves the object's defining properties. That is an algebraic formulation of a geometric concept. ``Preserving the object's defining properties'' in this case means preserving the cube's shape (metric) and its orientation in space. So, how many geometric symmetries does the cube have? Our position in this class is that we don't really understand the cube if we cannot answer this question. Our collective mind will fester around this problem, until we have built up the machinery necessary to answer it. We will see that not only can we count the symmetries, we can describe the structure of this set, using the algebraic notion of group. We will name the symmetry group of the cube. For the last few hundred years, algebra has developed mostly around attempts to solve specific math problems, mostly in number theory and geometry. For example, algebra was used to solve the ancient problem of determining which polygons can be constructed using only a straight edge and compass. Gauss solved this problem in 1796, when he was five years old. We'll try to solve some of these types of problems. ``Abstract'' means, ``disassociated from any specific instance''. In the 20th century, certain basic algebraic objects were isolated and abstracted to form the core of abstract algebra: groups, rings, fields, and modules. These abstract objects appear in practically every area of pure mathematics. We'll start with groups, then start on rings. We'll do modules and fields in Math 522.
NS205 Intro to Quantitative Methods Course Description An examination of advanced algebra techniques in the business setting, including linear systems, polynomials, exponential and logarithmic functions, as well as introduction to probability and statistics. The primary quantitative course required for MNS 407. (Students who have taken college algebra (MTH 215) within the last three years are exempt from this course.) Learning Outcomes Construct and solve (algebraically, graphically, and statistically) models for a variety of business problems. Apply mathematical ideas and express them graphically and numerically. Solve equations, inequalities, and systems of equations. Explain and work with polynomials, polynomial functions, rational expressions, quadratic equations and functions.
For courses in Elementary Number Theory for math majors, for mathematics education students, and for Computer Science students. This introductory undergraduate text is designed to entice a wide variety of majors into learning some mathematics, while teaching them to think mathematically at the same time. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used forMore... proving theorems rather than on specific
Summary The Student Support Edition of Basic College Mathematics, 8/e, brings comprehensive study skills support to students and the latest technology tools to instructors. In addition, the program now includes concept and vocabulary review material, assignment tracking and time management resources, and practice exercises and online homework to enhance student learning and instruction. With its interactive, objective-based approach, Basic College Mathematics provides comprehensive, mathematically sound coverage of topics essential to the basic college math course. The Eighth Edition features chapter-opening Prep Tests, real-world applications, and a fresh design--all of which engage students and help them succeed in the course. The Aufmann Interactive Method (AIM) is incorporated throughout the text, ensuring that students interact with and master concepts as they are presented. Table of Contents Note: Each chapter begins with a Prep Test and concludes with a Chapter Summary, a Chapter Review, and a Chapter Test Chapters 2-12 include Cumulative Review Exercises AIM for Success Whole Numbers Introduction to Whole Numbers Addition of Whole Numbers Subtraction of Whole Numbers Multiplication of Whole Numbers Division of Whole Numbers Exponential Notation and the Order of Operations Agreement Prime Numbers and Factoring Focus on Problem Solving: Questions to Ask Projects and Group Activities: Order of Operations Patterns in Mathematics Search the World Wide Web Fractions The Least Common Multiple and Greatest Common Factor Introduction to Fractions Writing Equivalent Fractions Addition of Fractions and Mixed Numbers Subtraction of Fractions and Mixed Numbers Multiplication of Fractions and Mixed Numbers Division of Fractions and Mixed Numbers Order, Exponents, and the Order of Operations Agreement Focus on Problem Solving: Common Knowledge Projects and Group Activities: Music Construction Fractions of Diagrams Decimals Introduction to Decimals Addition of Decimals Subtraction of Decimals Multiplication of Decimals Division of Decimals Comparing and Converting Fractions and Decimals Focus on Problem Solving: Relevant Information Projects and Group Activities: Fractions as Terminating or Repeating Decimals Ratio and Proportion Ratio Rates Proportions Focus on Problem Solving: Looking for a Pattern Projects and Group Activities: The Golden Ratio; Drawing the Floor Plans for a Building The U.S. House of Representatives Percents Introduction to Percents Percent Equations: Part I Percent Equations: Part II Percent Equations: Part III Percent Problems: Proportion Method Focus on Problem Solving: Using a Calculator as a Problem-Solving Tool Using Estimation as a Problem-Solving Tool Projects and Group Activities: Health
In the Real World Differential equations, believe it or not, show up everywhere. There are math departments studying how differential equations relate to biology papers using differential equations for the study of diseases, books relating differential equations and economics and whole journals devoted to differential equations and their applications. Calculus classes usually introduce differential equations, but don't do a whole lot with them. To get more, there are usually differential-equations-only classes offered after calculus.
0130962023 Books are BRAND NEW WITH BRAND NEW PAGES!! Textbooks78 will ship books out within 24 hours and we will provide a free tracking # with each order! We are consumer driven ...andRead moreShow Less More About This Textbook Overview This book helps readers develop the quantitative literacy skills and savvy needed to function effectively in society and the workplace. It focuses on "mathematical modeling" and the use of elementary mathematics--e.g., numbers and measurement, algebra, geometry, and data exploration--to investigate real-world problems and questions. It assumes no technology other than the use of graphing calculators, and provides a comprehensive technology support system on an accompanying CD-ROM and web site. Linear Functions and Models. Quadratic Functions and Models. Natural Growth Models. Exponential and Trigonometric Models. Polynomial Models and Linear Systems. Optimization Problems. Bounded Growth Models. For anyone wanting to develop proficiency in mathematical modeling. Related Subjects Meet the Author MARY ELLEN DAVIS Georgia Perimeter College, received her Master of Arts degree in mathematics from the University of Missouri-Columbia in 1976. She has taught mathematics at the secondary level and at Georgia State University and the University of Birmingham (England). She joined the mathematics department at Georgia Perimeter College (then DeKalb College) in 1991 and has taught a wide range of courses from college algebra to calculus and statistics. She was instrumental in the piloting and implementing of the college's Introduction to Mathematical Modeling course in 1998. She was selected as a Georgia Governor's Teaching Fellow in 1996, and in 1999 received a GPC Distance Education Fellowship to develop web-based materials for applied calculus. C. HENRY EDWARDS (Ph.D. University of Tennessee) Emeritus professor of mathematics at the University of Georgia, Edwards recently retired after 40 years of undergraduate classroom teaching at the universities of Tennessee, Wisconsin, and Georgia. Although respected for his diverse research interests, Edwards' first love has always remained teaching. Throughout his teaching career he has received numerous college- and university-wide teaching awards, including the University of Georgia's honoratus medal in 1983 and its Josiah Meigs award in 1991. In 1997, Edwards was the first university-level faculty recipient of the Georgia Board of Regents newly-instituted state-wide award for teaching excellence. A prolific author, Edwards is co-author of well-known calculus and differential equations textbooks and has written a book on the history of mathematics, in addition to several instructionalcomputer manuals. During the 1990s, Edwards has worked on three NSF-supported projects that fostered a better integration of technology into the mathematics curriculum. The last three years of his long teaching career were devoted principally to the development of a new technology-intensive entry-level mathematics course on which this new textbook is based. Additional information is provided on his web page Read an Excerpt calculus course courses). Preface calculuscourse courses
Numerical analysis provides the theoretical foundation for the numerical algorithms we rely on to solve a multitude of computational problems in science. Based on a successful course at Oxford University, this book covers a wide range of such problems ranging from the approximation of functions and integrals to the approximate solution of algebraic, transcendental, differential and integral equations. Throughout the book, particular attention is paid to the essential qualities of a numerical algorithm - stability, accuracy, reliability and efficiency. The authors go further than simply providing recipes for solving computational problems. They carefully analyse the reasons why methods might fail to give accurate answers, or why one method might return an answer in seconds while another would take billions of years. This book is ideal as a text for students in the second year of a university mathematics course. It combines practicality regarding applications with consistently high standards of rigour. less
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Algebra Textbook Online Students are constantly looking for their algebra textbook online. There are a variety of reasons for this: the student forgot their book, the student needs additional supplementary material, or maybe the student simply did not want to have to carry around their algebra textbook. After all, many if not all of the mathematics courses in modern academia are accompanied with extremely large and heavy textbooks. As new math students learn the fundamentals of algebra each year, they join their fellow students in seeking out their algebra textbook online. Learning a new subject is difficult and algebra is no exception. Of all the subjects you can find in high school or university, the one subject you will almost always hear a majority of students complaining about is that of math. Regardless of what exactly math is being taught, students seem to always have some kind of learning difficulties for the subject. Having an algebra textbook online makes its much easier for new algebra students to break through the learning barriers that are holding them back in their algebra class at pace and convenience that traditional algebra textbooks don't typically provide. Algebra textbooks online cover the basics of algebra: constants, variables, coefficients, terms, equations and quadratic equations. Algebra students will commonly tell you that within the first week of class they realized that having their textbook was essential to their success. Being able to have a reference to all of the topics that instructors are teaching to their students is indispensable to learning not just any type of math, but for just about every academic subject. In algebra particularly, there are so many terms and special equations that students need to learn and memorize that it is amazing students have done so well for so long without having access to an algebra textbook online. Online learning is growing tremendously and will only continue to grow as students continue their education. Algebra students love to be able to use an algebra textbook online whether they are at school or at home for studying and completing assignments. Often, an algebra textbook online will be formatted in a similar manner to its traditional hardback counterpart. Usually the first few pages will be a table of contents detailing the various algebra topics that the online textbook provides. The last few pages, like most textbooks, feature an index; however, algebra textbooks almost always have reference diagrams, lists, and charts for students to use that allow them to stream line the way that they learn and complete assignments. Unfortunately, traditional hardback algebra textbooks were made much heavier because of all this reference material that – while extremely helpful – causes many students to complain about the weight of their algebra textbook. With an algebra textbook online this caveat is no longer an issue as an algebra textbook can be any number of pages and still be just as acceptable and even more helpful to the students. Algebra involves the use of many calculations and with algebra textbooks online being utilized more by students and instructors alike, some school districts have begun incorporating new electronic portable computing devices for academic use. For algebra students this is good news because not only can they now access their algebra textbook online using these new technologies, but they also have access to a range of different calculating tools to help them with their studies. Publishers are hearing the call for algebra textbooks online and there should be no surprise as more students use their algebra textbook online, in years to come. Related Resources There are many free Algebra textbooks online that you can read and use for your studies? Please visit our list of free mathematics etextbooks here for a variety of free algebra textbooks to choose from.
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more
Book DescriptionEditorial Reviews From the Back Cover Brush up on algebra and trig concepts and get a glimpse of calculus Understand the principles and problems of pre-calculus Getting ready for calculus, but feel confused? Have no fear! This unintimidating, hands-on guide walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. You'll understand the concepts — not just the number crunching — and see how to perform all tasks, from graphing to tackling proofs. Apply the major theorems and formulas Graph trig functions like a pro Find trig values on the unit circle Tackle analytic geometry Identify function limits and continuity About the Author Krystle Rose Forseth graduated from the University of California, Santa Cruz, where she majored in mathematics with an emphasis in education. She has been tutoring math for eight years and teaching for three. Currently, she is the head of the math department at Fusion Learning Center and Fusion Academy, where she teaches mathematics and oversees the math instructors. Teaching students math has made Krystle a more compassionate individual, and her enthusiasm for the subject makes learning fun. Christopher Burger graduated with a Bachelor of Arts degree in mathematics from Coker College in Hartsville, South Carolina, with minors in art and theatre. He has taught math for more than 10 years and tutored math subjects ranging from basic math to calculus for 20 years. Currently, he is the director of independent studies for Fusion Learning Center and Fusion Academy in Solana Beach, California, where he not only teaches students one-on-one, but also writes curriculum, oversees a staff of 35 teachers, and maintains high levels of academic rigor within the school. Christopher takes teaching and connecting with his students very seriously, and he believes he makes a difference not only in their math education, but in their lives. Michelle Rose Gilman is proud to be known as Noah's mom (Hi, Noah!). A graduate from the University of South Florida, Michelle found her niche early, and at 19, she was working with emotionally disturbed and learning-disabled students in hospital settings. At 21, she made the trek to California, where she found her passion for helping teenagers become more successful in school and life. What started as a small tutoring business in the garage of her California home quickly expanded and grew to the point where traffic control was necessary on her residential street. Today, Michelle is the founder and CEO of Fusion Learning Center and Fusion Academy, a private school and tutoring/test prep facility in Solana Beach, California, serving more than 2,000 students each year. She is the author of The ACT For Dummies and other books on self-esteem, writing, and motivational topics. Michelle has overseen dozens of programs over the last 20 years, focusing on helping kids become healthy adults. She currently specializes in motivating the unmotivated adolescent, comforting their shellshocked parents, and assisting her staff of 35 teachers. The aim of this book is to introduce the subjects of pre-calculus in an easy, yet complete way. For the most part it accomplish it's objective, nevertheless, it has some typos and errors that will cause the student to get confused (i.e. the first time the student is shown the formula for the difference of cubes the book says something like this (a-b)(a^2+ab+b^3) which is wrong and in latter examples the formula takes the correct form of (a-b)(a^2+ab+b^2) which causes some confusion). Overall, this is a very good book specially for a review of the subjects. p. 107 #4 "Figure 5.5 illustrates this last step, which yields the parent log's graph". If you look at the graph, it is labeled f(x)=log^x. The parent log being graphed is f(x)=logx. Two very different equations. It seems the authors break their own rule in the first chapter on PEMDAS. This is a simple problem, but they wrote it out wrong in answers where they perform addition and subtraction first before multiplication. I think it should be the other way around. i.e. Ex 4 - Ch 1. abs(5 * 1 - 4 + 6) = 7 not 9. Every monkey knows multiplication is performed first. I'm not sure what the authors were thinking starting with subtraction first???I'm kinda blushing here...I punched this into my TI-89 Titanium and I get the same answer as I had come up with for the nominator. Calculators follow PEMDAS rule, so am I missing something here? Please, put an errata up people and have a forum, otherwise your books are useless. This is a refresher for me as I'm going in for calculus next fall. Mistakes as such would flunk a student in those courses... I ordered this book because I was struggling in my pre-calculus class. It was a great help to my understanding of whatever I had trouble with. Some examples could be made easier to understand but overall the book was worth buying. Too much is assumed, leaving important steps in the process either missing or not well-explained. It is a rare exception to the generally high-quality "dummy" books we have become accustomed to (and expect).
MATH 108, 109 Concepts of Mathematics for Teachers I/II (3, 3) This two-part sequence is designed for the elementary education major as an introduction to selected topics in mathematics. Topics include sets and set operations, number and numeration systems and their operations, algorithms, measurement, reasoning and problem solving, patterns and relations, geometry, probability and statistics. Open only to and required for students preparing to teach at the elementary school level. (108 Fall, 109 Spring) MATH 110 Prelude to Mathematical Thought (3) This course serves as an introduction to mathematical reasoning. Emphasis will be on reading and interpreting problems as well as developing fundamental problem-solving strategies. In addition, the course will familiarize students with mathematical notation and develop the writing skills needed to explain solutions with precision. Possible topics include puzzle problems, algebraic and logical reasoning, pattern recognition, and counting techniques. This course may be used to prepare for MATH 111. (Fall) MATH 111 Mathematical Thought and Problem-Solving (3) This course provides students with a mathematical approach to solving problems as well as an introduction to the nature of mathematics. The course seeks to improve facility with computations, mathematical notation, logical reasoning, and verbal expression of mathematical concepts. Content is selected from classical and modern areas of mathematics such as geometry, number theory, algebra, graph theory, fractals, and probability. The delivery of the content takes on a variety of forms including in-class activities, projects, discovery learning, and lecture. (Fall and Spring) MATH 114 Precalculus Mathematics (3) Designed to prepare students for the calculus sequence or for MATH 115. Topics include set theory, inequalities, systems of equations, basic analytic geometry, functions, logarithms and trigonometry. This course satisfies the core math requirement. Prerequisite: MATH 101 or placement by department. (Fall and Spring) MATH 247 Calculus I (4) An introduction to the fundamental concepts of differential and integral calculus with an emphasis on limits, continuity, derivatives and integrals of elementary functions. Applications to curve sketching, max-min values, related rates and areas will be given. Derivatives and integrals of elementary transcendental functions. Prerequisite:MATH 114 or placement by department. Students that placed into Math 101/102 may not take Calculus until they have completed MATH 114. (Fall and Spring) MATH 377 Foundations of Geometry (3) Survey of geometries emphasizing the axiomatic/deductive approach. Includes finite geometries, fundamental concepts of Euclidean geometry in the plane and higher dimensions, some theorems leading to the modern synthetic approach, constructions and transformations, history of the parallel postulate and non-Euclidean geometries. Understanding and writing clear and consistent proofs are major course objectives. Prerequisite: MATH 228 and MATH 248 or permission of instructor. (Fall, even years) MATH 436 Elementary Number Theory Elementary number theory with a focus on both history and theory. Topics include the Euclidean Algorithm, Diophantine equations, the Fundamental Theorem of Arithmetic, congruences, number-theoretic functions, primitive roots, continued fractions, and the theorems of Fermat, Wilson, and Euler. Prerequisites: MATH 228 and MATH 247 or permission of instructor. (On a rotating basis) MATH 447 Introduction to Real Analysis I (3) A rigorous development of the fundamental concepts of analysis, including the real number system, functions, sequences, limits, continuity, convergence, differentiation, integration and series. Prerequisite: MATH 249 or permission of instructor. (Spring, odd years) MATH 489 Modeling and Simulation (3) Emphasis on the study of models and their applications to other disciplines. Topics may include population growth, epidemics, scheduling problems, predator-prey interaction, transportation, economic and stochastic models. Prerequisite: MATH 248 and CSCI 120 or permission of instructor. (Same as CSCI 489.) (As needed) MATH 398 Independent Study (1-3) Independent study in an area of mathematics selected to meet a student's interest or need. Permission of the instructor, department chair and dean for academic affairs required. (As needed) MATH 492, 493 Practicum (1-3) An opportunity to gain practical experience in a work-related program. The nature of the work experience and the number of credits must be approved in advance by the department chair and the dean for academic affairs. (As needed) MATH 495, 496 Seminar I, II (1, 1) A course designed to enhance the comprehension of the fundamental concepts of higher mathematics and to develop an understanding of their organization. The course may involve applying ideas and techniques learned in earlier classes to solve mathematical and applied problems, and it may also involve directed reading and study in contemporary publications. (Spring) MATH 497 Undergraduate Research in Mathematics (1-3) Under the supervision of a faculty instructor, students conduct research on mathematical questions posed by the student or the instructor. Work may be done individually or in teams as determined by the instructor. The course prerequisites and enrollment limitation vary with the instructor and topic. Prerequisites: permission of instructor. (As needed) MATH 499 Special Topics in Mathematics (3) An intensive study or continuation of a field of mathematics. Some possible topics are differential equations, advanced complex number theory, or harmonic analysis. (As needed)
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Can low achieving mathematics students succeed in the study of linear inequalities and linear programming through real world problem based instruction? This study sought to answer this question by comparing two groups of low achieving mathematics...
St. Olaf College Context 1. What is the status of Quantitative Reasoning programming on your campus? Two of the most important features of Quantitative Reasoning programming at St. Olaf are (1) our recently-revised general education requirement in Abstract and Quantitative Reasoning (AQR), and (2) the majors and concentrations offered by our Department of Mathematics, Statistics, and Computer Science (MSCS). All students at St. Olaf must complete at least one AQR course, defined as a course which "develops analytic thinking skills through systematic focus on abstract and quantitative reasoning." Consistent with the overall design of our GE curriculum, AQR courses may be offered by any department, so long as they are approved by our Curriculum Committee as meeting the criteria for AQR accreditation. These criteria are as follows: "1. Abstract reasoning is the study of structures and patterns that arise in quantitative or computational settings. Quantitative reasoning is the use of formal structures and methods to model and analyze phenomena in the natural and human-made worlds. An AQR course should include elements of both of these reasoning activities. 2. An AQR course should develop students' problem-solving proficiency through analytic thinking, not merely routine calculation. An AQR course should develop skills and ideas beyond what is typically attained in secondary school. 3. An AQR course should incorporate multiple elements of abstract or quantitative reasoning (e.g., symbolic, geometric, and numerical perspectives; data analysis and statistical inference; visualization; algorithms and formal models)." ( Not surprisingly, at present all courses meeting the AQR requirement are either mathematics, statistics, or computer science courses. However, the newly-revised requirement has just been implemented, and it is hoped that more departments, particularly those in the social and natural sciences, will be able to offer courses satisfying the AQR requirement. Our department of Mathematics, Statistics, and Computer Science is among the strongest in liberal arts institutions nationwide. The department's website is a very fair portrait of its programming and how that programing is generally perceived by students and faculty alike: Mathematics is all of those things--and more--at St. Olaf, where the mathematics program is recognized nationally for innovative and effective teaching. Our program was cited as an example of a successful undergraduate mathematics program by the Mathematical Association of America (Models That Work, Case Studies in Effective undergraduate Mathematics Programs) and St. Olaf ranks sixth in the nation as a producer of students who went on to complete Ph.D.'s in the mathematical sciences (Report on Undergraduate Origins of Recent [1991-95] Science and Engineering Doctorate Recipients). The majors and concentrations offered by our MSCS department include a major in mathematics, which students pursue on a contract basis; a recently-initiated major in computer science; and a concentration in statistics. A signature element of the department's program is its commitment to active and applied learning. For example, for many years, the department has offered a January term Mathematics Practicum to its mathematics majors, in which students work for a month in five-person teams on real industrial problems and present their results to scientists and executives of the company that posed the problem. Recent Practicum topics include: * Time-Efficient Suturing During Cardiac Surgery * Estimation of Minimum Freight Car Needs * Optimal Positioning of Manufacturing Equipment * Load Factors for Airline Scheduling * Federal Fairness Test for Benefit Plans The programming in Statistics includes an NSF-funded Center for Interdisciplinary Research (CIR), which brings together undergraduate statistics students supervised by statistics faculty with faculty and students from other disciplines to share in the excitement and challenge of working across the traditional academic boundaries to collaborate on research. Projects have ranged from biology to psychology, economics, and linguistics, and have included work with the St. Olaf Office of Evaluation and Assessment to investigate the development of tools for assessing information literacy. Finally, mathematics programming at St. Olaf makes possible a variety of opportunities for study abroad, consistent with the larger college culture supporting international and off-campus study. 2. What are the key learning goals that shape your current programming or that you hope to achieve? Intended learning outcomes for the Abstract and Quantitative Reasoning (AQR) requirement: Students will demonstrate: 1. An ability to recognize and employ patterns, structures, and models appropriate to particular theoretical or applied problems, as well as derive and understand properties of patterns, structures, and models themselves; 2. An ability to apply abstract and quantitative reasoning to solve problems in novel contexts. 3. An ability to approach problems from multiple perspectives, employing a variety of strategies. Intended learning outcomes for the major in Mathematics: Students will demonstrate: 1. the ability to understand and write mathematical proofs. 2. the ability to use appropriate technology to assist in the learning and investigation of mathematics. 3. appreciation of mathematics as a creative endeavor. 4. the ability to use mathematics as a tool that can be used to solve problems in disciplinary and interdisciplinary settings. 5. the ability to effectively communicate mathematics and other quantitative ideas in written and oral forms. Intended learning outcomes for the major in Computer Science: Students will demonstrate: 1. the ability to solve problems that require creative reasoning with levels of abstraction. 2. competence in core computer science topics, represented by I. the ability to reason about structured computer systems, and II. the ability to design good algorithms. 3. the capacity for identifying, analytically discussing, and creatively addressing ethical issues in realistic computing systems. Intended learning outcomes for the concentration in Statistics: Students will demonstrate: 1. the ability to formulate statistical models based on research questions. To that end, students will demonstrate communication skills to assist non-statistical collaborators in addressing research questions through statistical models. 2. the ability to apply flexible approaches to modeling by graphically exploring data, choosing appropriate analyses from a variety of statistical methods and implementing analyses with proficient use of technology. 3. the ability to interpret results correctly and make inferences consistent with the study design. Students will be able to communicate results effectively orally and in written form to researchers and non-technical audiences alike without overstatement, acknowledging the limitations. 4. appreciation for the interdisciplinary nature of statistics in both academia and industry. 3. Do you have QR assessment instruments in place? If so, please describe: Two of our principal instruments are the Collegiate Learning Assessment (CLA) and a set of supplementary questions about quantitative reasoning which we included in our 2007 administration of the Higher Education Data Sharing Consortium (HEDS) senior survey. In addition, there are a few items related to quantitative reasoning in the National Survey on Student Engagement (NSSE) and the HEDS Alumni Survey. The CLA data we receive does not isolate information about quantitative reasoning; instead, it provides a score on students' abilities to sift through and present relevant evidence in support of a solution to a complex and ambiguous problem. Most of the scenarios provided to students in the CLA performance task include some hypothetical quantitative information which students must evaluate for relevance and then cite appropriately in their proposed solutions. So we see the CLA assessment data as relevant to the assessment of quantitative reasoning, but it is indirect. We anticipate expanding our assessment of quantitative reasoning through a new Teagle grant we have secured in partnership with Carleton and Macalester. The purpose of this Teagle grant is to make systematic improvements in students' ability to make effective arguments, using assessment data to inform classroom-level innovations and then to assess the extent to which those innovations yield improvements in student learning. We expect faculty from many disciplines to participate in this project, and we are confident that some faculty will develop innovations that improve students' abilities to use quantitative data skillfully in making effective arguments. 4. Considering your campus culture, what challenges or barriers do you anticipate in implementing or extending practices to develop and assess QR programming on your campus? We face the same challenges that all institutions of higher education face in growing our culture of assessment: (1) Limited faculty and staff time to frame questions, seek data systematically, and use data to inform practice; (2) the fact that high-quality assessment is just plain hard to do; it is a difficult thing to gather data that points conclusively to the effectiveness of specific educational practices, especially since so many of our learning outcomes are not fully realized until long after students have left the institution and are not available for direct measurement of those outcomes; (3) the fact that a culture of assessment requires different ways of thinking about student learning and making academic decisions. Even when we have good assessment data, it is not easy to make sure that the people who need it to inform their practice have access to it and know how to use it. 5. Considering your campus culture, what opportunities or assets will be available to support your QR initiatives? (1) A growing number of faculty are interested in, engaged with, and supportive of assessment. In particular, the principal faculty committee responsible for the college curriculum has a strong contingent of colleagues who are advocates for assessment. (2) Every member of our Dean's Council – the associate deans who lead the various academic divisions of the faculty and who work with the Provost and Dean of the College on budget, personnel, and the academic program – has had significant engagement with assessment in the past few years. Some have participated in another Teagle grant intended to foster a culture of assessment; one is a member of the Assessment Subcommittee of the college curriculum committee; and one has helped recruit students to participate in the CLA. The Assistant Provost, also a member of the Dean's Council, is a strong supporter of assessment and has provided excellent leadership in grant-seeking and organizational development in the Office of Institutional Research and Evaluation, the administrative unit of the college with principal responsibility for leading the college's program of assessment. (3) The Office of Institutional Research and Evaluation has also provided effective leadership and support in growing St. Olaf's culture of assessment. The office has two directors, a staff member who leads the institutional research work of the office, and a faculty member who leads the assessment work. This partnership across roles has strengthened faculty ownership of the college's assessment program while educating faculty about the resources we already have available in IR to support effective assessment. Assessment features strongly in the vision of the office, which is "to be a national leader among liberal arts institutions in institutional research and the assessment of student learning" ( IR&E is delivering on that vision: its directors are leaders of two inter-institutional grants which will enhance assessment on campuses in addition to our own, and they have been innovators in developing assessment instruments, stimulating appreciation for the value of assessment, and making assessment data readily available to faculty. The interim Director of Evaluation and Assessment, responsible for leading St. Olaf's assessment program during the 2008 calendar year while the long-term director is on sabbatical, is also chair of the Department of Chemistry, and is thus particularly well-positioned to enhance the assessment of quantitative reasoning. (4) We have enough assessment data that we are not starting from scratch. We can build on things we are already doing, rather than starting from ground zero. That is the idea behind the new Systematic Improvements Teagle grant – build on what we already have to enhance a specific student learning outcome and assess the extent to which we are successful.
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Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).

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