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Yes, the Mathematics Department operates a Math Study Area which is open to students wanting extra help in mathematics. This extra help opportunity is available to students on a drop in basis whenever they have a free in their schedule. A mathematics teacher will be there to provide this help.
For students without frees, there is one Mathematics teacher available to give extra help at lunch in the library on days 1, 2, and 6 in our 8 day cycle (as of 1st semester in 2009-2010). These days may change in 2nd semester.
Should extra help be needed beyond these times, the student should see their teacher to arrange a mutually convenient time.
All students have been advised of this above information!
Some may need occasional reminding!
How much homework is assigned in Math?
All students in Academic Math courses are expected to complete approximately 30 minutes of homework between classes. Assignments will also be given and are to be completed by the assigned dates.
All students in advanced or IB courses will be expected to complete more than 30 minutes of homework between classes. Assignments will also be given and are to be completed by the assigned dates.
Homework is an important component of these courses so that students have the opportunity to independently practice and develop their skills and understanding in Mathematics. In class students are usually working with other students. By doing homework, students will find out what they can do on their own and identify any areas which need further explanation. This independent study should allow students to build confidence in their abilities!
Not Taking Math 10 Academic Until Second Semester?
The Park View Mathematics Department has put together a unit with grade nine material to help you review some basic concepts. Copies can be picked up during Parent Teacher interviews or from any math teacher.
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Introductory Algebra - 2nd edition
ISBN13:978-0077281120 ISBN10: 0077281128 This edition has also been released as: ISBN13: 978-0073406091 ISBN10: 0073406090
Summary: Introductory Algebraoffers a refreshing approach to the traditional content of the course. Presented in worktext format,Introductory Algebrafocuses on solving equations and inequalities, graphing, polynomials, factoring, rational expressions, and radicals. Other topics include quadratic equations and an introduction to functions and complex numbers. The text reflects the compassion and insight of its experienced author team with features developed to address the specific needs of dev...show moreelopmental level students. ...show less
2nd Edition Paperback slight water damage, still very good book, may have wear and/or considerable writing, ships fast!!!, textbook only unless specified previously
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BookSleuth Danville, CA
Fast Shipping ! Used books may not include access codes, CDs or other supplements.
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Still wrapped, some shelf wear WN-10
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Prealgebra - With Cd (paper) - 5th edition 5eis appropriate for a 1-sem course in Prealgebra, and was written to help students effectively make the transition from arithmetic to algebra. To reach this goal, Martin-Gay introduces algebraic concepts early and repeats them as she treats traditional arithm...show moreetic topics, thus laying the groundwork for the next algebra course your students will take. ...show less21 +$3.99 s/h
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SellBackYourBook Aurora, IL
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Willow Grove Calculus what I've done in Matlab has centered around my physics research and to a lesser extent my geometry. So in that context, I've programmed dynamical models of quantized space (the "arrow shadow model") to prove certain of my symmetry claims and I've done some substantial work modelling J-S...
...Graphs of lines and inequalities, variables, and slope are ideas that each carry weight as one progresses through higher level math courses. Unfortunately, algebra is often seen as problematic due to the massive amount of symbols used. I combat this by helping my students see what the algebraic notation is describing (on a graph for example) so that it makes sense to them draw mostly from my imagination of all sorts of things. Things I draw are robots, cyborgs, aliens, mis-figured creatures, and spaceships. Since I draw from my imagination, every drawing is unique.
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A discussion of advances and developments in the field of number theory arising from finite fields. It emphasizes mean-value theorems of multiplicative functions, the theory of additive formulations, and the normal distribution of values from additive functions. more...... more...
This undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. The capstone of the book is a brief presentation of the Riemann zeta function and of the significance of the Riemann Hypothesis. more...
This book represents the highlights of a conference hosted at ETH Zurich. It will appeal to a broad spectrum of number theorists. However, it is also for research students interested in the state of number theory at the start of the 21st century and in possible developments for the future. more...
This valuable book focuses on a collection of powerful methods of
analysis that yield deep number-theoretical estimates. Particular
attention is given to counting functions of prime numbers and
multiplicative arithmetic functions. Both real variable ("elementary")
and complex variable ("analytic") methods are employed. more...
This textbook is intended to serve as a one-semester introductory course in number theory and in this second edition it has been revised throughout and many new exercises have been added. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject. more...
Introduction to Modern Number Theory surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems, the central ideas of modern theories are exposed. Some topics covered include non-Abelian generalizations of class field theory, recursive computability... more...
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A graphing calculator is required for this course. No particular model is required, but theTI-89 and/or TI-83 will be used for demonstrations.
It is advisable to have a notebook in which to keep homework assignments, class notes, handouts, and returned quizzes.
A pencil is the preferred writing instrument. You may take class notes and do homework with an instrument of your choosing, but it is expected that you will do all tests and quizzes in pencil.
Classwork & Homework
In order to master most mathematical concepts or processes, practice is required. Homework is assigned almost every night to give you opportunities to practice. You should do the homework with the idea of engaging the concepts and doing the assignment well, not just to get it done. You should keep your homework and class notes in a notebook of some kind. Organization is one of the keys to success for a student.
You will be divided into groups, and your first task each day is to check homework within your group. We will discuss any problems that cannot be done by anyone in a group. While you are checking your homework within your group, I will visit each group to see that you have done your work. Each time you haven't completed your assignment, a zero (0) will be recorded. Beginning with the third zero and for every zero thereafter, one (1) point will be deducted from your quarter average.
At times, I may take up your homework and grade it. I sometimes take a grade on your homework by giving a short quiz asking you to copy your solutions to three or four problems from the assignment.
Grading
Daily work and major tests are factors used in computing a grade for each quarter. Class work, homework, quizzes, and other types of daily (or short term) activities are used to determine the daily average for the quarter. Two to four major tests will be given during the quarter, and each will weigh equally in determining a test average for the quarter. The quarter grade is computed using the formula:
Qtr Grade = (2/3)*(test average) + (1/3)*(daily average)
Students are required to take a comprehensive examination at the end of each semester. The semester grade is computed from the quarter grades and the examination grade using the formula:
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The course presents to students knowledge on basic numerical methods:
matrix operations, solving systems of linear algebraic equations and regression. Another part of the lecture deals with polynomial interpolation and solution of one-dimensional nonlinear equations.
After successful passing of the course the students should be able to
- list and describe basic numerical methods lectured
- successfully apply these methods for solving a specified problem.
Syllabus
1) Number representation in a computer,precision, accuracy.
Errors in numerical algorithms,
propagation of the errors.
Stability of the algorthims.
Ill-posed methods.
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Tailored to both the specification and the tier, this Student Book delivers exactly what students and teachers need to cover the unit in exactly the right depth.
Synopsis:
* Supports teachers' understanding of AO2 and AO3 through clearly labelled AO2/3 questions in the exercises. * Packed with graded questions reflect the level of demand required, so students and teachers can see their progression. * Includes worked examples throughout the book to break the maths down into easy chunks. * Uses feedback to highlight common errors .
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The materials below were developed for both graduate and undergraduate courses taught under the title "Game Theory." The philosphy of these courses is to cover as broad a range of topics with a game theory flavor as possible. It it generally easier to get a deeper knowledge of a subject X where one has seen the key ideas and some of the major results than it is to start off reading in a book devoted only to the content of topic X.
Game Theory, here, refers primarily to the analysis using mathematical tools of conflict situations as they arise in business, economics, and political science. The subject was "put on the map" by the work of the mathematician John Von Neumann and the economist Oskar Morgenstern. However, there will also be a brief treatment of what have come to be called combinatorial games. Nim is perhaps the best known of such games. These ideas are related to a mathematical understanding of games such as chess and Go.
Here is a handout "sampler" which illustrates some questions in the spirit of mathematical modeling that deal with game theory and fairness situations. The handout is similar but not identical to others found below.
Here is a sampler of problems and situations which are "informed" by ideas from the theory of games. Many game theory problems and models arise from the broader arena of fairness models. Some of the examples I give have strong ties to game theory and/or were pioneered by game theorists but belong to the domain of fairness models. Many fairness problems have lots of "emotional" content (e.g. should women in the military be able to serve in front line combat operations; LSBG issues, access to abortion, etc.) but here I try to restrict myself to game theory and fairness problems which are amenable to mathematical insights, though "political" and "emotional" fairness issues are no less important (and sometimes profit from mathematical analysis).
Here is some course information, as well as links to blogs that deal with game theory, economics, fairness and equity, computational issues, market design, and mechanism design. There is also a link to the American Mathematical Society Feature Column which has a variety of articles dealing with game theory and fairness related issues.
This note suggests some questions for you related to game theory and equity situations. The questions raise ways to get students in K-12 thinking about the role games play in society and ways that mathematics might be used to get insight into games.
Here are some practice problems involving the use of dominant strategy analysis for zero-sum games. Other issues are finding optimal pure strategies versus optimal mixed strategies in zero-sum games. Solving these problems involves the use of elementary probability theory as well as factoring and/or the algebra of linear equations.
Here are some notes about matrix games which contrast the behavior of games where a dominant strategy analysis can be carried out compared with what a game that does not allow dominant strategy analysis.
Graph theory, the study of geometrical diagrams consisting of dots and lines, has proved to be very valuable in all branches of mathematics. A very abbreviated primer of this subject (one page!) is provided.
Here are some practice problems involving the basic ideas about non-zero-sum games. Non-zero-sum games are much more subtle than the zero-sum ones. One needs to understand the difference between pure strategies and mixed strategies, the notion of an equilibrium or stable outcome. These questions try to give insights into these ideas.
Non-zero-sum games lack many of the more transparent nice properties of zero-sum games. One thing that does get retained is the notion of Nash equilibrium. Unfortunately, though, there is no guarantee that playing a Nash equilibrium strategy is an appealing choice, since in many non-zero-sum games there will be outomces that are "Pareto improvements" over a Nash equilibrium. This means that there are outcomes that at least one player prefers and which are not worse for the other player.
Here are some practice problems involving the basic ideas about voting. To construct a mathematical modeling of voting requires one have candidates (choices), voters, ballots (to allow the voters to express their preferences about the choices), and election decision methods (how to use the ballots to decide on a winner, or a ranking for choices). Elections come up in deciding on a governor, a president or a major, but also in choosing what to serve at the company picnic, best picture of the year, or who to hire from a list of candidates for a job. Pioneering work was done by Borda, Condorcet, Charles Dodgson (Lewis Carroll) and Kenneth Arrow. This is a very appealing topic for teaching arithmetic, logic, basic graph theory, etc.
Mathematical modeling is one of the practice standards for the CCSS-M (Common Core State Standards in Mathematics). However, in many people's minds the reason for teaching modeling is further practice with mathematical techniques - solving linear equations, solving quadratic equations, etc. I belive in a broader concept of what modeling is about. In part, it is a way to encourage people to see mathematics as a subject that emphasizes various themes - fairness, optimization, information, etc. While this essay is not directly related to the content of this course I hope that it offers a perspective on how the theory of games and fairness situations might fit into curriculum at all levels.
When one has geographic districts or countries which have very different populations but one needs to have a legistlature to represent the different "locales" it seems reasonable to use a system where each representative casts a different number of votes. It is tempting to have the weights proportional to population but it is easy to see that this can create representatives who are never members of minimal winning coalitions. Such players are traditionally called dummies. So it makes more sense to have weights so that population is proportional to some measure of power. One way of doing this is the use the Banhaf Power, as is done in NY State. The European Union has made extensive use of weighted voting schemes.
Here are a few "practice" problems for a "final" summatory examination. The emphasis is on being able to do "typical" algorithmic problems that come up in zero-sum games, non-zero sum games, methods of deciding elections, power of players in weighted voting games, assigning an integer number of seats based on "population" (apportionment), the Gale/Shapley deffered acceptance algorithm, bankruptcy, etcHere are some notes about apportionment, giving claimants a non-negative number of seats in a parliament of fixed size h. The problem arose naturally in two versions. In the US each "state" has to be given at least one seat, while in Europe, each party must be assigned a block of seats in a way that represents its strengh in the election but there is no guarantee that a party that got a small vote will get a seat.
Here are some additional notes about apportionment. A review of the Adams, Jefferson and Webster Methods considered as divisor methods is given and then it is described how to do the calculations for these methods usings the rank index approach to the computations. I call this the "table" method. This approach is computationally more straightforward.
Here are some practice problems involving the Webster, Adams, Jefferson, and Hamilton methods of apportionment. See if you can get a feel for why Adams rewards smaller states while Jefferson rewards larger states.
Here are some notes about the many different approaches to solving bankruptcy problems. The basic idea is that one has claimants who are owed money from an estate of size E but the claims add to more than E. What amounts should be given to each claimant so that the amounts returned to the claimants sums to E and the amounts given are fair? Many different approaches to fairness come into play and can lead to very different allocations to the claimants. Axioms can be created which try to characterize which properties the different methods to resolve bankruptcy problems obey.
Teachers often hear the terms, exercises, problem solving, and mathematical modeling. While there is some similarity between these terms they connote rather different things. This essay tries to distinguish between the terms in an informal way, and some of the examples used come from the realm of fairness problems.
While carrying out this particular poll is probably of interest only to me and the students who took my game theory course in the summer of 2012, I think it is valuable for teachers to design polls of this kind to help them get insight into the relative appeal of different topics they treat in courses that they teachA voting game is a way specifying how action gets taken in a legislative body by indicating "winning coalitions" - coalitions that can get some action accomplished. One way of doing this is to have each player cast a number of votes called the weight of that player. There is a quota Q, and any collection of players whoses weights sum to Q or more are winning. However, one can have positive weight in such a voting game without haviing positive "power." This happens when a player is not a member of any "minimum winning coalition." A coalition that wins, but when any player is dropped from the coaliton no longer wins. Such a player is called a dummy. There are many power indices which try to measure the "power" of the players in a voting game either in terms of the coalitions the players are members of or the weights they cast. The best known of these are the Coleman Index, the Shapley-Shubik Index, and the Banzhaf Index.
Here is a cost allocation problem, typical of many, where towns or people come together to fund a project that will bring them all benefit but would be two expensive for one person (town) to do on its own. So the issue is developing a way to share the costs that is fair.
Here are a few "practice" problems about bankruptcy problems. The idea is that one has claims against an asset which is too small to pay off all of the claims. What is a fair way to distribute the asset? Just as with elections there are many reasonable systems which often differ in their distributions.
Here are a few "practice" problems about election methods. Methods that have either been used in the "real world" or are appealing because a reason that they might produce a "good" winner can be put forward are: pluarality, run-off, sequential run-off, Borda Count, Coombs, Bucklin, and Condorcet. Different methods can produce the same or different winners. There are a lot of topics in the traditional k-12 curriculum that can be taught and/or motivated with elections issues.
Here are a few "practice" problems involving the Gale-Shapley, deferred acceptance algorithm, that finds a stable matching between the two sides of a two-sided market. There are suprisingly many appicatiions of this elegant algorithm. With cleverness one can significantly extend the domain of applicability of this algorithm. The "vanilla" version requires no indifference and that not getting "married" is not an option. Depending on whether the men propose or the women propose one gets two potentially different stable marraiges. However, in many cases there are many other stable marriages.
Here are some notes about two-sided markets. These problems are often couched in terms of pairing off equal numbers and boys and girls in "marriage" in some "stable" way. Applications include matching doctors to hospitals when the finish medical school, school choice, college admissions, etc. The major algorithm here is called the Gale-Shapley, deferred acceptance algorithm.
Some of the materials treated on this blog are at the research frontier but this in part is why monitoring what is said is especially interesting. Since game theory is quick starting compared with many other parts of mathematics even if the technical stuff is not accessible you can get an idea of the kind of issues that drives research in game theory.
Here are more "practice" problems about games that deal with very basic ideas but they also hint at ways that one can use these ideas, say, in an algebra class, or when teaching about mathematical modeling. Being able to simplify games usings using dominant row/column and looking for saddle points is worth practicing so that you can understand the implications of these ideas.
Here are a few "practice" problems about zero-sum games that deal with very basic ideas but they also hint at ways that one can use these ideas, say, in an algebra class, or when teaching about mathematical modeling.
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User the cheese stands alone - MathOverflowmost recent 30 from of an Elementary Differential Geometry CourseThe Cheese Stands Alone2011-03-11T23:43:42Z2012-05-05T15:46:16Z
<p>I have to admit I'm not sure if this is an appropriate question. It's related to research in math education, but not directly to math.</p>
<p>I've found that in talking to professional physicists and engineers, most of them find some use for differential geometry nowadays. One theoretical physicist went as far as to say you could "do nothing serious without it." Yet at most schools (at least the few I've looked at) differential geometry is reserved for graduate students in math and advanced math undergraduates. No schools I looked at had an elementary differential geometry class in, say, a similar style as the calculus sequence. Some of the people I talked to also expressed a lot of difficulty in learning it for the first time on their own. I myself am taking an advanced graduate course in General Relativity, and a good portion of the difficulty of the students is in misuderstanding the fundamental concepts of differential geometry.</p>
<p>To cover differential geometry rigorously, of course one needs quite a bit of advanced mathematics, including topology and analysis. But universities teach elementary calculus classes, most of which are not terribly rigorous, but are sufficient for the purposes of non-mathematicians. Linear algebra, multivariate calculus, and a bit of differential equations would (in my mind) be sufficient to teach a course for engineers. You might argue that one needs to know the theory of manifolds first, but I see this as analagous to studying calculus without really knowing the structure of $\mathbb{R}$.</p>
<p>From my viewpoint, differential geometry is the logical extension of calculus. Based on it's huge (and growing) impact on applied disciplines, It seems logical to have a course in it for engineers and physicists, which I would put immediately after the final semester of calculus (assuming the students have also had linear algebra).</p>
<p>So my question is this: Are there specific instances, either textbooks or courses at a university, of differential geometry classes taught with the intent of being useful for engineers and scientists, which assume only basic calculus knowledge and linear algebra? (Obviously, there are books like "Differential Geometry for Physicists," but I really mean something that would be used by mathematicians teaching such a course). If so, how successful have these courses/books been? If not, or if the attempts have been unsuccessful, is there any particular reason as to why it is not feasable/common?</p>
by The Cheese Stands Alone for Finding the degree of minimal polynomialsThe Cheese Stands Alone2011-04-17T02:37:11Z2011-04-17T18:48:24Z<p>No. Some conditions are needed on the $a_i$ and $p_i$. For instance, take n=2, $a_1 = a_2 = 2$, $p_1 = p_2 = 2$. Then $x = 2 \sqrt{2}$, which has minimal polynomial $x^2 - 8$. As an even simpler example, n=1, $a_1 = 2$, $p_1 = 4$, then $x$ is rational.</p>
<p>For a less trivial example, take $a_1= 4$, $a_2 = 6$, $p_1=p_2=2$. Check that this has a polynomial of degree 12. In fact, this isn't really true at all. </p>
<p>One can, however, prove that the degree of the minimal polynomial is at most $\prod a_n$, which is an easy exercise in field theory. Any graduate algebra textbook covering Galois theory will be more than sufficient to prove this; just remember the degree of the minimal polynomial is the same as the dimension of the extension field viewed as a vector space over the base field.</p>
<p>EDIT:</p>
<p>After much miscommunication on my part, we've reached the following results:</p>
<p>Suppose $a_1,\ldots,a_n$ are pairwise relatively prime positive integers, $p_1, \ldots, p_n$ integers such that $\sqrt[a_i]{p_i}$ is of degree $a_i$ for each i. Then $\sqrt[a_1]{p_1} + \cdots + \sqrt[a_n]{p_n}$ is of degree $\displaystyle \prod_{i=1}^n a_i$.</p>
<p>The condition that each $\sqrt[a_i]{p_i}$ is met (by Eisenstein Criterion) should there be a prime $q_i$ such that $q_i | p_i$ and $q_i^2 \not{|} p_i$ for each i.</p>
surface area among convex subsets of the unit sphere of a given volumeThe Cheese Stands Alone2011-03-25T04:04:09Z2011-03-25T04:04:09Z
<p>The following problem is listed in Steven Lay's "Convex Sets and Their Applications" (1982) as unsolved (paraphrased):</p>
<p>Let $B$ be the unit ball in $\mathbb{R}^3$ and $0 < V < \pi$. Define $\mathcal{F}$ as the family of all convex subsets of $B$ with volume $V$. Find the member of $\mathcal{F}$ with maximum surface area.</p>
<p>The conjectured answer (as of 1982) is $B \cap$ { $(x_1,x_2,x_3)\in \mathbb{R}^3 | |x_1| < c $} for appropriately chosen $c$.</p>
<p>Does anyone know if this problem has been solved, or if any progress has been made on it? I couldn't find any recent references in the literature to it.</p>
of $\mathbb{R}^n$ spanned by the image of convex $(n-1)$-polyhedra under the face-counting mapThe Cheese Stands Alone2011-03-16T21:35:24Z2011-03-17T14:12:46Z
<p>Fix $n \in \mathbb{N}$. A convex polyhedron $C$ in $\mathbb{R}^n$ is the convex hull of finitely many points with nonempty interior. For $H$ a supporting hyperplane, ie $C$ is contained in one of the two closed half-spaces bounded by $H$, we call $H \cap C$ a $j$-face of $C$, where $j$ is the affine dimension of $H \cap C$. By convention, $\varnothing$ is called a $-1$-face of $C$ and $C$ an $n$-face of itself.</p>
<p>Define a function $F$ from the set of convex polyhedra to $\mathbb{R}^{n+2}$ by coordinates, so that $F(C) = (a^C_{-1}, ..., a^C_n)$, where $a^C_j$ is the number of $j$-faces of $C$ for $j=-1,...,n$. Let $W$ be the affine subspace of $\mathbb{R}^{n+2}$ generated by $\operatorname{im} F$.</p>
<p>It's clear that $a^C_{-1}=1$ and $a^C_n=1$. Euler's formula $\displaystyle \sum_{j=-1}^n (-1)^j a^C_j = 0$ (which may be more familiar as the Euler characteristic $V+E-F=2$ in the case of $n=3$) is a third affine relation between the $a^C_j$'s. Hence, $\operatorname{dim}W \le n-1$. </p>
<p>Is it always true for any n that $\operatorname{dim}W = n-1$? Put differently, for any $n$, are the three equations above the only affine relationships that must be satisfied by $a^C_j$'s for all convex polyhedra $C \subset \mathbb{R}^n$, or is there some $n$ in which there is another relation?</p>
<p>I seem to recall an affirmative answer to this, but I can't remember how it was solved or where I found it.</p>
by The Cheese Stands Alone for Erdos distance problem n=12The Cheese Stands Alone2011-03-11T23:07:24Z2011-03-11T23:07:24Z<p>I wish I could get images to work, but here goes my poor explanation:</p>
<p>Take a regular hexagonal lattice with distance 1 between nearest neighbors, and choose a 15-point equilateral triangle in this lattice (15 is a triangular number). Remove the 3 verticies of the triangle. You'll be left with 12 points and 5 distinct distances.</p>
<p>edit: just checked the OEIS reference, and it's available on google books. The picture you want is on page 200 at <a href=" rel="nofollow">
by The Cheese Stands Alone for Is there a mathematical axiomatization of time (other than, perhaps, entropy)?The Cheese Stands Alone2011-03-09T01:46:31Z2011-03-09T03:58:28Z<p>A bit tangential, but I'm surprised no one mentioned time in general relativity (Ken Knox discussed special relativity, but the case in general relativity is subtly different). This discusses relativity from a perspective closer to a physicist's, since it's a bit more elementary (and hence easier to understand) in my view. As discussed at the end, relativity in general is somewhat incompatible with statistical mechanics, at least under the standard approximations, so this is almost surely not what you're looking for, but it may be of use to someone.</p>
<p>In GR, space-time is a 4-manifold which is endowed with a Lorentzian metric $g_{ab}$, which is a rank 2 covariant tensor. The scalar product of two vectors $V$ and $W$ is then $g_{ab}V^aW^b$, from which we can compute things as in special relativity (where the metric is $\eta_{ab}$, which is $0$ if $a \ne b$, -1 if $a=b$ and $x^a$ is a spacial coordinate (i.e. x,y,z), and 1 if $a=b$ and $x^a=t$ is the time coordinate).</p>
<p>If a vector (field) V satisfies $g_{ab}V^aV^b >0$, we call it time-like, and similarly for trajectories based on their tangent vectors. These are the possible trajectories for particles with positive mass. Any time-like trajectory corresponds in the limit of low mass and low speed to a local frame of reference in which it is the 'time', and if the trajectory is a geodesic then the frame is inertial. Particles with 0 mass (i.e. light) have null trajectories, with $g_{ab}V^aV^b =0$. If $g_{ab}V^aV^b <0$, the vector is space-like. Particles with no forces (other than gravity) acting on them travel on geodesics. The trajectory of a massive particle is called it's world line. The sign conventions here are often reversed, so care is advised. For a time-like path, we can compute its length by $ds^2 = g_{ab} x^a x^b$, where $x$ are your coordinates.</p>
<p>For any observer, they observe themself as stationary, and traveling forward in time with unit speed. Time for that observer corresponds exactly to the length of their trajectory. That observer can even set up a local set of coordinates in which the metric is approximately $\eta_{ab}$, provided all masses are sufficiently far away. Locally, then, time behaves like in special relativity. The big difference between special and general relativity is that the latter has no inertial reference frames in general, so observers can only measure times in at their own location.</p>
<p>To summarize, time is another coordinate in space-time, just like space, but it isn't universal (it's observer dependent), and the only real restrictions on it is that it must be the part of the metric that has positive signature (in the sense above).</p>
<p>Since this was no-doubt useless in explaining the difference between time in special and general relativity, I recommend Hughston & Tod's <em>An introduction to general relativity</em>. It's fairly light reading and has an introduction to special relativity, but it's rigorous enough for mathematicians reading it. A number of other books at a higher level are available, of which Hawking & Ellis, Wald, and Misner, Thorne, & Wheeler are all good references.</p>
<p>However, if you're looking for a formulation in which the second law of thermodynamics is provable, or even where entropy is defined, relativity is the wrong place to look. Even in special relativity, concepts like thermal equilibrium depend on a particular reference frame, so entropy may be defined in one inertial frame but not another. There has historically been great debate over how temperature (which is the thermodynamic conjugate of entropy) should transform under Lorentz transformations, and it's still not totally resolved.</p>
by The Cheese Stands AloneThe Cheese Stands Alone2011-09-04T02:28:44Z2011-09-04T02:28:44ZInsofar as theoretical computer science is a part of mathematics, Chomsky should qualify. Despite being a linguist, his work helped shape modern theoretical computer science. by The Cheese Stands AloneThe Cheese Stands Alone2011-07-21T18:30:31Z2011-07-21T18:30:31Z@Gerry My solution can be modified to accomodate this fairly easily, concatenating all the entries of A and B into A'(1,1) and setting B' to be a matrix with all entries -1. by The Cheese Stands AloneThe Cheese Stands Alone2011-07-20T18:22:15Z2011-07-20T18:22:15ZTo expand on Emil's comment a little bit, we need to know if you want to exclude similarity over $\mathbb{Q}$, $\mathbb{C}$, $\mathbb{F}_2$, or what. In any case, it isn't clear to me whether A' and B' must both contain only zeros and ones, or if they can contain integers. If the latter, an easy solution is to view the matrix A as encoding an integer via decimal expansion, and stick that in the (1,1) position of A', then choose the remaining diagonal entries to make sure A and A' have different traces. by The Cheese Stands AloneThe Cheese Stands Alone2011-06-20T13:09:38Z2011-06-20T13:09:38ZIt seems to me that the "conventional mathematics of infinity" was started by Cantor much less than 400 years ago. Of course, there is a concept of infinity in calculus, but that's something totally different in my book. by The Cheese Stands AloneThe Cheese Stands Alone2011-06-05T06:20:26Z2011-06-05T06:20:26ZIf you're going to choose a language like C which is really only useful for computational programs (and software design, but I assume most mathematicians don't do too much of that), you might as well say Fortran rather than C. It's still the language of choice in most hard sciences. by The Cheese Stands AloneThe Cheese Stands Alone2011-04-17T18:39:26Z2011-04-17T18:39:26Z@Georges Elencwajg that's exactly what I meant. Sorry for the terrible miscommunication. I've added this to the answer, with all the hypotheses clearly stated. by The Cheese Stands AloneThe Cheese Stands Alone2011-04-17T18:37:38Z2011-04-17T18:37:38ZIn this case, the proof I had in mind is by induction on n on the stronger statement that if $a$ and $b$ are of degree $k$, $m$, for $m,n$ relatively prime, the degree of $a+b$ is $km$. I was under the impression this was a well-known result, but it may be incorrect. If it is true, then Eisenstein gives that $x^{a_i} - p_i$ is the minimal polynomial for $sqrt[a_i]{p_i}$, so $sqrt[a_i]{p_i}$ is of degree $a_i$.
I suppose we need the much stronger condition that $gcd(a_i,a_j)=1$ for $i \not{=} j$ rather than just $gcd(a_1,\ldots,a_n)=1$ for this. I blame this obvious mistake on lack of sleep. by The Cheese Stands AloneThe Cheese Stands Alone2011-04-17T18:05:11Z2011-04-17T18:05:11ZPerhaps it will be helpful to enunciate my claim fully. Suppose $p_i$ are integers, $a_i$ natural numbers, for $i=1,\ldots,n$, such that for each $p_i$ there is a prime $q_i$ such that $q_i|p_i$ and $q_i ^2 \not{|} p_i$, and additionally that $gcd(ai,\ldots ,an)=1$. Then I claim that $\sqrt[a_1]{p_1} + \cdots + \sqrt[a_n]{p_n}$ has degree $a_1 \cdots a_n$ over $\mathbb{Q}$. All the other claims I made, the "proofs" I thought I had were flawed except in the case n=2. Please disregard them. by The Cheese Stands AloneThe Cheese Stands Alone2011-04-17T06:57:13Z2011-04-17T06:57:13ZScratch what I said above; it's not true unless each $p_i^{1/a_i}$ has minimal polynomial of degree $a_i$. This holds in the case that some prime q divides $p_i$, but $q^2$ doesn't divide $p_i$, by the Eisenstein Criterion. With more advanced arguments, a little bit stonger statements can be made.
It fails in the case where we take $\sqrt{4}$. Another similar case is that the kth root of unity $e^{i2 \pi /k}$ satisfies the polynomial $a^{k−1}+\cdots+x+1$, which is of degree k−1. But when $p_i^{1/a_i}$ is of degree $a_i$ over $\mathbb{Q}$, the above should hold. by The Cheese Stands AloneThe Cheese Stands Alone2011-04-17T03:00:45Z2011-04-17T03:00:45ZThere are a good number of cases where your statement will hold, but it's not true in general. If the $a_i$ and the $p_i$ aren't related in any obvious way, it's probably true for small values of n, but I'd still check. Wolfram alpha ( can compute the minimal polynomials in sufficiently small examples, and most computational algebra engines can do it for arbitrary numbers of the form you want. by The Cheese Stands AloneThe Cheese Stands Alone2011-04-17T02:58:40Z2011-04-17T02:58:40ZI'm not totally sure what having $gcd(a_n,p_n)=1$ gives you (or conditions regarding pairs $(a_n,p_n)$), but if you have $gcd(a_1,...,a_n)=1$ then your statement holds. If $gcd(p_1, \ldots ,p_n)=1$ and $a_1= \cdots = a_n$, it should also hold. I'm not sure about the general case for $gcd(p_1, \ldots ,p_n)=1$, though I wouldn't be surprised if your statement held then. by The Cheese Stands AloneThe Cheese Stands Alone2011-04-01T05:30:44Z2011-04-01T05:30:44ZIf all you need is an estimate, I'd run a Monte Carlo code. Something like:
1) Randomly generate a position
2) Check if it's a valid position or not
3) Check under what rotations, reflections the position is invariant.
4) Repeat
Once you have a lot of trials (a computer can easily do a few billion), sum the reciprocals of the numbers from step 3 of those positions which were valid. Then estimate from this the probability that a randomly chosen configuration fits your criteria, and multiply by the number of positions (3^125), for a decent estimate. by The Cheese Stands AloneThe Cheese Stands Alone2011-03-29T00:54:33Z2011-03-29T00:54:33ZI'll attempt to justify some of why traces are important, though not solely from a physics perspective. The trace of a matrix $M$ comes naturally (as does the determinant) from the characteristic polynomial $p(\lambda) = \det(\lambda I-M)$. Namely, for any algebraically closed field, it is the sum of the roots of $p(\lambda)$ counted with multiplicity (the determinant is the product). This is clearly coordinate-independent and a rather fundamental quantity. The usual definition lacks any intuition, but is more useful for generalizing to arbitrary matrix rings and for efficient computation. by The Cheese Stands AloneThe Cheese Stands Alone2011-03-28T12:08:34Z2011-03-28T12:08:34ZFrom my limited experience, these types of geometry courses are not generally taught for math majors. Typically, they are more directed towards math education majors or engineers who are trying to get a math minor. Non-euclidean geometries are certainly mentioned occasionally, but are not always explored with any depth.
The reason for this is that geometry isn't totally necessary anymore. It would be difficult to do anything serious without, say, knowing what a group is, but geometry has become a niche topic. Graduate students typically learn some advanced geometry, but undergrads rarely do. by The Cheese Stands AloneThe Cheese Stands Alone2011-03-27T19:40:38Z2011-03-27T19:40:38ZI'm sure the solution is known for the 2D problem, but I can't find a source for the solution at the moment. I'm fairly sure the maximum achieved by the 2D analogue of the conjectured set, but not totally sure.
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Hi, This morning I began working on my math assignment on the topic Intermediate algebra. I am currently not able to complete the same since I am unfamiliar with the fundamentals of binomial formula, decimals and quadratic formula. Would it be possible for anyone to assist me with this?
Algebrator is one of the best resources that can render help to people like you. When I was a novice, I took support from Algebrator . Algebrator covers all the basics of Remedial Algebra. Rather than using the Algebrator as a step-by-step guide to solve all your homework assignments, you can use it as a tutor that can offer the basics of dividing fractions, multiplying matrices and binomials. Once you understand the basics, you can go ahead and solve any tough assignments on Algebra 2 in no time.
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Algebrator is a user friendly product and is surely worth a try. You will find lot of interesting stuff there. I use it as reference software for my math problems and can swear that it has made learning math much more enjoyable.
I'm sorry. I should have included the connection our first time around: I don't have any knowledge about a test copy, but the recognized sellers of Algebrator , as opposed to some suppliers of imitation software, put up an entire satisfaction guarantee. Hence, you can order the official copy, test the package and send it back if one is not gratified by the performance and functionality. Even though I think you are gonna love this program, I am very interested in learning from anyone should there be something for which the software doesn't excel. I don't desire to recommend Algebrator for something it cannot do. Only the next one discovered will likely be the first one!
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mathlab Mystatlab Student Access Kit For Ad Hoc Valuepacks
Prealgebra : An Integrated Approach
Student Solutions Manual for Prealgebra An Integrated Approach
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Summary
This book helps to better prepare students for higher-level math courses by integrating basic algebraic concepts early and continuing to revisit those concepts throughout. The friendly composition and the many pedagogical features are designed to help with student comprehension. The modern, relevant applications increase student motivation by immersing them in truly genuine and realistic mathematical situations.
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Math Made Nice - N - Easy, Book #1 - 99 edition
Summary: Almost everyone needs some math in everyday life, at work, in a career, for study, for shopping, for paying bills. dealing with a bank, in sports, using credit cards, etc. This series of books simplifies the learning, understanding, and use of math, making it non- threatening, interesting, and even fun. The series develops math skills in an easy-to-follow sequence ranging from basic arithmetic to pre-algebra and beyond. These books draw on material developed by the U...show more.S. Government for the education of government personnel with limited math and technical backgrounds. Volume I covers number systems, sets, integers, fractions, and decimalsVery good condition - book only shows a small amount of wear!
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An introduction to variables. The number-line is labeled and the different types of numbers are defined. Students manipulate simple equations, and practice constructing equations based on real world applications.
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FX Draw - Professional Quality Mathematical Diagrams
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books.google.ch - This... Geometry
Computational Geometry:
This and techniques from computational geometry are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. For students this motivation will be especially welcome. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement. All the basic techniques and topics from computational geometry, as well as several more advanced topics, are covered. The book is largely self-contained and can be used for self-study by anyone with a basic background in algorithms. In the second edition, besides revisions to the first edition, a number of new exercises have been added.
Bewertungen von Nutzern
Review: Computational Geometry: Algorithms and Applications
Review: Computational Geometry: Algorithms and Applications
Nutzerbericht - Willy Van den driessche - Goodreads
Beauty is the first test. This is a very beautiful book (form) with a beautiful contents. The book explains in a very throrough way some of the fundamental algoritms in "computational geometry". You ...Vollständige Rezension lesen
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Math Level L: Logarithms, Calculus
Level L marks the beginning of calculus. Students begin by studying logarithmic functions, followed by basic differentiation and definite and indefinite integration. The level concludes with an analysis of applications of integration, including areas, volumes, velocity and distance.
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Specification
Aims
The programme unit aims to discuss ordinary differential equations with applications to physical situations using Matlab to illustrate some of the ideas and methods.
Brief Description of the unit
The unit will be in 3, approximately 11 lecture sections. The first part on first order ordinary differential equations; the second part on motion in space; and the final part on second order ordinary differential equations.
Learning Outcomes
On completion of this unit successful students will be able to solve first order and second order linear problems and first order separable equations analytically. Use substitution methods and power series methods to find solutions. Be able to investigate solutions using direction fields and Euler's method. Have used Matlab as a mathematical tool and used differential equations to solve problems in mechanics and other applications.
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MAT-115 Intermediate Algebra
The course affords a transition between elementary algebra and college algebra, and provides a solid foundation in the basic algebraic concepts, including linear equations and inequalities, quadratic equations, graphing, rational expressions, functions, exponents, radicals, parabolas and systems of linear equations.
Advisory: It is advisable to have completed elementary algebra. Students are permitted to have scientific (nongraphing) calculators in examinations. Programmable calculators are not permitted in examinations.
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Algebra 2 - 03 edition
Summary: Applications with "Real" Data Since the graphics calculator is recommended, students experience excitement as they use "real" data in Algebra 2. Students investigate and extend relevant applications through engaging activities, examples, and exercises. Graphics Calculator Technology In Algebra 2, the graphics calculator is an integral tool for presenting, understanding, and reinforcing concepts. To assist student...show mores in using this tool, a detailed keystroke guide is provided for each example and activity at the end of each chapter. Functional Approach Algebra 2 examines functions through multiple representations, such as graphs, tables, and symbolic notation. Working with transformations (investigating how functions are related to each other and their parent functions) prepares students for advanced courses in mathematics by developing an extensive, workable knowledge of functions. ...show less
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Georgia's mathematics standards have been designed to achieve a balance among concepts, skills, and problem solving. The curriculum stresses rigorous concept development, presents realistic and relevant tasks, and keeps a strong emphasis on computational skills. At all grades, the curriculum encourages students to reason mathematically, to evaluate mathematical arguments both formally and informally, to use the language of mathematics to communicate ideas and information precisely, and to make connections among mathematical topics and to other disciplines. Grade-level information, including Standards and resources, can be found at
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10 flagrant grammar mistakes that make you look stupidVersion 1.0 May 23, 2006By Jody Gilbert These days, we tend to communicate via the keyboard as much as we do verbally. Often, we're in a hurry, quickly dashing off e-mails with typos, grammatical sho
I. Relational Programming (Ch 9) A. Motivation- MOTIVATIONS FOR RELATIONAL PROGRAMMING - 1. programming is difficult, expensive Why is programming so hard? What approaches might solve this problem? What are the steps in building a computer system
Bacteria Are Beautifulby Dianne K. NewmanAbove: The widespread use of antibacterial chemicals in common household products could be doing more harm than good (Annals of Internal Medi-cine, 2004, 140, 321329).Below: In contrast to the bacteriophobia of
Transparencies to accompany Rosen, Discrete Mathematics and Its ApplicationsSection 9.4Section 9.4 Connectivity We extent the notion of a path to undirected graphs. An informal definition (see the text for a formal definition): There is a path v0, v1, v
Transparencies to accompany Rosen, Discrete Mathematics and Its ApplicationsSection 9.3Section 9.3 Representing Graphs and Graph Isomorphism We wish to be able to determine when two graphs are identical except perhaps for the labeling of the vertices. W
Transparencies to accompany Rosen, Discrete Mathematics and Its ApplicationsSection 9.2Section 9.2 Graph Terminology and Special Types of Graphs Undirected Graphs Definition: Two vertices u, v in V are adjacent or neighbors if there is an edge e between
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Learning basic mathematics is easy and engaging with this combined text/workbook! BASIC COLLEGE MATHEMATICS is infused with Pat McKeague's passion for teaching mathematics. With years of classroom experience, he knows how to write in a way that you will understand and appreciate. McKeague's proven "EPAS" approach (Example, Practice Problem, Answer, and Solution) moves you through each new concept with ease while helping you break up problem solving into manageable steps. Real-world applications in every chapter of this user-friendly book highlight the relevance of what you are learning. And studying is easier than ever with the book's multimedia learning resources, including a Digital Video Companion CD-ROM and access to Basic MathematicsNOW, a personalized online learning companion.
Want the streamlined approach to statistics? ESSENTIALS OF STATISTICS FOR BUSINESS AND ECONOMICS, ABBREVIATED EDITION explains updated statistical methods in simple ways. This Homework Edition isPat McKeague's eighth edition of INTEMEDIATE ALGEBRA is the book for the modern student like you. Like its predecessors, the eighth edition is clear, concise, and patient in explaining the concepts. ...
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Book Description: Practical Mathematics utilizes a problem-solving approach to teaching mathematics. The text includes over 2,000 problems many of which require the use of a calculator that cover arithmetic, algebra, geometry, and trigonometry. Both English and metric units are used throughout the text.
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Mathematics
Subjects
Tutored
What Is Calculus?
By this point, you should be familiar with using functions and solving equations (and systems of equations) involving real numbers using the techniques you learned in Algebra. With Calculus we are allowed to do things that we are not allowed to do in Algebra using two mathematical constructs in particular: infinity, which is larger than every number, and the infinitesimal, which is smaller than every number.
Calculus I, AP Calculus AB
You remember from Algebra that the slope of a line can be determined using the change in y divided by the change in x. The slope is simply one number that tells you the direction the line is going. However, other functions do not have a definite slope. You can tell if a function is increasing or decreasing by looking at a graph, and if you were to pick two values of x, say a and b, you could find what's known as the average rate of change between a and b just by finding the change in y and dividing it by the change in x.
Sometimes, however, we want to find something called the instantaneous rate of change, or the direction in which a function is going at one value of x. We cannot use the average rate of change formula for one point, because the changes in y and in x are both 0, and we are not allowed to divide by 0. What we can do, however, is look at the average rate of change from a to some variable b, and see what happens to the average rate of change as we move b closer to a. As b moves closer to a, we see that both the changes in y and in x are approaching 0, and we can see what happens to that rate the closer they get. This process is called differentiating a function, or taking the derivative of a function.
Another thing you will learn to do in Calculus I is to find the area bound by a function. You know how to find the area of a rectangle by multiplying it's length by width, but if you want to find the area bound between a curve and the x-axis, you will learn to use a process called Riemann integration.
Calculus II, AP Calculus BC
Calculus II continues what you've learned in Calc I, and in most cases it will cover three topics:
Techniques of integration, where you will learn how to integrate more complicated functions, and how to choose what technique to use by looking at a function.
Intro to differential equations, where you learn to make mathematical models using a differential equation, or an equation with a derivative in it.
Infinite sequences and series. An infinite sequence is an ordered list of numbers, and here you will learn how to tell whether or not a sequence converges. An infinite series is the sum of all the terms in a sequence, and you will also learn how to tell whether or not a series will converge. You will also learn how to represent some functions, such as sin(x), as a special kind of infinite series called a power series.
Calculus III
Calc III, often called Multivariable Calculus or Vector Calculus, involves applying the techniques learned in Calc I and II to functions of more than one variable. You will learn how to take a partial derivative, which differentiates a multivariable function with respect to only one variable, and you will learn multiple integration, which involves integrating a multivariable function with
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The Mathematics Curriculum
This collection was produced by the Schools Council Project: The Mathematics Curriculum - A Critical Review. The Project was initiated by the Mathematics Committee of the Schools Council as a result of letters having been received from teachers asking for guidance on the vast amount of new mathematical literature which had been produced for schools during the 1960s. The Project was set up in 1973 and was based at the Shell Centre for Mathematical Education at Nottingham University.
The fundamental aim of the Project was to help teachers to perform their own critical appraisal of the then current mathematics syllabuses and teaching apparatus for secondary school students in the 11 to 16 age range, with the objective of making, for them, optimal choices.
Although the Project was not intended to be an exercise in curriculum development, it was almost inevitable and certainly desirable that a review of existing syllabuses should lead to a consideration of the possibility of a synthesis of 'modern' and 'traditional'. The authors believed that such a synthesis was possible and, indeed, sorely needed. So, although they did not attempt to spell out an optimum syllabus, they tried to identify the important ideas and skills which should be represented at school, and to show how so-called modern and traditional topics were related.
It was hoped that one of the lessons which would emerge was that the two could be integrated in a unified presentation of mathematics and its applications. The project leaders also felt that the debate surrounding numeracy might lead to syllabus revision and if so, it must be informed by sound mathematical and pedagogical considerations, to which end these books were devoted Mathematics Curriculum book, first published in 1978, begins by giving a short account of what 'algebra' means, to inform those readers whose mathematical training did not take in the most recently included topics of school syllabuses, and to refresh the memories of those who were exposed to these ideas. Objectives…
This Mathematics Curriculum book, first published in 1978, examines 'combinatorics', a new term at the time that needed definition. Two aspects were considered, the first being permutations and combinations and the second being a possible set-up (or configuration) and asking if examples of such configurations exist and…
This book first published in 1977 is about graphs, their drawing, their interpretation, their development and their use. It discusses the teaching of graphs from their early introduction and as far as the beginnings of integral and differential calculus. The book also places the teaching of graphs in an historical context as it reviews…
This Mathematics Curriculum book, first published in 1977, put the teaching of geometry in the historical context of the time. In particular, the author sets out to show that the introduction of modern transformation geometry does not rule out the teaching of the more traditional Euclidean-type proofs, and indicates some of the many…
This Mathematics Curriculum book, first published in 1977, intended to act as an aid and catalyst in co-operation between teachers of mathematics and teachers of other subjects which use mathematics. It looks at areas of applied mathematics that demonstrate its usefulness and are of genuine interest to other subject areas. A knowledge…
This Mathematics Curriculum book was first published in 1978 in a response to the many changes in the mathematics curriculum and resources available. It was intended to help teachers make sense of the changes and choose appropriate resources. The book provides a historical context for mathematics education at the time as it reviews…
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Khan Academy Video Course: Calculus
Description
Welcome to calculus, the study of change. Through limits, functions, derivatives, integrals, and infinite series, calculus provides systems to examine change and make predictions based on what we can calculate. This course covers topics that you'd find in first and second-semester calculus courses at the college level, and serves as a gateway to advanced mathematical analysis. A solid understanding of pre-calculus is essential for success. Good luck!
Note that given the length of these lessons, you may want to adjust your settings to receive one or two lessons a week.
Opening Lines (Experimental)
Today's Calculus lesson (in video) from the Khan Academy is: Introduction to Limits: To view other Khan Academy videos, you can find them at their website here: Enjoy! P.S. Note that given the length of these lessons, you may want to adjust your settings to receive one or two lessons a ...
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Features: - Outcomes at the start of every chapter - A dynamic full colour design that clearly distinguishes theory, examples, exercises, and features - Carefully graded exercises with worked examples and solutions linked to each - Cartoons offering helpful hints - Working mathematically strands that are fully integrated. These also feature regularly in challenging sections designed as extension material which also contain interesting historical and real life context - A Chapter review to revise and consolidate learning in each chapter - Speed skills sections to revise and provide mental arithmetic skills - Problem solving application strategies with communication and reasoning through an inquiry approach - a comprehensive Diagnostic test providing a cumulative review of learning in all chapters, cross referenced to each exercise - Integrated technology activities - Literacy skills develop language skills relevant to each chapter - Fully linked icons to accompanying CD-ROM.
The student CD-ROM accompanying this textbook can be used at school or at home for further explanation and learning.
Each CD-ROM contains: - Interactive diagnostic text – perfect revision for all Stage 4 work. The regenerative nature of the program allows for an almost limitless number of varied tests of equal difficulty. This test can be used prior to commencing Stage 5 work. - Dynamic geometry activities using WinGeom and Cabri software for student investigations - Using technology with formatted Excel spreadsheets - Full textbook with links to the above.
A.Kalra and J. Stamell
Connections Maths is a comprehensive, full colour, 6-book series that meets all the requirements of the new Years 7–10 course. The series will engage, motivate and support students of all abilities. The page design, the vibrant use of colour, and the range of photos and cartoons make each page an interactive learning experience. Each textbook is accompanied by a student CD-ROM.
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The
Online Encyclopedia and Dictionary
Algebra
Algebra is a branch of mathematics of Arabian origin or transmission which may be roughly characterized as a generalization and extension of arithmetic, in which symbols are employed to denote operations, and letters to represent number and quantity; it also refers to a particular kind of abstract algebra structure, the algebra over a field.
Greek Mathematician Diophantus around 200 AD, often referred to as the "father of algebra", is best known for his Arithmetica , a work on the solution of algebraic equations and on the theory of numbers.
The word "Algebra" itself comes from the name of the treatise first written by a persian mathematician Al-Khwarizmi 700 AD, who wrote a treatise titled: Kitab al-mukhtasar fi Hisab Al-Jabr wa-al-Moghabalah meaning The book of summary concerning calculating by transposition and reduction. The word Al-jabr (which Algebra is derived from) means "reunion", "connection" or "completion".
In advanced studies axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural topology compatible with algebraic structure. The list includes
Cubic equations
Cubic equations are written in the form y = ax3 + bx2 + cx + d. In this form, there are three x-intercepts. When graphed, the line will start going up, then curve to go down, then switch again to go up. (the opposite can occur with negative variables)
a is the coefficient of the variable cubed
b is the coefficient of the variable squared
c is the coefficient of the variable
d is the non-variable
Exponential equations
Exponential equations are written in the form y = mx + b.
Factoring trinomials
Simple factoring
Trinomials are algebraic expressions consisting of three unlike terms, such as x2 + 3x + 2. They can be factored using the "FOIL" technique. You factor the expression by using two sets of parentheses, each consisting of two terms, where the first, outside, inside, and last numbers of both sets multiplied together and added equal the trinomial. E.g.,
The last numbers in each set of parenthesis have another relationship. When multiplied together, they always equal the last number (3 times 2 equals 6), and when added, they equal the coefficient of the variable (3 plus 2 equals 5). The coefficient is the number in front of the variable that you multiply it by. This is because they're both multiplied by the variable, and then added.
Two variables
Sometimes, you get expressions such as:
3x2 + 8xy + 4y2.
In this situation, the factored form will look like:
(3x + 2y)(x + 2y).
3x times x is 3x2, 3x times 2y is 6xy, 2y times x is 2xy, and 2y times 2y is 4y2. This time, the coefficients of x have to be multiplied with the coefficient of x2, and same with x.
Symbols
Depending on whether the numbers are added or subtracted, you may need to use different symbols in the parenthesis.
If you add the mx and add the b, the symbols are both plus.
If you add the mx and subtract the b, the symbols are one plus and one minus.
If you subtract the mx and add the b, the symbols are both minus
If you subtract the mx and subtract the b, the symbols are one plus and one minus.
Symbolic method
The symbolic method is a way to figure out a variable when it's on both sides of the equation. E.g.,
3x + 25 = 5x + 5
The first step is to isolate the variable. By subtracting 3x from both sides, you get 25 = 2x + 5.
The second step is to get only the variable on one side. To do this, you subtract 5 from both sides to get 20 = 2x.
The last step is to get just 1 x. Divide both sides by the coefficient, in this case 2, and you have 10 = x.
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I mean I would definitely take the class because I like math, but if it really isn't going to help me in my major or reinforce my understanding of engineering, I shouldn't be wasting $1600 on taking it.
That's what I mean. If you like math then go for it. It's not required for engineering. Be sure that you can do well in it. This is abstract algebra, not regular algebra. If you have not taken differential equations and linear algebra then don't do it. This is a tough math sequence for math majors.
If I were to take this class, it would be ideal to do it during the summer, when I have a lot of time to study for the class. Sorry about the confusion: I am attending UCLA in the fall, but am asking about 110 at berkeley.
Assuming that you have completed all of the freshman and sophomore math courses needed for your engineering major, you effectively have slots for six quarters of additional free electives while you are at UCLA. You can certainly use some of them to take any additional math courses of interest -- you won't be taking any additional workload since the additional math course would substitute for a required math course that you have already completed.
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Descrição do produto
Descrição do produto
"Math, Better Explained" is a clear, intuitive guide to math topics essential for high school, college and beyond. Whether you're a student, parent, or teacher, this book is your key to unlocking the aha! moments that make math truly click -- and make learning enjoyable.
The book intentionally avoids mindless definitions and focuses on building a deep, natural intuition so you can integrate the ideas into your everyday thinking. Its explanations on the natural logarithm, imaginary numbers, exponents and the Pythagorean Theorem are among the most-visited in the world.
The book is written as the author wishes math was taught: with a friendly attitude, vivid illustrations and a focus on true understanding. Learn right, not rote!
Selected testimonials:
"I have several books on calculus (Calculus for Dummys, Math for the Millions, etc. etc. - never was able to read them) but your explanation is what I have needed all these years." - D. Hogg, Former Principal
"This is a great explanation! I am 49 years old and have never known what e is all about. It is thanks to your article that I get it and now can explain it to my son who is 13 years old..." - C. Dhaveji
"I've been following you for nearly two years...I find the intuitive approach to the subject and lucid writing unparalleled." - D. Ezell
About the Author
Kalid Azad graduated from Princeton University and has been writing professionally for over a decade, from chapters in the best-selling "How to Program" textbooks (from Deitel, Inc.) to technical whitepapers for Microsoft, Corp. Kalid has tutored math since high school (99% percentile for SAT/GRE/GMAT) and is enamored with finding the clearest, most intuitive insights on seemingly-complicated topics.
4.0 de 5 estrelasAuthor helps demystify concepts and paves way for a better understanding of math.10 de dezembro de 2011
Por Romanos Piperakis - Publicada na Amazon.com
A lot of math concepts come from the need to solve everyday problems but unfortunately are taught in the classroom in distilled, abstract form that transcends the specific examples it was originally invented for. As such, the intuition behind the abstract concepts is sacrificed at the expense of generalization.
The author makes the very useful and widely overlooked point that the intuition behind abstract concepts or theorems can be recovered by reversing the direction from abstract to specific and explaining how one can arrive at applied/concrete instances from the pure/distilled forms.
As such the author is not only helping the reader to demystify math (making it more accessible to anyone with common sense) but paves the way for developing modeling skills where those concepts can be successfully applied to real-world problems.
This is a very worthwhile undertaking and I would personally love to see the author extend this effort to more concepts e.g. determinants, adjoint/hermitian matrices, regression and function approximation etc. This type of explanations could serve as supplementary math texts or - in the form of wiki where others could share the burden of providing their intuition - as a very valuable reference.
Only gave 4 stars for the somewhat random selection of examples to be explained but would recommend this book without any reservations.
6 de 6 pessoas acharam a avaliação abaixo útil
5.0 de 5 estrelasA fantastic explanation of the math essentials9 de dezembro de 2011
Por Antonio Cangiano - Publicada na Amazon.com
I love the quality and teaching approach! The writing is clear, friendly and builds a deep understanding with excellent diagrams. The author takes your mind beyond rote details and makes you understand the essence of each math concept -- even experienced math fans will come away with valuable insights. For example, the metaphors about e, i and pi make it easy to understand Euler's formula, considered one of the most beautiful (and baffling) identities.
I've reviewed many books while running math-blog.com: very few are this approachable and entertaining while helping you truly learn. Highly recommended!
4 de 4 pessoas acharam a avaliação abaixo útil
5.0 de 5 estrelasExcellent book!9 de dezembro de 2011
Por Brendan J McWilliams - Publicada na Amazon.com
I followed this series on his website and was always impressed that it focused on the concepts rather than rote memorization or acceptance of ideas blindly.
I bought it and was very happy with it. I only hope he does more of these. Possibly with programming or other field.
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Course Description
(P)
This course covers basic arithmetic, introductory concepts in algebra,
and problem solving techniques. Specific topics include addition,
subtraction, multiplication and division of signed numbers, percentage, and
applications of these skills. The course introduces algebraic concepts,
including algebraic operations of polynomials, solving equations, formulas,
and an introduction to solving word problems. (Prerequisite: MATH C020
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Buy now
Detailed description * Highlights from the history of geometry are intertwined with explanations on how to read and write proofs. * This is the first book to present the works of Euclid and Hilbert in addition to other geometers in chronological order, all in an effort to show how the subject matter developed over time. * Hints and both partial and complete solutions are included at the end of the book as an aid for selected exercises. * An important contribution to the teaching of geometry, this book proves that learning to read and write proofs is a crucial aspect of the subject. * This book develops ideas with careful attention to logic and follows the development of the field through time. * An Instructor's Solutions Manual is available upon request.
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Higher
Higher revision classes run most evenings after school. Now that the prelims are completed, which will be returned before the February break, you need to identify your own action plans and focus your revision in a structured manner. Target setting discussions will also be happening in class to help you work out which areas you should be looking at again. We will start straight after prelim leave working towards the second NAB on addition formula and circles, before starting Unit 3 topics
Remember, it is a fast paced course and it's easier to keep up than catch up!
Any after school revision classes should suppliment your work at home and in lessons. Please start revision early and use for past paper questions. These also come with marking schemes so you can check your work as you go along. You all should have access to SCHOLAR too so can you please make use of all the materials you have been given in class and via the internet.
The pace will continue to be fast, so be prepared to keep up with your class work and homework exercises.
National Course Specification – General Information
Mathematics (Higher)
This course consists of three mandatory units as follows:
D321 12
Mathematics 1 (H)
1 credit (40 hours)
D322 12
Mathematics 2 (H)
1 credit (40 hours)
D323 12
Mathematics 3 (H)
1 credit (40 hours)
In common with all courses, this course includes 40 hours over and above the 120 hours for the component units. This may be used for induction, extending the range of learning and teaching approaches, support, consolidation, integration of learning and preparation for external assessment. This time is an important element of the course and advice on its use is included in the course details.
.
RECOMMENDED ENTRY
While entry is at the discretion of the department, candidates will normally be expected to have attained one of the following:
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MSCC Lab Assistant:
Course Description
Development and application of numerical methods and algorithms
to problems in the applied sciences and engineering. Applied
linear algebra and introduction to numerical methods. Emphasis
on use of conceptual methods in engineering, mathematics, and
science.
Other references:
There are many other "numerical analysis" or "numerical methods"
books that cover similar material. If you are having trouble understanding
a concept, look through some other books in the library.
Learning Objectives and Instructor Expectations
The main goal of the course is to introduce approximate numerical methods
for
solving mathematical equations that cannot be solved exactly by
hand. Such problems arise constantly in science, engineering, finance,
computer graphics, and elsewhere.
We will study several basic
numerical algorithms, how to implement them, and how to analyze their
behavior mathematically.
You should also become adept at using the MATLAB language for
numerical
problem solving. MATLAB has many built-in functions for solving
particular problems and you will learn how to use these.
You should also gain an understanding of how they work, why they sometimes
don't work, and how to use them intellegently.
Schedule and Homework
Follow links in the table below to obtain a copy of the homework in
PostScript (.ps) or
Adobe
Acrobat (.pdf) format, and also for associated scripts or data files.
For additional information regarding viewing and printing the
homework and solution sets,
click here.
Grading
There will be 8 homework assignments. These will usually be due on Fridays
in class, but check the schedule above for due dates.
Each homework will be worth 25 points and the
lowest homework score will be dropped, so 175 points are possible on
homework.
There will be two midterms, each worth 75 points,
The course will culminate with a final project that will be due during exam
week, worth 50 points.
A total of 375 points are possible in the course.
You may view your homework and exam
grades on-line.
Before doing so for the first time, you must
request a password.
When typing in your student number, delete any leading zeros (e.g. if your
student number is 0012345 type in 12345).
Tutorials
Instructions on using
PCTeX and a sample file for homework, if you want to learn this
mathematical typesetting system for doing your homework.
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Exam duration:
Aid:
Evaluation:
Qualified Prerequisites:
General course objectives:
To provide the student with a solid framework for understanding and applying a number of geometric shapes and techniques as they are used in engineering and architectural design contexts as exemplified below.
For ship building engineers: Propeller geometries via deformations of standard profiles and the construction of ship hulls.
For architectural engineers: Classical geometric concepts and basic operations for shape design and form description in plane and space.
To apply 2x2 and 3x3 matrices and their properties to analyze simple geometric constructions in plane and space and thereby obtain and practice the essential understanding of coordinate transformation techniques.
To define and to calculate precise modifications of a given geometric object.
To apply computer experiments as an integrated part of the course for illustrations, learning, and calculations.
Learning objectives:
A student who has met the objectives of the course will be able to:
Calculate on a vectorial basis the area and volume
Apply matrix calculus to construct and and analyze deformations of basic objects and explain the induced change in area and volume
Find parametrizations of simple geometric objects in plane and space
Calculate and explain the notions of area and volume for parametrized objects
Apply simple parametrizations or other representations to construct triangulations of surfaces and domains in space and compare the respective areas and volumes
Apply basic kinematic concepts to analyse simple motions in the plane
Calculate and explain the notions of arclength and curvature for curves in the plane
Apply extrudition, offsetting, and projection to construct new geometric objects from old ones
Course literature:
Remarks:
The course provides a basic foundation for the understanding of the geometric operations which are applied in FEM modelling, machine element design, architectural engineering, and sculptural design
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From time to time, not all images from hardcopy texts will be found in eBooks, due to copyright restrictions. We apologise for any inconvenience.
Description
For a one-semester or two-quarter calculus course covering multivariable calculus for mathematics, engineering, and science majors.
Briggs/Cochran is the most successful new calculus series published in the last two decades. The authors' decades of teaching experience resulted in a text that reflects how students generally use a textbook–i.e., they start in the exercises and refer back to the narrative for help as needed. The text therefore builds from a foundation of meticulously crafted exercise sets, then draws students into the narrative through writing that reflects the voice of the instructor, examples that are stepped out and thoughtfully annotated, and figures that are designed to teach rather than simply supplement the narrative. The authors appeal to students' geometric intuition to introduce fundamental concepts, laying a foundation for the rigorous development that follows.
To further support student learning, the MyMathLab course features an eBook with 700 Interactive Figures that can be manipulated to shed light on key concepts. In addition, the Instructor's Resource Guide and Test Bank features quizzes, test items, lecture support, guided projects, and more.
This book covers chapters multivariable topics (chapters 9—15) of Calculus for Scientists and Engineers: Early Transcendentals, which is an expanded version of Calculus: Early Transcendentals by the same authors.
Table of contents
9. Sequences and Infinite Series
9.1 An overview
9.2 Sequences
9.3 Infinite series
9.4 The Divergence and Integral Tests
9.5 The Ratio, Root, and Comparison Tests
9.6 Alternating series
10. Power Series
10.1 Approximating functions with polynomials
10.2 Properties of Power series
10.3 Taylor series
10.4 Working with Taylor series
11. Parametric and Polar Curves
11.1 Parametric equations
11.2 Polar coordinates
11.3 Calculus in polar coordinates
11.4 Conic sections
12. Vectors and Vector-Valued Functions
12.1 Vectors in the plane
12.2 Vectors in three dimensions
12.3 Dot products
12.4 Cross products
12.5 Lines and curves in space
12.6 Calculus of vector-valued functions
12.7 Motion in space
12.8 Length of curves
12.9 Curvature and normal vectors
13. Functions of Several Variables
13.1 Planes and surfaces
13.2 Graphs and level curves
13.3 Limits and continuity
13.4 Partial derivatives
13.5 The Chain Rule
13.6 Directional derivatives and the gradient
13.7 Tangent planes and linear approximation
13.8 Maximum/minimum problems
13.9 Lagrange multipliers
14. Multiple Integration
14.1 Double integrals over rectangular regions
14.2 Double integrals over general regions
14.3 Double integrals in polar coordinates
14.4 Triple integrals
14.5 Triple integrals in cylindrical and spherical coordinates
14.6 Integrals for mass calculations
14.7 Change of variables in multiple integrals
15. Vector Calculus
15.1 Vector fields
15.2 Line integrals
15.3 Conservative vector fields
15.4 Green's theorem
15.5 Divergence and curl
15.6 Surface integrals
15.6 Stokes' theorem
15.8 Divergence theorem
Features & benefits
Topics are introduced through concrete examples, geometric arguments, applications, and analogies rather than through abstract arguments. The authors appeal to students' intuition and geometric instincts to make calculus natural and believable.
Figures are designed to help today's visually oriented learners. They are conceived to convey important ideas and facilitate learning, annotated to lead students through the key ideas, and rendered using the latest software for unmatched clarity and precision.
Comprehensive exercise sets provide for a variety of student needs and are consistently structured and labeled to facilitate the creation of homework assignments by inspection.
Review Questions check that students have a general conceptual understanding of the essential ideas from the section.
Basic Skills exercises are linked to examples in the section so students get off to a good start with homework.
Applications present practical and novel applications and models that use the ideas presented in the section.
Additional Exercises challenge students to stretch their understanding by working through abstract exercises and proofs.
Examples are plentiful and stepped out in detail. Within examples, each step is annotated to help students understand what took place in that step.
Quick Check exercises punctuate the narrative at key points to test understanding of basic ideas and encourage students to read with pencil in hand.
The MyMathLab course for the text features the following:
More than 7,000 assignable exercises provide you with the options you need to meet the needs of students. Most exercises can be algorithmically regenerated for unlimited practice.
Learning aids include guided exercises, additional examples, and tutorial videos. You control how much help your students can get and when.
700 Interactive Figures in the eBook can be manipulated to shed light on key concepts. The figures are also ideal for in-class demonstrations.
Interactive Figure Exercises provide a way for you make the most of the Interactive Figures by including them in homework assignments.
A "Getting Ready for Calculus" chapter, with built-in diagnostic tests, identifies each student's gaps in skills and provides individual remediation directly to those skills that are lacking.
Ready-to-Go Courses designed by experienced instructors miminize the start-up time for new MyMathLab users.
Guided Projects, available for each chapter, require students to carry out extended calculations (e.g., finding the arc length of an ellipse), derive physical models (e.g., Kepler's Laws), or explore related topics (e.g., numerical integration). The "guided" nature of the projects provides scaffolding to help students tackle these more involved problems.
The Instructor's Resource Guide and Test Bank provides a wealth of instructional resources including Guided Projects, Lecture Support Notes with Key Concepts, Quick Quizzes for each section in the text, Chapter Reviews, Chapter Test Banks, Tips and Help for Interactive Figures, and Student Study Cards.
This book is an expanded version of Calculus: Early Transcendentals by the same authors. It contains an entire chapter devoted to differential equations and additional sections on other topics (Newton's method, surface area of solids of revolution, and hyperbolic functions). Most sections also contain additional exercises, many of them mid-level skills exercises.
Author biography
William Briggs has been on the mathematics faculty at the University of Colorado at Denver for twenty-three years. He received his BA in mathematics from the University of Colorado and his MS and PhD in applied mathematics from Harvard University. He teaches undergraduate and graduate courses throughout the mathematics curriculum with a special interest in mathematical modeling and differential equations as it applies to problems in the biosciences. He has written a quantitative reasoning textbook, Using and Understanding Mathematics; an undergraduate problem solving book, Ants, Bikes, and Clocks; and two tutorial monographs, The Multigrid Tutorial and The DFT: An Owner's Manual for the Discrete Fourier Transform. He is the Society for Industrial and Applied Mathematics (SIAM) Vice President for Education, a University of Colorado President's Teaching Scholar, a recipient of the Outstanding Teacher Award of the Rocky Mountain Section of the Mathematical Association of America (MAA), and the recipient of a Fulbright Fellowship to Ireland.
Lyle Cochran is a professor of mathematics at Whitworth University in Spokane, Washington. He holds BS degrees in mathematics and mathematics education from Oregon State University and a MS and PhD in mathematics from Washington State University. He has taught a wide variety of undergraduate mathematics courses at Washington State University, Fresno Pacific University, and, since 1995, at Whitworth University. His expertise is in mathematical analysis, and he has a special interest in the integration of technology and mathematics education. He has written technology materials for leading calculus and linear algebra textbooks including the Instructor's Mathematica Manual for Linear Algebra and Its Applications by David C. Lay and the Mathematica Technology Resource Manual for Thomas' Calculus. He is a member of the MAA and a former chair of the Department of Mathematics and Computer Science at Whitworth University.
Bernard Gillett is a Senior Instructor at the University of Colorado at Boulder; his primary focus is undergraduate education. He has taught a wide variety of mathematics courses over a twenty-year career, receiving five teaching awards in that time. Bernard authored a software package for algebra, trigonometry, and precalculus; the Student's Guide and Solutions Manual and the Instructor's Guide and Solutions Manual for Using and Understanding Mathematics by Briggs and Bennett; and the Instructor's Resource Guide and Test Bank for Calculus and Calculus: Early Transcendentals by Briggs, Cochran, and Gillett. Bernard is also an avid rock climber and has published four climbing guides for the mountains in and surrounding Rocky Mountain National Park.
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Function Notation - A basic description of function notation and a few examples involving function notation. For more free math videos, visit In this video, we officially define a function. We discuss the difference between a relation and a function. We talk about...
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This consumer math course will show you how to use your basic math skills in real life situations such as buying items, budgeting your money, investing, and paying taxes. You will begin with a basic review of number skills then move on to numbers in jobs, salaries, taxes, insurance, and transportation costs.
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This guide is an informal and accessible introduction to plane algebraic curves. It also serves as an entry point to algebraic geometry, which is playing an ever-expanding role in areas ranging from biology and chemistry to robotics and cryptology.
By keeping the exposition simple and readily understandable, and by introducing abstract concepts with concrete examples and pictures, the book offers readers a lucid overview of the subject. It can also be used as the text in an undergraduate course on plane algebraic curves, or as a companion to algebraic geometry at the graduate level.
...there exist curves--many with very simple defining polynomials--that bend, twist and contort so much that in order to fit in the plane, they must have self-intersections and/or kinks. Such points are rare (accounting for their name "singularities"), but rare or not, questions arise:
What do curves look like around singularities?
Are some singularities easily understood, while others are more complicated?
How is their number and type related to the amount of twisting and contorting of the curve?
For a curve with singularities, what happens to Bézout's theorem?
For a curve with singularities, what happens to that remarkably simple genus formula?
Can you transform a curve with singularities into a curve without singularities?
Keith Kendig (Cleveland State University) is the author of two other MAA books: Conics and Sink or Float? Thought Problems in Math and Physics. He serves on the editorial boards of Mathematics Magazine and the Spectrum series of MAA books.
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Tags: wolfram.
Wolfram Mathematica 8.0 Windows | 926.5 MB What is Mathematica? Almost any job related to the calculation results, and that is what Mathematica is not-from building a web of hedge funds, - - download rapidshare appz megaupload fileserve mediafire warez li
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What Is Mathematica? Almost any workflow involves computing results, and that's what Mathematica does—from building a hedge fund trading website or publishing interactive engineering textbooks to developing embedded image recognition algorithms or - The
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Math 20F
MATLAB Assignments
This is the gateway page for the Math 20F MATLAB
assignments. This portion of Math 20F is an introduction to using
computer software for topics in linear algebra. The labs will
explore not
only how to use the software but also how linear algebra arises in
practice. The material will also give additonal
persepective and
add to topics covered in lecture.
There are three main packages for general purpose
scientific computation: Maple, Mathematica, and MATLAB, known as the
3Ms. While this course focuses solely on MATLAB, all three
packages
are similiar and if you know one, you can usually pickup another one
within an hour or two. Many students enrolled in Math 20F will
use one
of these packages (or more specialized software of a similiar nature)
heavily in their major and in future careers.
Basic information for using MATLAB on campus ACS
computers
can be found on the What
You Need To Know About MATLAB portion of 20F page, while due dates
for homeworks can be found on the MATLAB Assignment Due
Dates page. In general,
though, each lab is self-contained and can be completed without prior
MATLAB experience, although some assignments depend on commands learned
in previous assignments.
The following optional
labs expand upon many of the topics from class. Feel free to take
a look and see if any of the topics interest you. After each the
prerequisite needed to complete the lab is listed.
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Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
their education.
Math 375 Week 11.1 Introduction to GroupsIntroductionAbstract algebra is the study of structures that certain collections of `objects' or `sets' possess. You have already had a taste of this in Math 204, linear algebra, or in CS 221, discrete stru
Math 375 Week 33.1 The Center and CentralizerDEFINITION 1There are two subgroups of any group G that are easily de ned and easily confused If G is a group then the center of G is the setC G = fa 2 Gj ax = xa 8x 2 Gg:Note that the center consi
Math 375 Week 66.1 Homomorphisms and IsomorphismsDEFINITION 1 Let G1 and G2 groups and let : G1 ! G2 be a function. Then is a group homomorphism if for every a; b 2 G we haveab = a b: Notice that the operation on the left is occurring in G1 while
Math 375 Week 99.1 Normal Subgroups
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Chestnut Mountain Precalculus is built off a healthy curiosity for the world around us. These are concepts that when explained properly can prompt a further desire for learning in the future. Let's get the ball rolling.
...Physical
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cheap scientific calculator that does matrix operations
cheap scientific calculator that does matrix operations
I do not have a calculator that does matrix stuff that would be great for my classes this semester. I don't want to spend $100+ on a graphing calculator since I use matlab at home strictly now. Im looking to solve simultaneous equations, matrix operations like det,eigen,mult,...etc
There is an excellent software calculator called Mathwizard that does Matrices,algebra,calculus,scientific calculator and plot graph.check it out. you can also find mobile applications that does matrices,algebra,calculus,differential equations , and plot graphs
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Contemporary Precalculus : A Graphing Approach - 5th edition
Summary: Respected for its detailed guidance in using technology, CONTEMPORARY PRECALCULUS: A GRAPHING APPROACH, Fifth Edition, is written from the ground up to be used with graphing calculators that you may be using in your precalculus course. You'll appreciate that the text has also long been recognized for its careful, thorough explanations and its presentation of mathematics in an informal yet mathematically precise manner. The authors also emphasize the all-important ''why?'' of mathemat...show moreics--which is addressed in both the exposition and in the exercise sets by focusing on algebraic, graphical, and numerical perspectives. ...show less
5. EXPONENTIAL AND LOGARITHMIC FUNCTIONS. Radicals and Rational Exponents. Special Topics: Radical Equations. Exponential Functions. Special Topics: Compound Interest and the Number e. Common and Natural Logarithmic Functions. Properties of Logarithms. Special Topics: Logarithmic Functions to Other Bases. Algebraic Solutions of Exponential and Logarithmic Equations. Exponential, Logarithmic, and Other Models. Discovery Project: Exponential and Logistic Modeling of Diseases.
8. TRIANGLE TRIGONOMETRY. Trigonometric Functions of Angles. Alternate: Trigonometric Functions of Angles. Applications of Right Triangle Trigonometry. The Law of Cosines. The Law of Sines. Special Topics: The Area of a Triangle. Discovery Project: Life on a Sphere.
9. APPLICATIONS OF TRIGONOMETRY. The Complex Plane and Polar Form for Complex Numbers. DeMoivre� s Theorem and nth Roots of Complex Numbers. Vectors in the Plane. The Dot Product. Discovery Project: Surveying.259.62
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Triangular Arrays with Applications
0199742944
9780199742943, proof-techniques, and problem-solving techniques.While a good deal of research exists concerning triangular arrays and their applications, the information is scattered in various journals and is inaccessible to many mathematicians. This is the first text that will collect and organize the information and present it in a clear and comprehensive introduction to the topic. An invaluable resource book, it gives a historical introduction to Pascal's triangle and covers application topics such as binomial coefficients, figurate numbers, Fibonacci and Lucas numbers, Pell and Pell-Lucas numbers, graph theory, Fibonomial and tribinomial coefficients and Fibonacci and Lucas polynomials, amongst others. The book also features the historical development of triangular arrays, including short biographies of prominent mathematicians, along with the name and affiliation of every discoverer and year of discovery. The book is intended for mathematicians as well as computer scientists, math and science teachers, advanced high school students, and those with mathematical curiosity and maturity. «Show less,... Show more»
Rent Triangular Arrays with Applications today, or search our site for other Koshy
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Book Description: This volume presents students with problems and exercises designed to illuminate the properties of functions and graphs. The 1st part of the book employs simple functions to analyze the fundamental methods of constructing graphs. The 2nd half deals with more complicated and refined questions concerning linear functions, quadratic trinomials, linear fractional functions, power functions, and rational functions. 1969 edition.
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Integral Calculus
It is not an exaggeration that the fort of Mathematics is Calculus and the most important part of it is Integral Calculus. It is quite a lot scoring and should be taken very seriously in the preparation of IIT JEE, AIEEE, DCE, EAMCET and other engineering entrance examinations. The Calculus is said to be complete only if you have mastered the topic of Integral Calculus.
Since the prerequisite to the preparation of Integral Calculus is the study of Differential Calculus, we can judge a student in Calculus by seeing his comfort level and proficiency in Integral Calculus. The importance of Integral Calculus is not just restricted to Mathematics but it is of profound importance in the major part of Physics and Physical Chemistry. The major portions of study of Integral Calculus include Indefinite Integral and Definite Integral and they are rightly termed as tools which are further used in its applications under the topics of Area. The topic of Area is the one which used the most in Physics and should be taken very seriously. The next topic of Differential Equations is also quite important. Since the syllabus do not demand differential equations of higher order so it is easy for the students to have proficiency in the subject.
It is true that most of the students fear from Integral calculus, but all the high rank holders in IIT JEE are always very comfortable in the topic. It is advised to do a practice in Integral Calculus at an early stage as it is one of the new topics which students study while preparing for IIT JEE, AIEEE, DCE, EAMCET and other engineering entrance examinations.
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Student Responsibilities:
One cannot benefit from or contribute to a class discussion or activity unless one is physically present (this a necessary condition, not a sufficient one). Attendance is required. Call me (796-3658) if you will not be in class. A valid excuse is necessary to miss class. Unexcused absences may lower your grade for the course.
Assigned readings of the texts and handouts need to be done if meaningful discussion can occur.
Your active participation makes the course go. Math is not a spectator sport. Assigned problems and textbook exercises are ways for you to develop problem solving skills and reflect on your learning. Do the problems when they are assigned.
Content
I. Logic and Proofs
A. Propositions and Connectives
B. Conditional and Biconditionals
C. Quantifiers
D. Basic Proof Methods I
E. Basic Proof Methods II
F. Proofs Involving Qunatifiers
II. Set Theory
A. Basic Concepts
B. Set Operations
C. Extended Operations and Indexing
D. Induction
III. Relations
A. Cartesian Products
B. Equivalence Relations
C. Renaming
D. Partitions
IV. Functions
A. Functions as Relations
B. Constructions
C. One-to-One, Onto Functions
V. Cardinality (if time permits)
A. Equivalent Sets
B. Infinite Sets
C. Countable Sets
Evaluation
I will use a 90 – 80 – 70 – 60 framework for grading. I will give you written assignments on Thursday and these will be due to the following Thursday. You may consult each other but the write-up is your responsibility. I suggest you go to separate rooms to write up your answers.
There will be three exams in class. I need to see what you can do all by yourself. The dates will be determined during the first week of class.
A Note to You
Mathematics can be an intellectual adventure, a powerful tool, and a creative experience Some of you may have had mathematics courses that were based on the transmission, or absorption, view of teaching and learning. In this view, students passively It is one way to make sense of the world.
Consequently, I have three goals when I teach. The first is to help you develop mathematical structures that are more complex, abstract, and powerful than the ones you currently possess so that you will be capable of solving a wide variety of meaningful problems. The second is to help you become autonomous and self- motivated Hopefully.
As
So
You may find this experience frustrating at times. Persevere! Eventually I hope you will own personally the mathematical ideas you once knew unthinkingly or only peripherally (and sometimes anxiously). I want you to become competent and confident using mathematical ideas and techniques.
In training a child to activity of thought, above all things we must beware of what I will call "inert ideas" - that is to say, ideas that are merely received into the mind without being utilized, or tested, or thrown into fresh combinations . . . Education with inert ideas is not only useless: it is, above all things, harmful. Except at rare intervals of intellectual ferment, education in the past has been radically infected with inert ideas . . . Let us now ask how in our system of education we are to guard against this mental dryrot. We enunciate two educational commandments, "Do not teach too many subjects," and again, "What you teach, teach thoroughly." . . . Let the main ideas which are introduced into a child's education be few and important, and let them be thrown into every combination possible. The child should make them his own, and should understand their application here and now in the circumstances of his actual life. From the very beginning of his education, the child should experience the joy of discovery. (Alfred North Whitehead, The Aims of Education)
Americans With Disability Act. If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me or Wayne Wojciechowski (MC 320, 796-3085) within ten days to discuss your accommodation needs.
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Traditionally, an Algebra 1 course focuses on rules or specific strategies for solving standard types of symbolic manipulation problems-usually to simplify or combine expressions or solve equations. For many students, symbolic rules for manipulation are memorized with little attempt to make sense of why they work. They retain the ideas for only a short time. There is little evidence that traditional experiences with algebra help students develop the ability to "read" information from symbolic expression or equations, to write symbolic statements to represent their thinking about relationships in a problem, or to meaningfully manipulate symbolic expressions to solve problems.
In the United States, algebra is generally taught as a stand-alone course rather than as a strand integrated and supported by other strands. This practice is contrary to curriculum practices in most of the rest of the world. Today, there is a growing body of research that leads many United States educators to believe that the development of algebraic ideas can and should take place over a long period of time and well before the first year of high school. Developing algebra across the grades and integrating it with other strands helps students become proficient with algebraic reasoning in a variety of contexts and gives them a sense of the coherence of mathematics. Transition to High School in Implementing CMP.
The Connected Mathematics program aims to expand student views of algebra beyond symbolic manipulation and to offer opportunities for students to apply algebraic reasoning to problems in many different contexts throughout the course of the curriculum. The development of algebra in Connected Mathematics is consistent with the recommendations in the NCTM Principles and Standards for School Mathematics 2000 and most state frameworks.
Algebra in Connected Mathematics focuses on the overriding objective of developing students' ability to represent and analyze relationships among quantitative variables. From this perspective, variables are not letters that stand for unknown numbers. Rather they are quantitative attributes of objects, patterns, or situations that change in response to change in other quantities. The most important goals of mathematical analysis in such situations are understanding and predicting patterns of change in variables. The letters, symbolic equations, and inequalities of algebra are tools for representing what we know or what we want to figure out about a relationship between variables. Algebraic procedures for manipulating symbolic expressions into alternative equivalent forms are also means to the goal of insight into relationships between variables. To help students acquire quantitative reasoning skills, we have found that almost all of the important tasks to which algebra is usually applied can develop naturally as aspects of this endeavor. (Fey, Phillips 2005)
There are eight units which focus formally on algebra. Titles and descriptions of the mathematical content for these units are:
Variables and Patterns
Introducing Algebra
Representing and analyzing relationships between variables, including tables, graphs, words, and symbols
Frogs, Fleas, and Painted Cubes
Quadratic Relationships
Examining the pattern of change associated with quadratic relationships and comparing these patterns to linear and exponential patterns, recognizing, representing, and analyzing quadratic functions in tables graphs, words, and symbols; determining and predicting important features of the graph of a quadratic functions, such as the maximum/minimum point, line of symmetry, and the x-and y-intercepts; factoring simple quadratic expressions
Say It With Symbols
Making Sense of Symbols
Writing and interpreting equivalent expressions; combining expressions; looking at the pattern of change associated with an expression; solving linear and quadratic equations
Linear Systems and Inequalities
Even though the first primarily algebra unit occurs at the start of seventh grade, students study relationships among variables in grade 6.
There also are opportunities in 6th and in 7th grade for students to begin to examine and formalize patterns and relationships in words, graphs, tables, and with symbols.
In Shapes and Designs (Grade 6), students explore the relationship between the number of sides of a polygon and the sum of the interior angles of the polygon. They develop a rule for calculating the sum of the interior angle measures of a polygon with N sides.
In Covering and Surrounding (Grade 6), students estimate the area of three different- size pizzas and then relate the area to the price. This problem requires students to consider two relationships: one between the price of a pizza and its area and the other between the area of a pizza and its radius. Students also develop formulas and procedures-stated in words and symbols-for finding areas and perimeters of rectangles, parallelograms, triangles, and circles.
In Bits and Pieces I, II and III (Grade 6), students learn, through fact families, that addition and subtraction are inverse operations and that multiplication and division are inverse operations. This is a fundamental idea in equation solving. They use these ideas to find a missing factor or addend in a number sentence.
In Data About Us (Grade 6), students repre- sent and interpret graphs for the relationship between variables, such as the relationship between length of an arm span and height of a person, using words, tables, and graphs.
In Accentuate the Negative (Grade 7), students explore properties of real numbers, including the commutative, distributive, and inverse properties. They use these properties to find a missing addend or factor in a number sentence.
In Filling and Wrapping (Grade 7), students develop formulas and procedures-stated in words and symbols-for finding surface area and volume of rectangular prisms, cylinders, cones, and spheres.
Developing Functions
In a problem-centered curriculum, quantities (variables) and the relationships between variables naturally arise. Representing and reasoning about patterns of change becomes a way to organize and think about algebra. Looking at specific patterns of change and how this change is represented in tables, graphs, and symbols leads to the study of linear, exponential, and quadratic relationships (functions).
Linear Functions
In Moving Straight Ahead, students investigate linear relationships. They learn to recognize linear relationships from patterns in verbal, tabular, graphical, or symbolic representations. They also learn to represent linear relationships in a variety of ways and to solve equations and make predictions involving linear equations and functions. Problem 1.3 illustrates the kinds of questions students are asked when they meet a new type of relationship or function-in this case, a linear relationship. In this problem students are looking at three pledge plans that students suggest for a walkathon.
Moving Straight Ahead. p. 9
Whereas many algebra texts choose to focus almost exclusively on linear relationships, in Connected Mathematics students build on their knowledge of linear functions to investigate other patterns of change. In particular, students explore inverse variation relationships in Thinking With Mathematical Models, exponential relationships in Growing, Growing, Growing, and quadratic relationships in Frogs, Fleas, and Painted Cubes. Examples are given below which illustrate the different types of functions students investigate and some of the questions they are asked about these functions. By contrasting linear relationships with exponential and other relationships, students develop deeper understanding of linear relationships.
Inverse Functions
In Thinking With Mathematical Models, students are introduced to inverse functions.
Thinking With Mathematical Models. p. 32
Exponential Functions
In Growing, Growing, Growing, students are given the context of a reward figured by placing coins called rubas on a chessboard in a particular pattern, which is exponential. The coins are placed on the chessboard as follows.
Place 1 ruba on the first square of a chessboard, 2 rubas on the second square, 4 on the third square, 8 on the fourth square, and so on, until you have covered all 64 squares. Each square should have twice as many rubas as the previous square.
In this problem students use tables, graphs, and equations to examine exponential relationships and describe the pattern of change for this relationship.
Growing, Growing, Growing. p. 7
Quadratic Functions
In Problem 1.3 from Frogs, Fleas and Painted Cubes, students use tables, graphs, and equations to examine quadratic relationships and describe the pattern of change for this relationship.
Frogs, Fleas and Painted Cubes. p. 10
As students explore a new type of relationship, whether it is linear, quadratic, inverse, or exponential, they are asked questions like these:
What are the variables? Describe the pattern of change between the two variables.
Describe how the pattern of change can be seen in the table, graph, and equation.
Decide which representation is the most helpful for answering a particular question. (see Question D in Problem 1.3 Frogs and Fleas and Painted Cubes above)
Describe the relationships between the different representations (table, graph, and equation).
Compare the patterns of change for different relationships. For example, compare the patterns of change for two linear relationships, or for a linear and an exponential relationship.
After students have explored important relationships and their associated patterns of change and ways to represent these relationships, the emphasis shifts to symbolic reasoning.
Equivalent Expressions
Students use the properties of real numbers to look at equivalent expressions and the information each expression represents in a given context and to interpret the underlying patterns that a symbolic statement or equation represents. They examine the graph and table of an expression as well as the context the expression or statement represents. The properties of real numbers are used extensively to write equivalent expressions, combine expressions to form new expressions, predict patterns of change, and to solve equations. Say It With Symbols pulls together the symbolic reasoning skills students have developed through a focus on equivalent expressions. It also continues to explore relationships and patterns of change. Problem 1.1 in Say It With Symbols introduces students to equivalent expressions.
Say It With Symbols. p. 6
In Problem 2.1 students revisit Problem 1.3 from Moving Straight Ahead (see above) to combine expressions. They also use the new expression to find information and to predict the underlying pattern of change associated with the expression.
Say It With Symbols. p. 24
Solving Equations
Equivalence is an important idea in algebra. A solid understanding of equivalence is necessary for understanding how to solve algebraic equations. Through experiences with different functional relationships, students attach meaning to the symbols. This meaning helps student when they are developing the equation-solving strategies integral to success with algebra.
In CMP, solving linear equation is an algebra idea that is developed across all three grade levels, with increasing abstraction and complexity. In grade six, students write fact families to show the inverse relationships between addition and subtraction and between multiplication and division. The inverse relationships between operations are the fundamental basis for equation solving. Students are exposed early in sixth grade to missing number problems where they use fact families. Below is a description of fact families and a few examples of problems where students use fact families to solve algebraic equations in grades 6 and 7. These experiences precede formal work on equation solving.
In Bits and Pieces II (Grade 6), Bits and Pieces III (Grade 6), and Accentuate the Negative (Grade 7), students use fact families to find missing addends and factors.
Bits and Pieces II. p. 22
Bits and Pieces III. p. 28
Accentuate the Negative. p. 30
In Variables and Patterns (Grade 7), students solve linear equations using a variety of methods including graph and tables. As students move through the curriculum, these informal equation- solving experiences prepare them for the formal symbolic methods, which are developed in Moving Straight Ahead (Grade 7), and revisited throughout the five remaining algebra units in eighth grade.
Moving Straight Ahead. p. 85
Say It With Symbols (Grade 8), pulls together the symbolic reasoning skills students have developed through a focus on equivalent expressions and on solving linear and quadratic equations.
Say It With Symbols. p. 42
Shapes of Algebra (Grade 8), explores solving linear inequalities and systems of linear equations and inequalities. By the end of Grade 8, students in CMP should be able to analyze situations involving related quantitative variables in the following ways:
identify variables
identify significant patterns in the relationships among the variables
represent the variables and the patterns relating these variables using tables, graphs, symbolic expressions, and verbal descriptions
translate information among these forms of representation
Students should be adept at identifying the questions that are important or interesting to ask in a situation for which algebraic analysis is effective at providing answers. They should develop the skill and inclination to represent information mathematically, to transform that information using mathematical techniques to solve equations, create and compare graphs and tables of functions, and make judgments about the reasonableness of answers, accuracy, and completeness of the analysis.
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This is a theoretical course focusing of fundamental topics in modern
integer programming. The course will be based on lectures by the instructor,
with homework projects involving proofs. The course will also include a
comprehensive survey of linear programming.
Materials
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Maths Statistics Coursework
Students enrolled in mathematics courses should expect to have to perform some maths statistics courseworks during their courses at some point. There are many different types of maths statistics courseworks that students may have to perform, such as mathematical research, equations, and more. The particular type of maths statistics coursework may depend solely on the course that the student is taking.
Whenever a student needs to work on maths statistics coursework assignments, the student needs to begin by making sure that she or he understands the assignment requirements. In many cases, learners will receive an assignment sheet or instructions from a professor. Students should review these assignment requirements prior to beginning work in order to ensure that they understand what they will need to do. Students then need to ask professors any inquiries about the work if they require further clarification.
There are some occasions where students will not understand how to do a particular maths statistics coursework. For example, if a student does not understand how to use a scientific calculator, then the student may not be able to calculate the sine or cosine for some equations. In such situations, the student may need to seek additional help from an on-campus mathematics center or from a professor.
Students may also benefit by making sure that they enroll in the appropriate courses in the right order. For example, a student may have a hard time in a geometry course if the student has not already learned about algebra. Therefore, because mathematics is a progressive subject, students need to take the right courses in the right order.
Mathematics also encompasses many different fields of study. Each field of study may require learners to work on different types of maths statistics courseworks. For example, students enrolled in a theoretical mathematics course may have to perform a great deal of research, whereas students enrolled in a basic calculus course may have to perform equations as part of their coursework.
Coursework for all subjects requires many of the same variables. Students need to understand an assignment, develop a plan for completing the assignment, and make sure that they check their work. However, maths statistics courseworks may also require that students learn, essentially, a different language. Mathematics is a universal language that takes students time to learn how to speak. Therefore, students cannot rush too far ahead in the process of learning math without having a stable foundation and understanding of how to speak in mathematics terms.
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Book Description: Lie groups, Lie algebras, and representation theory are the main focus of this text. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory. Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.
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Math Review Tips
Buy the textbook. Too many people think that they can just borrow their book from other students, use the one in the Learning Center, or just do the class without it. There is no review tool which equals the textbook.
Ask specific questions. While doing homework, identify specific areas in which you're having difficulties. This can be crucial when trying to figure out what to study. It also can be extremely helpful when getting help from a tutor or professor.
Utilize the Learning Center. The Learning Center can be a great way to review or get help on the problems that you're having. With tutors and staff that are available to help you, you have many people that are willing to help you accomplish your goal.
Attend all classes. If you're not spending time in classes, you're most likely not learning very much. Math textbooks are hard to just sit down and study from scratch, and class periods can give you a great feel for the material as well as possibly answer questions you had.
Read the appropriate section before class. With a bit of preparation, class time can make much more sense with a bit of background in the topic before the professor talks about it.
Online Math Review Websites
Khan Academy - Thousands of lessons on topics ranging from arithmetic to calculus and beyond.
Spanish
Audio
Librivox - A website dedicated to creating and compiling audiobooks in the public domain. Here you will find hundreds of hours of recordings in the spanish language that you can download and listen to at your leisure. Be sure to check out "Las Fábulas de Esopo" – Aesop's Fables.
Grammar
Dictionaries
Word Reference - A very useful dictionary with definitions in English and Spanish, a verb conjugator, and great forums that address finer language points that no textbook could ever attempt. Diccionario de la Lengua Española - A terrific reference with definitions in Spanish.
French
Audio
Librivox - A website dedicated to compiling audiobooks in the public domain. Here you will find hundreds of hours of recordings in the french language that you can download and listen to at your leisure. Indo-European Languages - A terse but comprehensive review of French grammar and vocabulary with extensive audio examples.
Grammar
Indo-European Languages - A terse but comprehensive review of French grammar and vocabulary with extensive audio examples.
Dictionaries
Word Reference - A very useful dictionary with definitions in English and French, a verb conjugator, and great forums that address finer language points that no textbook could ever attempt.
Russian
Audio
Librivox - A website dedicated to compiling audiobooks in the public domain. Here you will find hundreds of hours of recordings in the russian language that you can download and listen to at your leisure.
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This chapter introduces the fundamental concepts and terminology of options. The relationships between options and the underlying securities are intuitively explained and how these relationship must be maintained to eliminate arbitrage is used to then motivate and setup the Black-Scholes PDE
This textbook provides an introduction to financial mathematics and financial engineering for undergraduate students who have completed a three or four semester sequence of calculus courses. It introduces the theory of interest, random variables and probability, stochastic processes, arbitrage, option pricing, hedging, and portfolio optimization. The student progresses from knowing only elementary calculus to understanding the derivation and solution of the Black–Scholes partial differential equation and its solutions. This is one of the few books on the subject of financial mathematics which is accessible to undergraduates having only a thorough grounding in elementary calculus. It explains the subject matter without "hand waving" arguments and includes numerous examples. Every chapter concludes with a set of exercises which test the chapter's concepts and fill in details of derivations.
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Everything you wanted to know about abstract algebra, but were afraid to buy
Sage is an open-source program for doing mathematics and is the ideal companion to Abstract Algebra: Theory and Applications. Sage is designed to be a free, open source alternative to Magma, Maple, Mathematica and Matlab. It includes many mature and powerful open-source tools for mathematics, such as GAP for group theory. With a strength in number theory, Sage also has excellent support for rings and fields.
Available here is an electronic version of the text in the Sage worksheet format. This version of the text has been supplemented with significant discussion for each chapter about the use of Sage for the topic at-hand. This discussion is followed by a selection of exercises. If you are curious about the nature or scope of this supplement, download the PDF version of just the supplemental material, but understand that in the worksheet version the code is executable and editable. The "exercises only" version is designed for students who will be submitting the exercises as worksheets and do not want all the text. Be sure to read the notes below.
Sage for Abstract Algebra, by Rob Beezer
For Sage Version 5.2 and AATA Annual Edition 2012-13
Notes
Download the collections of worksheets as a zip file to your local storage. Then use the Sage notebook upload function to upload the entire zip file - the notebook will unzip it for you. If Safari unzips the file automatically, look in the trash for the original zip file.
You can use a free notebook server by making an account at sagenb.org.
Links do not work between worksheets as the Sage notebook does not support this yet. In particular, tables of contents will not be useful. Just open the chapters you want to read.
If you have lots of Sage worksheets already, these may unzip into your notebook in the vicinity of worksheets you last used in August 2012, so look there if you do not see them, or sort your worksheets on a different column.
Examples have all been tested for accuracy with the Sage doctest framework using Sage Version 5.2.
There was once a tutorial here, and we have left the PDF available. It is now obsolete and is no longer being maintained. If you have a link pointing here, you might wish to adjust the description of what is actually available.
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An expose you to a variety of areas of mathematics, (b) solving simple linear equations, (c) exploring the mathematical model of simple and compounded interest rates, and learning how to use those ideas in solving the problems of loan payments, (d) exploring a few major concepts of Euclidean Geometry, focusing especially on the axiomatic-deductive nature of this mathematical system, including a variety of different proofs of the Pythagorean Theorem, (e) develop an ability to use deductive reasoning, in the context of the rules 1 2 MATH 155 WAY OF THINKING SYLLABUS - FALL 2002 of logic and syllogisms, i.e., learn how to make/recognize a valid argument, (f) some basics of probability and statistics . . . Mastering this material requires to learn how to reason mathematically, and also how to communicate mathematics. In learning how to do so (on exams, essays, portfolio, and in oral presentations), you will also develop a confidence in your ability to do mathematics.
Other benefits of this course include: cultural skills (appreciation of the history of mathematics and its role in today's world, learning how to handle simple loans, learning how to reason correctly and make a valid argument), appreciate the beauty and intellectual honesty of deductive reasoning, thereby adding to life value and aesthetic skills.
I encourage you to read the text at: the Viterbo critical thinking web page
Text: Robert Blitzer, Thinking Mathematically, Prentice-Hall, 2000• InternetBlackboard) software.
• Learning center.
• Library. Note that both a video set and a CD set that covers your textbook exists.
You can use either of these to hear a lecture again, or just to see/hear another explanation of a particular topic.
Grading: The final grade is based on homework, exams, presentations, portfolioMATH 155 WAY OF THINKING SYLLABUS - FALL 2002 3 Assignments: • Recommended practice: First 10, middle 5 and the last 5 problems from each Practice Exercises set in each section that we cover; at least one or two of the Application Exercises, at least one of the Writing in Mathematics Exercise, and at least two of the Critical
Thinking Exercises. These practice problems will not be graded. However, fell free to ask me for help with any difficulty you might have with those problems.
• Two essays, 20 points each:
(1) Autobiography: Introduce yourself to me in a 2-3 pages essay. State your name, and where (city/state) you are coming from.
The reason you are taking this course, and what mathematics courses you have had before. What was your experience from those courses and what are your expectations, if any, from this course?
This assignment is due Friday, January 18.
(2) World without mathematics: another 2-3 pages 20 points essay.
Try to imagine, and describe, a world without mathematics.
Due: Friday, January 25.
Homework At the end of each chapter, there is a Chapter Test. Each one of those tests will be due second class period after the corresponding chapter is covered, and each problem on the "test"is worth 1 point.
Exams There will be three in-class exams, worth 100 points each. An the exam 2. This makeup will be oral, and will apply to those under 90/100 points on the test, and is to be done within two weeks after the exam.
Final Exam Final exam is a 2-hour, cumulative exam, and is worth 200 points.
Portfolio
• In-class Presentation: The presentation of a proof of the Pythagorean Theorem found on the Internet. Typically, the explanations you will find on the Internet are a bit sketchy. So part of your job will be to make sure you really understand the proof you are going to present (including filling in the gaps, i.e., the reasons not entirely spelled out in the Internet write-up), and then 4 MATH 155 WAY OF THINKING SYLLABUS - FALL 2002 to clearly explain that proof to your class mates. Sometimes, some people, may find this part quite difficult. Of course, I am here to help you understand and overcome those difficulties, and so please do not hesitate to ask me for help. You should also be prepared for the questions from the audience (myself and/or other students), and it is expected that you listen closely to other presentations and ask any question you might have. The presentation will be worth 35 points. In addition to that, one certain problem for one of the exams, and for the final exam is going to be:
State and prove the Pythagorean Theorem.
Important University Policies: The links:
• Plagiarism:
- Viterbo policy statement on plagiarism, see also Student Planner and
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Encyclopedia of Mathematics & Society
Published by Salem Press
Presents articles showing the math behind our daily lives. Explains how and why math works, and allows readers to better understand how disciplines such as algebra, geometry, calculus, and others affect what we do every day
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This applet encompasses five different applets on different topics and at different levels. (The user can access each topic via Course Activity tab.) The first activity consists of a simple function grapher that graphs a pre-defined function from a large collection of functions and two user-defined functions. (Very easy zoom-in and zoom-out functionality that can be used to illustrate local linearity.) The second activity provides an interactive illustration of the Mean Value Theorem. The third part shows the Newton's method in action. The fourth activity shows Riemann sums and several numerical integration methods, including the midpoint and the trapezoid rules. Finally, the last activity demonstrates the definite integral in terms of areas. All activities are interactive: the user chooses a function, an interval, a starting point, the number of divisions, etc. depending on the activity.
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Binomial Expansion
Introduction and Summary
This chapter deals with binomial expansion; that is, with writing expressions of
the form
(a + b)n
as the sum of several monomials.
Prior to the discussion of binomial expansion, this chapter will present
Pascal's Triangle. Pascal's Triangle is a triangle in which each row has
one more entry than the preceding row, each row begins and ends with "1," and
the interior elements are found by adding the adjacent elements in the preceding
row. Section one will display part of Pascal's Triangle, and will provide a
formula for finding any element of any row in the triangle.
Pascal's Triangle is essential to the discussion of binomial expansion because,
as it turns out, the numbers in Pascal's Triangle are the coefficients of the
monomials in the expansion of
(a + b)n
. The monomials also have other
properties, which can be summed up in the Binomial Theorem. This theorem is
presented in section two. Using this theorem, we will be able to write out any
expansion of any binomial.
Binomial expansion has other uses besides those in algebra II. It is used in
statistics to calculate the binomial distribution. This allows statisticians to
determine the probability of a given number of favorable outcomes in a repeated
number of trials. Binomial expansion is also interesting from a mathematical
point of view--it gives mathematicians insight into the properties of
polynomials.
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Hey guys, so I suggested to start threads about discussions for courses, since they kinda get lost in the other forums.
How is this course going for you guys so far? Any advice from people who took it? I have an 84 after the first unit of piecewise/polynomial functions.
This is a thread for the discussion about grade 12 Advanced Functions(AP and Non AP). General discussions about the course, specific questions, studying guides, strategies etc can all be posted here.
Here is the Ontario Curriculum for this course: 1. Characteristics and Transformations 2. Polynomial and Rational Functions 3. Trigonometric functions and identities 4. Exponential and Logarithmic Functions 5. Combinations of functions and rate of change start. This course has been going pretty well so far. The graphing part kind of takes me long on the tests, so I can never finish them, but graphing is so important in this course.
We did polynomial functions, piecewise functions, and now we are on rates of change. I don't know what the easiest units are, but I know trig will be an average killer!Ur so true about MHF being the easiest of all three grade 12 math, haven't take calculus, but I have data and adv functions this semester. Adv functions like soo easy where in data, I find it quite hard and my teacher said its normal lol
If data management is taught as it should be, the application and extension of concepts should be harder than advanced functions.
At my school, MDM4U is the credit that we get for IB Math Studies, which is basically math for people who 1. are just bad at math OR 2. don't need math for university but need to take a math course to get their IB diploma. Needless to say, it's a super easy course.
At my school, Data Management is easily the easiest 4U math course. It's pretty much designed for students who need a 4U math credit but aren't particularly good at math.
Same at my school. No one I know thinks advanced functions is easier than data.
I dont actually need data, but i took it cuz i thought it's easy. Then i realize that was a mistake....i only have a 91% in data whereas 99% in advanced functions.
Wow, so you're saying you could have chosen another five courses to beat that mark in data? That's incredible. If not, it wouldn't necessarily be a 'mistake'.
yeh I already took CHI4U, BBB4M, CLN4U and they all beat my mark in data, plus English and MHF4U, there're 5 courses in total. Im positive i can get a high 90s in BAT4M next semester. Therefore, yeh i shouldn't have taken data....Hi, I'm taking Data Management right now. We learned permutations and combinations. Now we're learning statstics. I'm wondering if there's any way to improve my mark?
Getting an 83 in this course. The class average is 45% because the teacher is always putting at least one hardcore thinking question that is ten marks. Our school usually does well on the math contests, so I hope it will go well. I'm quite happy with my mark though, considering how hard she was. I just wish I didn't make careless mistakes on the tests. Losing 4 marks on questions because you got mixed up with speed and velocity or didn't know the definition or calculated the rate of change for 3 months instead of 2 months can cost a lot when the test is out of 35
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Why Choose MathMedia?
Why Choose MathMedia?
Item# Choose_MathMedia
Product Description
What are your specific needs? Do you have students with a variety of ability levels?
Do you have students with "holes" in their background?
Would you like to supplement your classroom teaching?
Do you have adult students preparing for the TABE or GED?
Are you, yourself, preparing for college placement exams?
Are you, yourself, preparing for the GED?
Would you like a resource to supplement your class work?
Do you need to access the "Cloud" for tablets?
Do you need to have ONLY local CD access?
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CMAT - Comprehensive Mathematical Abilities Test
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Based on state and local curriculum guides, and math education tools used in schools, the CMAT is a major advance in the accurate assessment of math taught in today's schools. Contains six core subtests (addition, subtraction, multiplication, division, problem solving, and charts, tables, & graphs) and six supplemental subtests.
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Lial series has helped thousands of students succeed in developmental mathematics through its friendly writing style, numerous realistic examples, extensive problem sets, and complete supplements package. In keeping with its proven track record, this revision includes a new open design, more exercises and applications, and additional features to help both students and instructors succeed. KEY MESSAGE: The Lial series has helped thousands of readers succeed in developmental mathematics through its approachable w... MOREriting style, relevant real-world examples, extensive exercise sets, and complete supplements package. The Real Number System; Linear Equations and Inequalities in One Variable; Linear Equations and Inequalities in Two Variables: Functions; Systems of Linear Equations and Inequalities; Exponents and Polynomials; Factoring and Applications; Rational Expressions and Applications; Roots and Radicals; Quadratic Equations For all readers interested in Beginning Algebra.
(Note: Each chapter ends with a Group Activity, Summary, Review Exercises, a Chapter Test and, with the exception of Ch. 1, a Cumulative Review). List of Applications. Preface. Feature Walkthrough. 1. The Real Number System.
The Addition Property of Equality. The Multiplication Property of Equality. More on Solving Linear Equations. Summary Exercises on Solving Linear Equations . An Introduction to Applications of Linear Equations. Formulas and Applications from Geometry. Ratios and Proportions. More About Problem Solving. Solving Linear Inequalities.
3. Linear Equations and Inequalities in Two Variables; Functions.
Reading Graphs; Linear Equations in Two Variables. Graphing Linear Equations in Two Variables. The Slope of a Line. Equations of a Line. Graphing Linear Inequalities in Two Variables. Introduction to Functions.
The Product Rule and Power Rules for Exponents. Integer Exponents and the Quotient Rule. Summary Exercises on the Rules for Exponents. An Application of Exponents: Scientific Notation. Adding and Subtracting Polynomials; Graphing Simple Polynomials. Multiplying Polynomials. Special Products. Dividing Polynomials.
6. Factoring and Applications.
The Greatest Common Factor; Factoring by Grouping. Factoring Trinomials. More on Factoring Trinomials. Special Factoring Rules. Summary Exercises on Factoring. Solving Quadratic Equations by Factoring. Applications of Quadratic Equations.
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COURSE DESCRIPTION
As suggested by the catalog description (below), this course provides an advanced perspective on the concepts in school algebra. Content focuses on problem solving in an algebraic context.
Algebraic reasoning incorporating the use of technology. This course includes investigations of patterns, relations, functions, and analysis, with a focus on representations and the relationships among them.
Note: This section is specially designed to leverage connections to high school mathematics.
STUDENT LEARNING OUTCOMES
Upon successful completion, students in the class will:
Algebraic Reasoning: Be able to describe the three habits of algebraic thinking and recognize examples of each in multiple contexts.
Functions: Recognize and describe the processes of doing and undoing functions, relations, and algorithms in a variety of contexts.
Linear Change: Recognize linear change in multiple forms and be able to find the equation of a linear relationship presented in numerical, graphical, verbal, or symbolic form.
Families of Functions: Identify linear, quadratic, exponential, and other common families of functions when represented in patterns, graphs, equations, and tables.
MAJOR COURSE REQUIREMENTS and ASSESSMENTS
Final course grades will be a weighted average of mean scores using the following weights:
Classwork 20%
Project 30%
Homework & Quizzes 20%
Final Exam 30%
Final weighted grades exceeding 90% will result in a letter grade of A. Those exceeding 80% will result in at least a B; ≥ 70% will result in at least a C; ≥ 60% will result in at least a D; below 60% will result in an F.
Classwork– participate in inquiry tasks, whole-class discussion, and group work activities during regularly scheduled class time.
Projects – First, select a challenging concept in school algebra and describe an approach from educational literature for teaching the concept. Then, create an inquiry-based activity for you to teach the concept. See the project guidelines for assessment details.
Final Exam – complete a comprehensive summative evaluation of your knowledge through a post-test. The final exam cannot be made-up if missed. If you have a conflict with the scheduled final exam time, please contact me at least one week prior to discuss scheduling options.
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What is SOLUTION OF CLASS 12 MATHS EXEMPLAR PROBLEM BOOKBY NCERT?
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These challenging books are designed for students with significant mathematical backgrounds, yet it can be appreciated by non-mathematicians; there are no maths formulae. As well as exploring the concepts of transformation and deformation, they introduce the idea of surfaces without thickness or boundary.
Waves, Diffusion and Variational Principles (MS324)
Variational Principles
Four books focus on three areas of applied mathematics. The first explores wave motion using vibrating strings and sound waves as examples. The second describes heat flow and the flow of particles which follow random walks. The third area introduces variational principles and calculus through simple problems.
In recent decades, mathematicians have increasingly employed computer- assisted algebra packages in their calculations. Maple is one of the more popular packages, used to expand functions as series, evaluate sums and integrals, solve differential equations and plot the results of calculations. Books and a CD-ROM introduce computer-assisted algebra techniques using Maple.
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specific... read moreFundamental Concepts of Geometry by Bruce E. Meserve Demonstrates relationships between different types of geometry. Provides excellent overview of the foundations and historical evolution of geometrical concepts. Exercises (no solutions). Includes 98 illustrations.
A Vector Space Approach to Geometry by Melvin Hausner This examination of geometry's correlation with other branches of math and science features a review of systematic geometric motivations in vector space theory and matrix theory; more. 1965Invitation to Geometry by Z. A. Melzak Intended for students of many different backgrounds with only a modest knowledge of mathematics, this text features self-contained chapters that can be adapted to several types of geometry courses. 1983 edition.
Advanced Euclidean Geometry by Roger A. Johnson This classic text explores the geometry of the triangle and the circle, concentrating on extensions of Euclidean theory, and examining in detail many relatively recent theorems. 1929 edition.
A Course in the Geometry of n Dimensions by M. G. Kendall This text provides a foundation for resolving proofs dependent on n-dimensional systems. The author takes a concise approach, setting out that part of the subject with statistical applications and briefly sketching them. 1961 edition.
Elementary Mathematics from an Advanced Standpoint: Geometry by Felix Klein This comprehensive treatment features analytic formulas, enabling precise formulation of geometric facts, and it covers geometric manifolds and transformations, concluding with a systematic discussion of fundamentals. 1939 edition. Includes 141 figures.
Euclidean Geometry and Transformations by Clayton W. Dodge This introduction to Euclidean geometry emphasizes transformations, particularly isometries and similarities. Suitable for undergraduate courses, it includes numerous examples, many with detailed answers. 1972 edition.
The Beauty of Geometry: Twelve Essays by H. S. M. Coxeter Absorbing essays demonstrate the charms of mathematics. Stimulating and thought-provoking treatment of geometry's crucial role in a wide range of mathematical applications, for students and mathematicians.
Challenging Problems in Geometry by Alfred S. Posamentier, Charles T. Salkind Collection of nearly 200 unusual problems dealing with congruence and parallelism, the Pythagorean theorem, circles, area relationships, Ptolemy and the cyclic quadrilateral, collinearity and concurrency, and more. Arranged in order of difficulty. Detailed solutions.
Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene F. Krause Fascinating, accessible introduction to unusual mathematical system in which distance is not measured by straight lines. Illustrated topics include applications to urban geography and comparisons to Euclidean geometry. Selected answers to problemsA Modern View of Geometry by Leonard M. Blumenthal Elegant exposition of the postulation geometry of planes, including coordination of affine and projective planes. Historical background, set theory, propositional calculus, affine planes with Desargues and Pappus properties, much more. Includes 56 figures.
Product Description:
specifics of the axiomatic method. Well-written and accessible, the text begins by acquainting students with the axiomatic method as well as a general pattern of thought. Subsequent chapters present in-depth coverage of Euclidean geometry, including the geometry of four dimensions, plane hyperbolic geometry, and a Euclidean model of the hyperbolic plane. Detailed definitions, corollaries, theorems, and postulates are explained incrementally and illustrated by numerous figures. Each chapter concludes with multiple exercises that test and reinforce students' understanding of the material.
Bonus Editorial Feature:
Clarence Raymond Wylie, Jr., had a long career as a writer of mathematics and engineering textbooks. His Advanced Engineering Mathematics was the standard text in that field starting in the 1950s and for many decades thereafter. He also wrote widely used textbooks on geometry directed at students preparing to become secondary school teachers, which also serve as very useful reviews for college undergraduates even today. Dover reprinted two of these books in recent years, Introduction to Projective Geometry in 2008 and Foundations ofGeometry in 2009.
The author is important to our program for another reason, as well. In 1957, when Dover was publishing very few original books of any kind, we published Wylie's original manuscript 101 Puzzles in Thought and Logic. The book is still going strong after 55 years, and the gap between its first appearance in 1957 and Introduction to Projective Geometry in 2008 may be the longest period of time between the publication of two books by the same author in the history of the Dover mathematics program. Wylie's 1957 book launched the Dover category of intriguing logic puzzles, which has seen the appearance of many books by some of the most popular authors in the field including Martin Gardner and, more recently, Raymond Smullyan.
Here's a quick one from 101 Puzzles in Thought and Logic:
If it takes twice as long for a passenger train to pass a freight train after it first overtakes it as it takes the two trains to pass when going in opposite directions, how many times faster than the freight train is the passenger train?
Answer: The passenger train is three times as fast as the freight train
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College Algebra covers all of the topics generally taught in a one-term college algebra course. It is completely Web-based, and is presented in a format that is optimal for online learning, whether you teach a fully online, hybrid, or traditional course.
Student Guide and Syllabus
The Student Guide clearly explains how the course content is organized so that students can focus on learning, not on trying to figure out where to find information.
Instructors customize the syllabus to fit their course, adding information such as a class schedule with important dates, any additional requirements or resources, and grading policies.
Chapter Pretests
Students take a pretest prior to beginning each chapter in order to self-assess their current knowledge of the chapter topics and help determine where to focus their efforts as they go through the chapter. Feedback pointing students to the relevant section for each question is provided. The pretest is delivered through and graded by the course management system.
Lessons, Animations with Audio, Interactive Exercises, Practice Problems, Videos and Section Quizzes
Each chapter is broken into sections. Within each section are lessons that contain animations with audio, interactive exercises, practice problem and answer sets, videos, and quizzes.
Each section begins with a list of specific learning objectives. Lessons give students a thorough, straightforward presentation of the material, broken into small "chunks" of content. There is no extraneous information, allowing students to focus on the key elements of the specific concepts being conveyed. Equivalent to a traditional textbook explanation and lecture presentation, lessons allow students to set their own pace as they move through the material.
Within lessons, extensive use of animation helps students visualize difficult concepts and problem-solving steps. Some lessons also contain interactive examples that guide students step-by-step through problem solving, with hints for completing each step. Students are able to consider each step before requesting a hint.
Many sections contain interactive exercises (Java applets) that help students take their learning one step further.
Problem solving tutorial videos (with audio) for every section allow students to watch an instructor hand-write and explain the solution to selected practice problems.
Practice problems and a complete answer set are provided for each section. Students work through problems related to the lesson. Answers are available, most with step-by-step solutions, for students to self-assess their progress and identify areas of difficulty.
NEW! Algorithmically generated problems with hints and full solutions are provided for each section.
Automatically-graded homework problems are assigned for each section. This allows the instructor to monitor individual student progress on a regular basis. It is delivered through and graded by the course management system.
Posttests
A cumulative posttest is provided for students' self-assessment, allowing them to seek help where needed before the chapter exam. Feedback pointing students to the relevant section for review is provided for each question. It is delivered through and graded by the course management system.
Chapter Exams
Two versions of a comprehensive end-of-chapter exam are available and delivered through and graded by the course management system. Also, a NEW algorithmic testing option is available for chapter, midterm, and final exams.
Test Banks
In addition to two versions of a Chapter Exam for each module, test banks are provided for each chapter. You can use these test banks to modify existing assessments, and/or to create midterm and final exams.
Course Management
We have partnered with BlackboardTM, eCollegeTM, and AngelTM, the leading content management systems in higher education, to provide you with the best assessment, communication, and classroom management tools widely available.
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Math Department at Glassboro High School is to produce students who can solve unique and complex problems independently and collaboratively, apply skills and knowledge to generate meaningful real world solutions, and to communicate them clearly in a variety of forms.
GHS 2013 JETS Team Ranks Second in NJ The GHS JETS (Junior Engineering Technology Society) Team of Logan Greer (captain), John Schneider, Nicholas Felker, Phillip Dang, Carolyn Provine, Lauren Yan, Autumn Brown and Allison Gilbert placed second in the state at this year's TEAMS (Test of Engineering Aptitude in Math and Science) Competition held by the TSA (Technology Student Association) at the Gloucester County Institute of Technology. The students have been invited to compete against winners from other states in the National Competition in Orlando this June.
Interactive Algebra/Geometry I (100)
This course is the first in a series of 5 courses designed to provide a common core of broadly useful math for all students. The curriculum allows math to become accessible and more meaningful. Modeling and the use of technology will be emphasized as well as engaging students in collaborating on tasks. The series is a unified curriculum that replaces the traditional Algebra-Geometry sequence. Interwoven strands of the NJ State Math Standards are featured so that the students will be better prepared to pass the High School Proficiency Assessment (HSPA) in grade 11. We strongly recommend that each student own a TI-83 or TI-84 graphing calculator!
NOTE:
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Interactive Algebra/Geometry II (126)
This course is the third in a series of 5 courses designed to provide a common core of broadly useful math for all students. Please see the description above.
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Interactive Algebra/Geometry III (127)
This course is the third in a series of 5 courses designed to provide a common core of broadly useful math for all students. Please see the description above.
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Interactive Algebra/Geometry IV (128)
This course is the fourth in a series of 5 courses designed to provide a common core of broadly useful math for all students. Please see the description above.
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Interactive Algebra/Geometry V (129)
This course is the final course in a series of 5 courses and will cover the remaining strandards addressed in Geometry and Algebra II and it will also prepare the students for college entrance exams such as the AccuPlacer and the SAT. A student who completes all five levels of Interactive Algebra/Geometry is recognized as having completed the standards for Algebra I, Geometry, and Algebra II.
Consumer Math(126)
This course includes a review of the fundaments of mathematics as applies to the cost of transportation, food, clothing, housing, taxation, insurance budgeting, banking, and investments. This is a course for Junior's and Senior's .
Prerequisite: Algebra I or Interactive Math with a grade of "70" or better.
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Algebra I (132)
This course is usually elected by students who plan to prepare for college, technical institutes, or other education beyond high school. Topics include singed numbers, algebraic expressions and operations, inequalities, graphs, factoring, ratios and proportions, quadratic equations, solving equations, and word problems, systems of equations, absolute value, matrices, probability, data analysis, and polynomials functions and relations, and basic geometry concepts. Students must earn a "70" or higher to take Algebra II or Geometry
This course may be elected following Algebra I or Geometry
Prerequisite: Algebra I with a grade of "70" or better.
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Geometry (136)
Geometry may be elected before or after Algebra II. Topics include deductive and inductive reasoning, constructions, geometric proofs, triangles, circles, parallels, perpendiculars and applications.
Prerequisite: Algebra I with a grade of "70" or better.
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Pre-Calculus (140)
This math course is designed to prepare students for Calculus. It is an opportunity to review all of the concepts presents in Algebra I, Calculators, a computer is available in the room for graphing functions.
Prerequisite: Interactive Math III/IV, Algebra II and Geometry with a grade of "80" or better.
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Honors Pre-Calculus (139)
This math course is designed to prepare students for Calculus. It is an opportunity to review all of the concepts presents in Algebra I Calculator, a computer is available in the room for graphing functions. Honors Pre-Calculus is a college level class geared for the student who will be taking a calculus course in college.
Prerequisite: Algebra II and Geometry with a grade of "90" or above.
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Advanced Placement Calculus (141) Fall semester
Calculus is an advanced This is an Advanced Placement weighted course that prepares the student for the AP Calculus Test of the College Board.
Prerequisite: Honors Pre-Calculus with a grade of "80" or better recommended or Pre-Calculus with a grade of "90" or better.
This course is weighted according to BOE Policy.
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Calculus (143) Fall semester
Calculus is a
Prerequisite: Pre-Calculus with a grade of "80" or better recommended.
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AP Statistics (163) Fall Semester
The purpose of the Advanced Placement course in statistics is to introduce students to the major concepts and tools for collecting, analyzing and drawing conclusions from data. Students are exposed to four broad conceptual themes 1) Exploring Data: Observing patterns and departure from patterns, 2) Planning to study: Deciding what and how to measure 3) Anticipating patterns: producing models using probability and simulation, 4) statistical reference: confirming models. Since Glassboro High School operates under block scheduling, it is proposed that this course by offered daily for one semester (5 credits) A TI-83 plus graphing calculator is required for the course
Students who successfully complete the course and examination may receive credit and/or advanced placement for an one-semester introductory college statistics course.
Prerequisites: Pre-Calc with a grade average of "70" or better.
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HSPA Math (123)
HSPA Math is a junior level course in mathematics for students who are at risk for failing the HSPA. Students who did not perform well in Pre-Algebra, Algebra I, Interactive Math I, II, Geometry, Algebra II OR who failed the GEPA are recommended fro Although students may continue in a math course beyond HSPA Math, most students will satisfy their math course requirements with the completion of HSPA Math. For this reason, topics covered in this course prepare the student for math that may be business or consumer related, while stressing the beginning algebraic techniques.
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SRA Math (111B)
SRA Math is a senior level course in mathematics for students who have failed the HSPA. Students who did not pass HSPA are recommended for Students will satisfactorily complete the SRA process. This course is a terminal math course. For this reason, topics covered in this course prepare the student for math courses that may be business or consumer related, while stressing beginning algebraic techniques
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Advanced Placement Calculus BC (144B) Spring Semester
This course is a continuation of the curriculum begun in AP Calculus AB/Fall. Concepts and applications in differential and integral calculus introduced in the AB section will be further developed. Topics will include, but are not limited to: The calculus of parametric, polar, and vector functions, L'Hopital's Rule, numerical solutions to differential equations using Euler's method, and the treatment and application of infinite series. Successful students will be prepared to take the AP exam for Calculus BC in May.
Prerequisite: Pre-Calculus with a grade of "80" or better recommended.
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Honors Algebra II (145B)
This course is for the advanced mathematics students. The student must have an "90" or higher in Algebra I. All of the topics in Algebra II will be covered, but at a faster pace, and with more enrichment/SAT type problems. In addition, topics on logarithms, exponential functions, and conic sections will be covered. Basic trigonometric identities will be introduced.
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Honors Geometry (137)
This course is for the advanced mathematics students. The student must have an "90" or higher in Algebra I. All of the topics in Geometry will be covered, but at a faster pace, and with more enrichment/SAT type problems. In addition, topics on fractal geometry, Non-Euclidean geometry, Navigation and Astronomy will be covered.
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DOUBLING COURSES: If a student wishes to double up in Algebra I, Algebra II or Geometry, it is important to discuss this with the current math teacher. Two main courses can be taken in one academic year; one in the fall and one in the spring.
DEPARTMENT GRADING POLICY: Grades are calculated through the end of each semester and will be based only on personal effort and performance, not desire or potential. Updated grade reports will be posted on the internet so students may continually monitor their progress. Student performance may be assessed in the following areas: homework, in class assignments (primarily book work and work sheets), quizzes, projects and tests. Final course grades include both quarter grades, and midterm and final exams.
NOTE: Every student should own his/her own graphing calculator since it is the one used in most math classes as well as in science classes, HSPA, SAT and even college. For this reason, we strongly suggest investing in this calculator.
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Tour of Symmetry Groups
This document is being developed to provide a guided tour
of symmetry groups using the Kali program.
Currently, only frieze groups are covered by the tour.
How to use this document
The pages in this document may be viewed with a WWW browser
but may be more useful if printed out in PostScript from a WWW browser.
A PostScript version of each section will eventually be provided
to avoid the hassle of having to save or print each page as PostScript
from the WWW browser.
The first two sections,
Types of Symmetry and Symmetry in Frieze Groups,
cover information that students should be familiar with
before they come into the Geometry Center for a tour.
Teachers may use the materials provided here or other equivalent material
in their classrooms before bringing the students into the Geometry Center,
so that the visit will be more productive.
The third section,
Using Kali to Explore Frieze Groups,
contains a bunch of exercises that students may do while
using the Kali program at the Geometry Center.
The exercises are divided
into groups based on their level of difficulty:
beginner, intermediate and advanced.
The last set of exercises, the
Pattern Gallery,
contains exercises in all ranges of difficulty.
The fourth section contains a very brief mention of
symmetries in the plane. This section may or may not be developed
into exercises at a later date.
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Chapter 4: TRANSFORMATIONS
Chapter 4: TRANSFORMATIONS
The chapter, Transformations is useful for both the visual learner and the abstract learner. Each lesson allows students to manipulate, derive and define mathematical terms through the use of Geometer's Sketchpad. When students work through each lesson they will be introduced and will utilize new GSP commands. Along with new commands, each student will be required to manipulate images in each lesson. Using a trial and error method and the students will then determine the correct transformation image from the file.
This chapter will require both teacher-directed and student-directed learning. Different modes of teaching may be used depending on how well students grasp the new GSP commands.
LESSON ONE - Investigation of Translations
The first lesson in the chapter, Transformations, is a good introductory lesson to the chapter itself. It introduces students to graphs and translating figures constructed using GSP. It incorporates several of the expectations from the grade seven strand "Geometry and Spatial Sense" in the Ontario Curriculum and opens the door for further exploration in the grade eight curriculum.
Investigation of Translations will enable the students to make observations about transformations based on constructed angles in GSP. GSP commands introduced in the lesson include: plot points, graph, translate and transform.
It is recommended that the teacher lead this lesson because of the new GSP commands and concepts used.
LESSON TWO Reflections follows the introductory lesson in this chapter. Students will be using graphs and will be plotting points using GSP functions previously delved into. A new function that will be introduced to students in GSP is the mirror command.
Students will be able to create a triangle on the graph and then visualize the reflection of this image in GSP. Similar to the lesson Investigation of Translations, this lesson is geared towards the grade seven mathematics curriculum.
It is suggested that students have a basic knowledge of mathematical terminology prior to this activity. This will ensure maximum understanding of the mathematical concepts being used in GSP. Since students are familiar with graphing in GSP, instructors may allow students to work independently to discover properties of geometric reflections.
LESSON THREE Rotations allows students to continue experimenting and discovering with geometric figures - this time students will make observations about rotations rather than reflections or translations. Students will investigate the different properties a rotation will have on a shape by using the GSP function rotate. Similar to the previous lessons in this chapter, students will plot specified points on a graph before beginning the transformation.
The suggested mode of learning will be a student-centred approach to this lesson. The students have worked with various GSP commands in the chapter and rotate is the only new command for this lesson. Students will then be able to hypothesize and make observations on their own.
LESSON FOUR - Investigation of Dilations
ONTARIO CURRICULUM Covered:
Grade 7: 7m24, 7m41, 7m47, 7m50, 7m51, 7m52, 7m62, 7m65
Grade 8: 8m55, 8m60, 8m68, 8m70
The lesson Investigation of Dilations delves into area and dilation. Dilation is a transformation that changes all dimensions by a factor k called the scale factor. For enlargements, k is greater than 1 and for reductions, k is between 0 and 1. Since dilations can either be an enlargement or reduction, it is advised that teacher direct this lesson and follow through the steps with the students to ensure maximum understanding of both the mathematical terminology and the GSP commands used. The GSP command that will be introduced and utilized in this specific lesson is dilate.
Along with a new GSP function that will be explored, students will also be required to problem solve independently throughout this activity. Students will make conclusions based on their research in this activity. It is essential for students to have background information regarding the mathematics exhibited in this lesson.
LESSON FIVE - Investigation of Glide ReflectionsThis lesson will incorporate the material learned throughout lessons in this chapter. This is a good culminating lesson as it reviews the GSP commands used and introduces the new commands mark vector, hide transformation and glide reflection. Students will have the opportunity to experiment with the GSP functions to glide both the specified transformation and a created transformation of their choice.
Considering this lesson is highly student-centred, the instructor may allow the students to make discoveries without their guidance.
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Our CCGPS Coordinate Algebra program is a complete set of course materials for Coordinate Algebra. The program provides multiple access points to the same standard, focuses on problem-based learning, and includes a variety of instructional components. The course design has benefited from direct input from Georgia teachers and the materials provide extensive support and scaffolding for teaching and learning. Please click on any of the bold, underlined links below for more information.
The Components
The Teacher Resource – has all of the materials to teach the course, including assessments, station activities and much more. $495.00
Student Resource Book – a resource available for students and their parents to use as a reference (at home or in school) when completing assignments and preparing for tests. $50.00
Online Assessments – In addition to pencil and paper forms included in the TRB, Pre-Assessments, Progress Assessments and Unit Assessments are available online, saving teacher grading time and providing immediate feedback to students. $35/student/year
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Math Class Formats
TMCC offers math classes in a variety of formats to accommodate varying student needs and preferences. Students should check with the mathematics department when in doubt as to the format of a particular class.
Lecture format. Class meets twice a week for one hour and fifteen minutes on one of the TMCC sites. Traditional and/or non-traditional learning/instruction methods may be used (lecture, group work, discovery modules, in-class exercises, question-and-answer sessions, etc.). A lecture math class may include an online component (for example, homework and quizzes).
Computer-based format (Math 95 and 96). These classes meet in a classroom equipped with computers. Students work with interactive software, completing homework and assessments on the computer. Faculty instruct on an individual and/or small group basis. Access to a computer outside of class time is required in order to complete coursework. Computer-based math classes are described in the TMCC class schedule as: "COMPUTER-BASED CLASS: ASSIGNMENTS WILL BE COMPLETED ON A COMPUTER. STUDENTS NEED COMPUTER ACCESS OUTSIDE CLASS TIME."
Online format. Syllabus, class notes, videos, homework, quizzes, practice tests, etc. are delivered online. Students interact with the instructor and with their classmates online. Students must come to the college to take their midterm and final exams (unless proctoring arrangements have been made with the instructor).
Hybrid format. Online class but meets on campus one day per week for discussion.
Self-paced lab format. Class meets twice a week for one hour and fifteen minutes in a math lab. Students work individually and at their own pace. Homework isn't collected. Students take exams after studying the appropriate sections of the textbook. The instructor helps students on an individual and/or small group basis.
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MATLAB is a computer program for people doing numerical computation, especially linear algebra (matrices). It began as a "MATrix LABoratory" program, intended to provide interactive access to the libraries Linpack and Eispack. It has since grown well beyond these libraries, to become a powerful tool for visualization, programming, research, engineering, and communication.
Matlab's strengths include cutting-edge algorithms, enormous data handling abilities, and powerful programming tools. Matlab is not designed for symbolic computation, but it makes up for this weakness by allowing the user to directly link to Maple. The interface is mostly text-based, which may be disconcerting for some users.
Matlab is packaged as a core program with several "toolboxes", sold separately. We will only cover the core package. The current version is Matlab 2012a.
The reader of this document should have at least a passing familiarity with linear algebra and be comfortable using computers. In order to be more broadly understood, we will not cover any engineering topics (e.g. signal processing, spectral analysis), though Matlab is commonly used for these tasks. No previous math software experience is necessary, though we will point out important differences between the various packages along the way.
MATLAB is available for many different kinds of computer platforms. A student edition is available from local bookstores for your personal Windows, Macintosh, and Linux systems.
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Job Title
Math Teacher
Summary
Teach courses
pertaining to mathematical concepts, statistics, and actuarial science and to
the application of original and standardized mathematical techniques in
solving specific problems and situations.
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The Basic Math DVD Series helps students build confidence in their mathematical knowledge, skills, and ability.
In this episode, the graphing calculator in introduced in the context of statistics. Students will learn how statistics can be used to analyze sets of data in order to measure tendency and variation. The concept of outlier is introduced, as well as the box plot, graphic displays of data, and the qualitative analysis of data. Grades 3-7. 30 minutes on DVD.
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Course Offerings in Mathematics
Below is a list of available courses offered by the Mathematics Department. Consult the Registrar's Office and the College Catalog for registration information.
MATH 109 - INTRO TO QUANTITATIVE REASONING
This course presents mathematical ideas in a real world context. Topics covered include critical thinking and problem solving, the mathematics of finance, basic statistical principles, mathematics and the arts, and the theory of voting.
Hours credit: 3. Students considering Curricular Studies should not register for this course, as students may not receive credit for both MATH 109 and MATH 208
MATH 113 - ELEMENTARY MATHEMATICAL MODELING
This course explores mathematical models of natural phenomena such as population growth and radioactive decay. Analysis of data using computer technology. Linear, quadratic, general polynomial, exponential, and logarithmic models will be discussed.
Hours credit: 3. Not open to students who have completed Mathematics 119R or above, except by departmental recommendation.
MATH 119 - PRECALCULUS
A study of the properties of various functions, including polynomial, trigonometric, exponential, and logarithmic. Analytic geometry of conic sections.
Hours credit: 3. Prerequisite: Mathematics 113 or the equivalent. Not open to students who have been placed into Mathematics 149 or above, except by permission of the Department.
MATH 149 - CALCULUS I
Limits, continuity, and differentiation of algebraic functions of one variable. Applications to curve sketching, optimization, and rates of change. The definite integral applied to finding the area under a curve.
Hours credit: 3. Prerequisite: Mathematics 119R or the equivalent.
MATH 150 - CALCULUS II
A continuation of Mathematics 149R. Volumes and surface area of solids of revolution. Lengths of curves. The logarithm and exponential functions. Techniques of integration. Areas in polar coordinates. Improper integrals, infinite series, and power series.
Hours credit: 3. Prerequisite: Mathematics 149R or permission of the Department.
MATH 227 - ELEMENTARY APPLIED STATISTICS
An introduction to statistics, including probability, binomial distributions, normal distributions, sampling theory, testing hypotheses, chi-square tests, and linear regression.
Hours credit: 3. Not open to students who have satisfactorily completed Mathematics 343. A student may receive credit for only one of these courses: MATH 227, POL 231, or PSYC 227. Offered second semester.
MATH 229 - ADVANCED MATHEMATICAL PROBLEM SOLVING
In this course, students will be expected to solve and present solutions to a collection of problems gathered from various mathematics competitions. Problem solutions may involve the techniques of classical algebra, geometry, calculus, and combinatorics.
Hours credit: 1.0. Prerequisite: Permission of the Instructor. May be repeated for credit up to a maximum of 4 hours.
MATH 250 - CALCULUS III
An introduction to vector calculus. Differential and integral calculus of more than one variable. Vector fields, including Green's, Stokes', and the Divergence Theorems.
Hours credit: 3. Prerequisite: Mathematics 150R and either MATH 241 or PHYS 115, or permission of the Department.
MATH 318 - FOUNDATIONS OF GEOMETRY
A study of modern geometries, including finite projective and Non-Euclidean geometries. Geometric transformations and synthetic geometry.
Hours credit: 3. Prerequisite: Mathematics 150R or permission of the Department. Offered alternate years.
MATH 331 - DIFFERENTIAL EQUATIONS
First order linear and non-linear equations, second and higher order linear equations, series solutions, Laplace transforms, and systems of linear differential equations. Applications, primarily to mechanics and population dynamics.
Hours credit: 3. Prerequisite: Mathematics 150R and 241, or the course may be taken concurrently with MATH 241 by permission of the Department.
MATH 353 - MATHEMATICAL MODELING
The construction and analysis of mathematical models to solve problems in the physical and social sciences. Dynamical systems are emphasized with a particular concentration on linear and non-linear discrete dynamical systems. Topics may include dimensional analysis, stability, chaos, and fractals.
Hours credit: 3. Prerequisites: MATH 150R and 241. Offered alternate years.
MATH 420 - NUMERICAL ANALYSIS
A study of algorithms for solving mathematical problems using computers. These problems include finding the roots of functions, solving systems of linear equations, interpolation, approximate integration, and solving differential equations. Implementation of these algorithms on the computer will be an important part of the course.
Hours credit: 3. Prerequisite: MATH 150R and 241. Prerequisite or corequisite: CSCI 156. Offered alternate years.
MATH 443 - INTRODUCTION TO ANALYSIS
A rigorous study of limits, continuity, differentiation, and integration of functions of a real variable. Hours credit: 3. Prerequisite: Mathematics 250 or permission of the Department.
MATH 494 - SENIOR SEMINAR
Readings on the history of mathematics from the seventeenth century through modern times. Oral reports by students and faculty on topics of interest in mathematics. Written reports are also required.
Hours credit: 3. Prerequisite: Mathematics 360 and 443.
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books.google.ch - This... Geometry
Computational Geometry:
This and techniques from computational geometry are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. For students this motivation will be especially welcome. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement. All the basic techniques and topics from computational geometry, as well as several more advanced topics, are covered. The book is largely self-contained and can be used for self-study by anyone with a basic background in algorithms. In the second edition, besides revisions to the first edition, a number of new exercises have been added.
Bewertungen von Nutzern
Review: Computational Geometry: Algorithms and Applications
Review: Computational Geometry: Algorithms and Applications
Nutzerbericht - Willy Van den driessche - Goodreads
Beauty is the first test. This is a very beautiful book (form) with a beautiful contents. The book explains in a very throrough way some of the fundamental algoritms in "computational geometry". You ...Vollständige Rezension lesen
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Because of the visual nature of these courses, this set is available only in video formats. It contains thousands of visual elements to enhance your learning experience, including step-by-step diagrams and solutions, animations, interactive practice problems, graphs, charts, and on-screen equations and text.
Your Guide to High School Math (Set)
COURSE DESCRIPTION
Your Guide to High School Math (Set)
Course
1
of
5:
Algebra I Professor
Professor James A. Sellers,
The Pennsylvania State University Ph.D., The Pennsylvania State University Algebra I is an entirely new course designed to meet the concerns of both students and their parents. These 36 accessible lectures make the concepts of first-year algebra—including variables, order of operations, and functions—easy to grasp. For anyone wanting to learn algebra from the beginning, or for anyone needing a thorough review, Professor James A. Sellers will prove to be an inspirational and ideal tutor. Open yourself up to the world of opportunity that algebra offers by making the best possible start on mastering this all-important subject.
Course Lecture Titles
36
Lectures
30
minutes/lecture
1.
An Introduction to the Course
Professor Sellers introduces the general topics and themes for the course, describing his approach and recommending a strategy for making the best use of the lessons and supplementary workbook. Warm up with some simple problems that demonstrate signed numbers and operations.
19.
Factoring Trinomials2.
Order of Operations20.
Quadratic Equations—Factoring3.
Percents, Decimals, and Fractions
Continue your study of math fundamentals by exploring various procedures for converting between percents, decimals, and fractions. Professor Sellers notes that it helps to see these procedures as ways of presenting the same information in different forms.
21.
Quadratic Equations—The Quadratic Formula4.
Variables and Algebraic Expressions22.
Quadratic Equations—Completing the Square5.
Operations and Expressions23.
Representations of Quadratic Functions6.
Principles of Graphing in 2 Dimensions24.
Quadratic Equations in the Real World7.
Solving Linear Equations, Part 125.
The Pythagorean Theorem8.
Solving Linear Equations, Part 226.
Polynomials of Higher Degree9.
Slope of a Line27.
Operations and Polynomials10.
Graphing Linear Equations, Part 128.
Rational Expressions, Part 111.
Graphing Linear Equations, Part 229.
Rational Expressions, Part 2
Continuing your exploration of rational expressions, try your hand at multiplying and dividing them. The key to solving these complicated-looking equations is to proceed one step at a time. Close the lesson with a problem that brings together all you've learned about rational functions.
12.
Parallel and Perpendicular Lines30.
Graphing Rational Functions, Part 1
Examine the distinctive graphs formed by rational functions, which may form vertical or horizontal curves that aren't even connected on a graph. Learn to identify the intercepts and the vertical and horizontal asymptotes of these fascinating curves.
13.
Solving Word Problems with Linear Equations31.
Graphing Rational Functions, Part 214.
Linear Equations for Real-World Data32.
Radical Expressions15.
Systems of Linear Equations, Part 133.
Solving Radical Equations16.
Systems of Linear Equations, Part 234.
Graphing Radical Functions17.
Linear Inequalities35.
Sequences and Pattern Recognition, Part 1
Pattern recognition is an important and fascinating mathematical skill. Investigate two types of number patterns: geometric sequences and arithmetic sequences. Learn how to analyze such patterns and work out a formula that predicts any term in the sequence
18.
An Introduction to Quadratic Polynomials36.
Sequences and Pattern Recognition, Part 2Course Lecture Titles
36
Lectures
30
minutes/lecture
1.
An Introduction to Algebra II
Professor Sellers explains the topics covered in the course, the importance of algebra, and how you can get the most out of these lessons. You then launch into the fundamentals of algebra by reviewing the order of operations and trying your hand at several problems.
19.
An Introduction to Polynomials
Pause to examine the nature of polynomials—a class of algebraic expressions that you've been working with since the beginning of the course. Professor Sellers introduces several useful concepts, such as the standard form of polynomials and their degree, domain, range, and leading coefficients.
2.
Solving Linear Equations
Explore linear equations, starting with one-step equations and then advancing to those requiring two or more steps to solve. Next, apply the distributive property to simplify certain problems, and then learn about the three categories of linear equations.
20.
Graphing Polynomial Functions
Deepen your insight into polynomial functions by graphing them to see how they differ from non-polynomials. Then learn how the general shape of the graph can be predicted from the highest exponent of the polynomial, known as its degree. Finally, explore how other terms in the function also affect the graph.
3.
Solving Equations Involving Absolute Values
Taking your knowledge of linear equations a step further, look at examples involving absolute values, which can be thought of as a distance on a number line, always expressed as a positive value. Use your critical-thinking skills to recognize absolute value problems that have limited or no solutions.
21.
Combining Polynomials
Switch from graphs to the algebraic side of polynomial functions, learning how to combine them in many different ways, including addition, subtraction, multiplication, and even long division, which is easier than it seems. Discover which of these operations produce new polynomials and which do not.
4.
Linear Equations and Functions
Moving into the visual realm, learn how linear equations are represented as straight lines on graphs using either the slope-intercept or point-slope forms of the function. Next, investigate parallel and perpendicular lines and how to identify them by the value of their slopes.
22.
Solving Special Polynomial Equations
Learn how to solve polynomial equations where the degree is greater than two by turning them into expressions you already know how to handle. Your "toolbox" includes techniques called the difference of two squares, the difference of two cubes, and the sum of two cubes.
5.
Graphing Essentials
Reversing the procedure from the previous lesson, start with an equation and draw the line that corresponds to it. Then test your knowledge by matching four linear equations to their graphs. Finally, learn how to rewrite an equation to move its graph up, down, left, or right—or flip it entirely.
23.
Rational Roots of Polynomial Equations
Going beyond the approaches you've learned so far, discover how to solve polynomial equations by applying two powerful tools for finding rational roots: the rational roots theorem and the factor theorem. Both will prove very useful in succeeding lessons.
6.
Functions—Introduction, Examples, Terminology
Functions are crucially important not only for algebra, but for precalculus, calculus, and higher mathematics. Learn the definition of a function, the notation, and associated concepts such as domain and range. Then try out the vertical line test for determining whether a given curve is a graph of a function.
24.
The Fundamental Theorem of Algebra
Explore two additional tools for identifying the roots of polynomial equations: Descartes' rule of signs, which narrows down the number of possible positive and negative real roots; and the fundamental theorem of algebra, which gives the total of all roots for a given polynomial.
7.
Systems of 2 Linear Equations, Part 1
Practice solving systems of two linear equations by graphing the corresponding lines and looking for the intersection point. Discover that there are three possible outcomes: no solution, infinitely many solutions, and exactly one solution.
8.
Systems of 2 Linear Equations, Part 2
Explore two other techniques for solving systems of two linear equations. First, the method of substitution solves one of the equations and substitutes the result into the other. Second, the method of elimination adds or subtracts the equations to see if a variable can be eliminated.
26.
Solving Equations Involving Radicals
Drawing on your experience with roots and radicals from the previous lesson, try your hand at solving equations with these expressions. Begin by learning how to manipulate rational, or fractional, exponents. Then practice with simple equations, while being on the lookout for extraneous, or "imposter," solutions.
9.
Systems of 3 Linear Equations
As the number of variables increases, it becomes unwieldy to solve systems of linear equations by graphing. Learn that these problems are not as hard as they look and that systems of three linear equations often yield to the strategy of successively eliminating variables.
27.
Graphing Power, Radical, and Root Functions
Using graph paper, experiment with curves formed by simple radical functions. First, determine the domain of the function, which tells you the general location of the graph on the coordinate plane. Then, investigate how different terms in the function alter the graph in predictable ways.
10.
Solving Systems of Linear Inequalities
Make the leap into systems of linear inequalities, where the solution is a set of values on one side or another of a graphed line. An inequality is an assertion such as "less than" or "greater than," which encompasses a range of values.
28.
An Introduction to Rational Functions
Shift your focus to graphs of rational functions—functions that are the ratio of two polynomials. These graphs are more complicated than those from the previous lesson, but their general characteristics can be quickly determined by calculating the domain, the x- and y-intercepts, and the vertical and horizontal asymptotes.
11.
An Introduction to Quadratic Functions
Begin your investigation of quadratic functions by visualizing what these functions look like when graphed. They always form a U-shaped curve called a parabola, whose location on the coordinate plane can be predicted based on the individual terms of the equation.
29.
The Algebra of Rational Functions
Combine rational functions using addition, subtraction, multiplication, division, and composition. The trick is to start each problem by putting the expressions in factored form, which makes the calculations go more smoothly. Leaving the answer in factored form also allows other operations, such as graphing, to be easily performed.
12.
Quadratic Equations—Factoring
One of the most important skills related to quadratics is factoring. Review the basics of factoring, and learn to recognize a very useful special case known as the difference of two squares. Close by working on a word problem that translates into a quadratic equation.
30.
Partial Fractions
Now that you know how to add rational expressions, try the opposite procedure of splitting a more complicated rational expression into its component parts. Called partial fraction decomposition, this approach is a topic in introductory calculus and is used for solving a wide range of more advanced math problems.
13.
Quadratic Equations—Square Roots
The square root approach to solving quadratic equations works not just for perfect squares, such as 3 × 3 = 9, but also for values that don't seem to involve squares at all. Probe the idea behind this technique, and also venture into the strange world of complex numbers.
31.
An Introduction to Exponential Functions
Exponential functions are important in real-world applications involving growth and decay rates, such as compound interest and depreciation. Experiment with simple exponential functions, exploring such concepts as the base, growth factor, and decay factor, and how different values for these terms affect the graph of the function.
14.
Completing the Square
Turn a quadratic equation into an easily solvable form that includes a perfect square—a technique called completing the square. An important benefit of this approach is that the rewritten form gives the coordinates for the vertex of the parabola represented by the equation.
32.
An Introduction to Logarithmic Functions
Plot a logarithmic function on the coordinate plane to see how it is the mirror image of a corresponding exponential function. Just like a mirror image, logarithms can be disorienting at first; but by studying their properties you will discover how they make certain calculations much simpler.
15.
Using the Quadratic Formula
When other approaches fail, one tool can solve every quadratic equation: the quadratic formula. Practice this formula on a wide range of problems, learning how a special expression called the discriminant immediately tells how many real-number solutions the equation has.
33.
Uses of Exponential and Logarithmic Functions
Delve deeper into exponential and logarithmic functions with the goal of solving a typical financial investment problem using the "Pert" formula. To prepare, study the change of base formula for logarithms and the special function of the base called e.
16.
Solving Quadratic Inequalities
Extending the exercises on inequalities from lecture 10, step into the realm of quadratic inequalities, where the boundary graph is not a straight line but a parabola. Use your skills analyzing quadratic expressions to sketch graphs quickly and solve systems of quadratic inequalities.
34.
The Binomial Theorem
Pascal's triangle is a famous triangular array of numbers that corresponds to the coefficients of binomials of different powers. In a lesson connecting a branch of mathematics called combinatorics with algebra, investigate the formula for each value in Pascal's triangle, the factorial function, and the binomial theorem.
17.
Conic Sections—Parabolas and Hyperbolas
Delve into the algebra of conic sections, which are the cross-sectional shapes produced by slicing a cone at different angles. In this lesson, study parabolas and hyperbolas, which differ in how many variable terms are squared in each. Also learn how to sketch a hyperbola from its equation.
35.
Permutations and Combinations
Continue your study of the link between combinatorics and algebra by using the factorial function to solve problems in permutations and combinations. For example, what are all the permutations of the letters a, b, c? And how many combinations of four books are possible when you have six to choose from?
18.
Conic Sections—Circles and Ellipses
Investigate the algebraic properties of the other two conic sections: ellipses and circles. Ellipses resemble stretched circles and are defined by their major and minor axes, whose ratio determines the ellipse's eccentricity. Circles are ellipses whose eccentricity = 1, with the major and minor axes equal.
36.
Elementary Probability
After a short introduction to probability, celebrate your completion of the course with a deck of cards. Can you use the principles of probability, permutations, and combinations to calculate the probability of being dealt different hands? As with the rest of algebra, once you know the rules, it's simplicity itself!
Whether you're a high-school student preparing for the challenges of higher math classes, an adult who needs a refresher in math to prepare for a new career, or someone who just wants to keep his or her mind active and sharp, there's no denying that a solid grasp of arithmetic and prealgebra is essential in today's world. In Professor James A. Sellers' engaging course, Mastering the Fundamentals of Mathematics, you learn all the key math topics you need to know. In 24 lectures packed with helpful examples, practice problems, and guided walkthroughs, you'll finally grasp the all-important fundamentals of math in a way that truly sticks.
Course Lecture Titles
24
Lectures
30
minutes/lecture
1.
Addition and Subtraction
This introductory lecture starts with Professor Sellers' overview of the general topics and themes you'll encounter throughout the course. Then, plunge into an engaging review of the addition and subtraction of whole numbers, complete with several helpful tips designed to help you approach these types of problems with more confidence.
13.
Exponents and Order of Operations
Explore a fifth fundamental mathematical operation: exponentiation. First, take a step-by-step look at the order of operations for handling longer calculations that involve multiple tasks—complete with invaluable tips to help you handle them with ease. Then, see where exponentiation fits in this larger process.
2.
Multiplication
Continue your quick review of basic mathematical operations, this time with a focus on the multiplication of whole numbers. In addition to uncovering the relationship between addition and multiplication, you'll get plenty of opportunities to strengthen your ability to multiply two 2-digit numbers, two 3-digit numbers, and more.
14.
Negative and Positive Integers
Improve your confidence in dealing with negative numbers. You'll learn to use the number line to help visualize these numbers; discover how to rewrite subtraction problems involving negative numbers as addition problems to make them easier; examine the rules involved in multiplying and dividing with them; and much more.
3.
Long Division
Turn now to the opposite of multiplication: division. Learn how to properly set up a long division problem, how to check your answers to make sure they're correct, how to handle zeroes when they appear in a problem, and what to do when a long division problem ends with a remainder.
15.
Introduction to Square Roots
In this lecture, finally make sense of square roots. Professor Sellers offers examples to help you sidestep issues many students express frustration with, shows you how to simplify radical expressions involving addition and subtraction, and reveals how to find the approximate value of a square root without using a calculator.
4.
Introduction to Fractions
Mathematics is also filled with "parts" of whole numbers, or fractions. In the first of several lectures on fractions, define key terms and focus on powerful techniques for determining if fractions are equivalent, finding out which of two fractions is larger, and reducing fractions to their lowest terms.
16.
Negative and Fractional Powers
What happens when you have to raise numbers to a fraction of a power? How about when you have to deal with negative exponents? Or negative fractional exponents? No need to worry —Professor Sellers guides you through this tricky mathematical territory, arming you with invaluable techniques for approaching these scenarios.
5.
Adding and Subtracting Fractions
Fractions with the same denominator. Fractions with different denominators. Mixed numbers. Here, learn ways to add and subtract them all (and sometimes even in the same problem) and get tips for reducing your answers to their lowest terms. Math with fractions, you'll discover, doesn't have to be intimidating—it can even be fun!
17.
Graphing in the Coordinate Plane
Grab some graph paper and learn how to graph objects in the coordinate (or xy) plane. You'll find out how to plot points, how to determine which quadrant they go in, how to sketch the graph of a line, how to determine a line's slope, and more.
6.
Multiplying Fractions
Continue having fun with fractions, this time by mastering how to multiply them and reduce your answer to its lowest term. Professor Sellers shows you how to approach and solve multiplication problems involving fractions (with both similar and different denominators), fractions and whole numbers, and fractions and mixed numbers.
18.
Geometry—Triangles and Quadrilaterals
Continue exploring the visual side of mathematics with this look at the basics of two-dimensional geometry. Among the topics you'll focus on here are the various types of triangles (including scalene and obtuse triangles) and quadrilaterals (such as rectangles and squares), as well as methods for measuring angles, area, and perimeter.
7.
Dividing Fractions
Professor Sellers walks you step-by-step through the process for speedily solving division problems involving fractions in this lecture filled with helpful practice problems. You'll also learn how to better handle calculations involving different notations, fractions, and whole numbers, and even word problems involving the division of fractions.
19.
Geometry—Polygons and Circles
Gain a greater appreciation for the interaction between arithmetic and geometry. First, learn how to recognize and approach large polygons, including hexagons and decagons. Then, explore the various concepts behind circles (such as radius, diameter, and the always intriguing pi), as well as methods for calculating their circumference, area, and perimeter.
8.
Adding and Subtracting Decimals
What's 29.42 + 84.67? Or 643 + 82.987? What about 25.7 – 10.483? Problems like these are the focus of this helpful lecture on adding and subtracting decimals. One tip for making these sorts of calculations easier: making sure your decimal points are all lined up vertically.
20.
Number Theory—Prime Numbers and Divisors
Shift gears and demystify number theory, which takes as its focus the study of the properties of whole numbers. Concepts that Professor Sellers discusses and teaches you how to engage with in this insightful lecture include divisors, prime numbers, prime factorizations, greatest common divisors, and factor trees.
9.
Multiplying and Dividing Decimals
Investigate the best ways to multiply and divide decimal numbers. You'll get insights into when and when not to ignore the decimal point in your calculations, how to check your answer to ensure that your result has the correct number of decimal places, and how to express remainders in decimals.
21.
Number Theory—Divisibility Tricks
In this second lecture on the world of number theory, take a closer look at the relationships between even and odd numbers, as well as the rules of divisibility for particular numbers. By the end, you'll be surprised that something as intimidating as number theory could be made so accessible.
10.
Fractions, Decimals, and Percents
Take a closer look at converting between percents, decimals, and fractions—an area of basic mathematics that many people have a hard time with. After learning the techniques in this lecture and using them on numerous practice problems, you'll be surprised at how easy this type of conversion is to master.
22.
Introduction to Statistics
Get a solid introduction to statistics, one of the most useful areas of mathematics. Here, you'll focus on the four basic "measurements" statisticians use when gleaning meaning from data: mean, media, mode, and range. Also, see these concepts at work in everyday scenarios in which statistics plays a key role.
11.
Percent Problems
Use the skills you developed in the last lecture to better approach and solve different kinds of percentage problems you'd most likely encounter in your everyday life. Among these everyday scenarios: calculating the tip at a restaurant and determining how much money you're saving on a store's discount.
23.
Introduction to Probability
Learn more about probability, a cousin of statistics and another mathematical field that helps us make sense of the seemingly unexplainable nature of the world. You'll consider basic questions and concepts from probability, drawing on the knowledge and skills of the fundamentals of mathematics you acquired in earlier lectures.
12.
Ratios and Proportions
How do ratios and proportions work? How can you figure out if a particular problem is merely just a ratio or proportion problem in disguise? What are some pitfalls to watch out for? And how can a better understanding of these subjects help save you money? Find out here.
24.
Introduction to Algebra
Professor Sellers reviews the importance of math in daily life and previews the next logical step in your studies: Algebra I (which involves variables). Whether you're planning to take more Great Courses in mathematics or simply looking to sharpen your mind, you'll be sent off with new levels of confidence.
Course Lecture Titles
36
Lectures
30
minutes/lecture
1.
An Introduction to Precalculus—Functions
Precalculus is important preparation for calculus, but it's also a useful set of skills in its own right, drawing on algebra, trigonometry, and other topics. As an introduction, review the essential concept of the function, try your hand at simple problems, and hear Professor Edwards's recommendations for approaching the course.
19.
Trigonometric Equations
In calculus, the difficult part is often not the steps of a problem that use calculus but the equation that's left when you're finished, which takes precalculus to solve. Hone your skills for this challenge by identifying all the values of the variable that satisfy a given trigonometric equation.
2.
Polynomial Functions and Zeros
The most common type of algebraic function is a polynomial function. As examples, investigate linear and quadratic functions, probing different techniques for finding roots, or "zeros." A valuable tool in this search is the intermediate value theorem, which identifies real-number roots for polynomial functions.
20.
Sum and Difference Formulas
Study the important formulas for the sum and difference of sines, cosines, and tangents. Then use these tools to get a preview of calculus by finding the slope of a tangent line on the cosine graph. In the process, you discover the derivative of the cosine function.
3.
Complex Numbers
Step into the strange and fascinating world of complex numbers, also known as imaginary numbers, where i is defined as the square root of -1. Learn how to calculate and find roots of polynomials using complex numbers, and how certain complex expressions produce beautiful fractal patterns when graphed.
21.
Law of Sines
Return to the subject of triangles to investigate the law of sines, which allows the sides and angles of any triangle to be determined, given the value of two angles and one side, or two sides and one opposite angle. Also learn a sine-based formula for the area of a triangle.
4.
Rational Functions
Investigate rational functions, which are quotients of polynomials. First, find the domain of the function. Then, learn how to recognize the vertical and horizontal asymptotes, both by graphing and comparing the values of the numerator and denominator. Finally, look at some applications of rational functions.
22.
Law of Cosines
Given three sides of a triangle, can you find the three angles? Use a generalized form of the Pythagorean theorem called the law of cosines to succeed. This formula also allows the determination of all sides and angles of a triangle when you know any two sides and their included angle.
5.
Inverse Functions
Discover how functions can be combined in various ways, including addition, multiplication, and composition. A special case of composition is the inverse function, which has important applications. One way to recognize inverse functions is on a graph, where the function and its inverse form mirror images across the line y = x.
23.
Introduction to Vectors
Vectors symbolize quantities that have both magnitude and direction, such as force, velocity, and acceleration. They are depicted by a directed line segment on a graph. Experiment with finding equivalent vectors, adding vectors, and multiplying vectors by scalars.
6.
Solving Inequalities
You have already used inequalities to express the set of values in the domain of a function. Now study the notation for inequalities, how to represent inequalities on graphs, and techniques for solving inequalities, including those involving absolute value, which occur frequently in calculus.
24.
Trigonometric Form of a Complex Number
Apply your trigonometric skills to the abstract realm of complex numbers, seeing how to represent complex numbers in a trigonometric form that allows easy multiplication and division. Also investigate De Moivre's theorem, a shortcut for raising complex numbers to any power.
7.
Exponential Functions
Explore exponential functions—functions that have a base greater than 1 and a variable as the exponent. Survey the properties of exponents, the graphs of exponential functions, and the unique properties of the natural base e. Then sample a typical problem in compound interest.
25.
Systems of Linear Equations and Matrices
Embark on the first of four lectures on systems of linear equations and matrices. Begin by using the method of substitution to solve a simple system of two equations and two unknowns. Then practice the technique of Gaussian elimination, and get a taste of matrix representation of a linear system.
8.
Logarithmic Functions
A logarithmic function is the inverse of the exponential function, with all the characteristics of inverse functions covered in Lecture 5. Examine common logarithms (those with base 10) and natural logarithms (those with base e), and study such applications as the "rule of 70" in banking.
26.
Operations with Matrices
Deepen your understanding of matrices by learning how to do simple operations: addition, scalar multiplication, and matrix multiplication. After looking at several examples, apply matrix arithmetic to a commonly encountered problem by finding the parabola that passes through three given points.
9.
Properties of Logarithms
Learn the secret of converting logarithms to any base. Then review the three major properties of logarithms, which allow simplification or expansion of logarithmic expressions—methods widely used in calculus. Close by focusing on applications, including the pH system in chemistry and the Richter scale in geology.
27.
Inverses and Determinants of Matrices
Get ready for applications involving matrices by exploring two additional concepts: the inverse of a matrix and the determinant. The algorithm for calculating the inverse of a matrix relies on Gaussian elimination, while the determinant is a scalar value associated with every square matrix.
10.
Exponential and Logarithmic Equations
Practice solving a range of equations involving logarithms and exponents, seeing how logarithms are used to bring exponents "down to earth" for easier calculation. Then try your hand at a problem that models the heights of males and females, analyzing how the models are put together.
28.
Applications of Linear Systems and Matrices
Use linear systems and matrices to analyze such questions as these: How can the stopping distance of a car be estimated based on three data points? How does computer graphics perform transformations and rotations? How can traffic flow along a network of roads be modeled?
11.
Exponential and Logarithmic Models
Finish the algebra portion of the course by delving deeper into exponential and logarithmic equations, using them to model real-life phenomena, including population growth, radioactive decay, SAT math scores, the spread of a virus, and the cooling rate of a cup of coffee.
29.
Circles and Parabolas
In the first of two lectures on conic sections, examine the properties of circles and parabolas. Learn the formal definition and standard equation for each, and solve a real-life problem involving the reflector found in a typical car headlight.
12.
Introduction to Trigonometry and Angles
Trigonometry is a key topic in applied math and calculus with uses in a wide range of applications. Begin your investigation with the two techniques for measuring angles: degrees and radians. Typically used in calculus, the radian system makes calculations with angles easier.
30.
Ellipses and Hyperbolas
Continue your survey of conic sections by looking at ellipses and hyperbolas, studying their standard equations and probing a few of their many applications. For example, calculate the dimensions of the U.S. Capitol's "whispering gallery," an ellipse-shaped room with fascinating acoustical properties.
13.
Trigonometric Functions—Right Triangle Definition
The Pythagorean theorem, which deals with the relationship of the sides of a right triangle, is the starting point for the six trigonometric functions. Discover the close connection of sine, cosine, tangent, cosecant, secant, and cotangent, and focus on some simple formulas that are well worth memorizing.
31.
Parametric Equations
How do you model a situation involving three variables, such as a motion problem that introduces time as a third variable in addition to position and velocity? Discover that parametric equations are an efficient technique for solving such problems. In one application, you calculate whether a baseball hit at a certain angle and speed will be a home run.
14.
Trigonometric Functions—Arbitrary Angle Definition
Trigonometric functions need not be confined to acute angles in right triangles; they apply to virtually any angle. Using the coordinate plane, learn to calculate trigonometric values for arbitrary angles. Also see how a table of common angles and their trigonometric values has wide application.
32.
Polar Coordinates
Take a different mathematical approach to graphing: polar coordinates. With this system, a point's location is specified by its distance from the origin and the angle it makes with the positive x axis. Polar coordinates are surprisingly useful for many applications, including writing the formula for a valentine heart!
15.
Graphs of Sine and Cosine Functions
The graphs of sine and cosine functions form a distinctive wave-like pattern. Experiment with functions that have additional terms, and see how these change the period, amplitude, and phase of the waves. Such behavior occurs throughout nature and led to the discovery of rapidly rotating stars called pulsars in 1967.
33.
Sequences and Series
Get a taste of calculus by probing infinite sequences and series—topics that lead to the concept of limits, the summation notation using the Greek letter sigma, and the solution to such problems as Zeno's famous paradox. Also investigate Fibonacci numbers and an infinite series that produces the number e.
16.
Graphs of Other Trigonometric Functions
Continue your study of the graphs of trigonometric functions by looking at the curves made by tangent, cosecant, secant, and cotangent expressions. Then bring several precalculus skills together by using a decaying exponential term in a sine function to model damped harmonic motion.
34.
Counting Principles
Counting problems occur frequently in real life, from the possible batting lineups on a baseball team to the different ways of organizing a committee. Use concepts you've learned in the course to distinguish between permutations and combinations and provide precise counts for each.
17.
Inverse Trigonometric Functions
For a given trigonometric function, only a small part of its graph qualifies as an inverse function as defined in Lecture 5. However, these inverse trigonometric functions are very important in calculus. Test your skill at identifying and working with them, and try a problem involving a rocket launch.
35.
Elementary Probability
What are your chances of winning the lottery? Of rolling a seven with two dice? Of guessing your ATM PIN number when you've forgotten it? Delve into the rudiments of probability, learning basic vocabulary and formulas so that you know the odds.
18.
Trigonometric Identities
An equation that is true for every possible value of a variable is called an identity. Review several trigonometric identities, seeing how they can be proved by choosing one side of the equation and then simplifying it until a true statement remains. Such identities are crucial for solving complicated trigonometric equations.
36.
GPS Devices and Looking Forward to Calculus
In a final application, locate a position on the surface of the earth with a two-dimensional version of GPS technology. Then close by finding the tangent line to a parabola, thereby solving a problem in differential calculus and witnessing how precalculus paves the way for the next big mathematical adventure.
Course Lecture Titles
36
Lectures
30
minutes/lecture
1.
A Preview of Calculus
Calculus is the mathematics of change, a field with many important applications in science, engineering, medicine, business, and other disciplines. Begin by surveying the goals of the course. Then get your feet wet by investigating the classic tangent line problem, which illustrates the concept of limits.
19.
The Area Problem and the Definite Integral
One of the classic problems of integral calculus is finding areas bounded by curves. This was solved for simple curves by the ancient Greeks. See how a more powerful method was later developed that produces a number called the definite integral, and learn the relevant notation.
2.
Review—Graphs, Models, and Functions
In the first of two review lectures on precalculus, examine graphs of equations and properties such as symmetry and intercepts. Also explore the use of equations to model real life and begin your study of functions, which Professor Edwards calls the most important concept in mathematics.
20.
The Fundamental Theorem of Calculus, Part 1
The two essential ideas of this course—derivatives and integrals—are connected by the fundamental theorem of calculus, one of the most important theorems in mathematics. Get an intuitive grasp of this deep relationship by working several problems and surveying a proof.
3.
Review—Functions and Trigonometry
Continue your review of precalculus by looking at different types of functions and how they can be identified by their distinctive shapes when graphed. Then review trigonometric functions, using both the right triangle definition as well as the unit circle definition, which measures angles in radians rather than degrees.
21.
The Fundamental Theorem of Calculus, Part 2
Try examples using the second fundamental theorem of calculus, which allows you to let the upper limit of integration be a variable. In the process, explore more relationships between differentiation and integration, and discover how they are almost inverses of each other.
4.
Finding Limits
Jump into real calculus by going deeper into the concept of limits introduced in Lecture 1. Learn the informal, working definition of limits and how to determine a limit in three different ways: numerically, graphically, and analytically. Also discover how to recognize when a given function does not have a limit.
22.
Integration by Substitution
Investigate a straightforward technique for finding antiderivatives, called integration by substitution. Based on the chain rule, it enables you to convert a difficult problem into one that's easier to solve by using the variable u to represent a more complicated expression.
5.
An Introduction to Continuity
Broadly speaking, a function is continuous if there is no interruption in the curve when its graph is drawn. Explore the three conditions that must be met for continuity—along with applications of associated ideas, such as the greatest integer function and the intermediate value theorem.
23.
Numerical Integration
When calculating a definite integral, the first step of finding the antiderivative can be difficult or even impossible. Learn the trapezoid rule, one of several techniques that yield a close approximation to the definite integral. Then do a problem involving a plot of land bounded by a river.
6.
Infinite Limits and Limits at Infinity
Infinite limits describe the behavior of functions that increase or decrease without bound, in which the asymptote is the specific value that the function approaches without ever reaching it. Learn how to analyze these functions, and try some examples from relativity theory and biology.
24.
Natural Logarithmic Function—Differentiation
Review the properties of logarithms in base 10. Then see how the so-called natural base for logarithms, e, has important uses in calculus and is one of the most significant numbers in mathematics. Learn how such natural logarithms help to simplify derivative calculations.
7.
The Derivative and the Tangent Line Problem
Building on what you have learned about limits and continuity, investigate derivatives, which are the foundation of differential calculus. Develop the formula for defining a derivative, and survey the history of the concept and its different forms of notation.
25.
Natural Logarithmic Function—Integration
Continue your investigation of logarithms by looking at some of the consequences of the integral formula developed in the previous lecture. Next, change gears and review inverse functions at the precalculus level, preparing the way for a deeper exploration of the subject in coming lectures.
8.
Basic Differentiation Rules
Practice several techniques that make finding derivatives relatively easy: the power rule, the constant multiple rule, sum and difference rules, plus a shortcut to use when sine and cosine functions are involved. Then see how derivatives are the key to determining the rate of change in problems involving objects in motion.
26.
Exponential Function
The inverse of the natural logarithmic function is the exponential function, perhaps the most important function in all of calculus. Discover that this function has an amazing property: It is its own derivative! Also see the connection between the exponential function and the bell-shaped curve in probability.
9.
Product and Quotient Rules
Learn the formulas for finding derivatives of products and quotients of functions. Then use the quotient rule to derive formulas for the trigonometric functions not covered in the previous lecture. Also investigate higher-order derivatives, differential equations, and horizontal tangents.
27.
Bases other than e
Extend the use of the logarithmic and exponential functions to bases other than e, exploiting this approach to solve a problem in radioactive decay. Also learn to find the derivatives of such functions, and see how e emerges in other mathematical contexts, including the formula for continuous compound interest.
10.
The Chain Rule
Discover one of the most useful of the differentiation rules, the chain rule, which allows you to find the derivative of a composite of two functions. Explore different examples of this technique, including a problem from physics that involves the motion of a pendulum.
28.
Inverse Trigonometric Functions
Turn to the last set of functions you will need in your study of calculus, inverse trigonometric functions. Practice using some of the formulas for differentiating these functions. Then do an entertaining problem involving how fast the rotating light on a police car sweeps across a wall and whether you can evade it.
11.
Implicit Differentiation and Related Rates
Conquer the final strategy for finding derivatives: implicit differentiation, used when it's difficult to solve a function for y. Apply this rule to problems in related rates—for example, the rate at which a camera must move to track the space shuttle at a specified time after launch.
29.
Area of a Region between 2 Curves
Revisit the area problem and discover how to find the area of a region bounded by two curves. First imagine that the region is divided into representative rectangles. Then add up an infinite number of these rectangles, which corresponds to a definite integral.
12.
Extrema on an Interval
Having covered the rules for finding derivatives, embark on the first of five lectures dealing with applications of these techniques. Derivatives can be used to find the absolute maximum and minimum values of functions, known as extrema, a vital tool for analyzing many real-life situations.
30.
Volume—The Disk Method
Learn how to calculate the volume of a solid of revolution—an object that is symmetrical around its axis of rotation. As in the area problem in the previous lecture, you imagine adding up an infinite number of slices—in this case, of disks rather than rectangles—which yields a definite integral.
13.
Increasing and Decreasing Functions
Use the first derivative to determine where graphs are increasing or decreasing. Next, investigate Rolle's theorem and the mean value theorem, one of whose consequences is that during a car trip, your actual speed must correspond to your average speed during at least one point of your journey.
31.
Volume—The Shell Method
Apply the shell method for measuring volumes, comparing it with the disk method on the same shape. Then find the volume of a doughnut-shaped object called a torus, along with the volume for a figure called Gabriel's Horn, which is infinitely long but has finite volume.
14.
Concavity and Points of Inflection
What does the second derivative reveal about a graph? It describes how the curve bends—whether it is concave upward or downward. You determine concavity much as you found the intervals where a graph was increasing or decreasing, except this time you use the second derivative.
32.
Applications—Arc Length and Surface Area
Investigate two applications of calculus that are at the heart of engineering: measuring arc length and surface area. One of your problems is to determine the length of a cable hung between two towers, a shape known as a catenary. Then examine a peculiar paradox of Gabriel's Horn.
15.
Curve Sketching and Linear Approximations
By using calculus, you can be certain that you have discovered all the properties of the graph of a function. After learning how this is done, focus on the tangent line to a graph, which is a convenient approximation for values of the function that lie close to the point of tangency.
33.
Basic Integration Rules
Review integration formulas studied so far, and see how to apply them in various examples. Then explore cases in which a calculator gives different answers from the ones obtained by hand calculation, learning why this occurs. Finally, Professor Edwards gives advice on how to succeed in introductory calculus.
16.
Applications—Optimization Problems, Part 1
Attack real-life problems in optimization, which requires finding the relative extrema of different functions by differentiation. Calculate the optimum size for a box, and the largest area that can be enclosed by a circle and a square made from a given length of wire.
34.
Other Techniques of Integration
Closing your study of integration techniques, explore a powerful method for finding antiderivatives: integration by parts, which is based on the product rule for derivatives. Use this technique to calculate area and volume. Then focus on integrals involving products of trigonometric functions.
17.
Applications—Optimization Problems, Part 2
Conclude your investigation of differential calculus with additional problems in optimization. For success with such word problems, Professor Edwards stresses the importance of first framing the problem with precalculus, reducing the equation to one independent variable, and then using calculus to find and verify the answer.
35.
Differential Equations and Slope Fields
Explore slope fields as a method for getting a picture of possible solutions to a differential equation without having to solve it, examining several problems of the type that appear on the Advanced Placement exam. Also look at a solution technique for differential equations called separation of variables.
18.
Antiderivatives and Basic Integration Rules
Up until now, you've calculated a derivative based on a given function. Discover how to reverse the procedure and determine the function based on the derivative. This approach is known as obtaining the antiderivative, or integration. Also learn the notation for integration.
36.
Applications of Differential Equations
Use your calculus skills in three applications of differential equations: first, calculate the radioactive decay of a quantity of plutonium; second, determine the initial population of a colony of fruit flies; and third, solve one of Professor Edwards's favorite problems by using Newton's law of cooling to predict the cooling time for a cup of coffee.
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Summary
This book emphasizes the intelligent application of approximation techniques to the type of problems that commonly occur in engineering and the physical sciences. Readers learn why the numerical methods work, what type of errors to expect, and when an application might lead to difficulties. The authors also provide information about the availability of high-quality software for numerical approximation routines. In this book, full mathematical justifications are provided only if they are concise and add to the understanding of the methods. The emphasis is placed on describing each technique from an implementation standpoint, and on convincing the reader that the method is reasonable both mathematically and computationally.
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Emphasizing proofs, fundamental mathematical concepts and techniques are investigated within the context of number theory and other topics. Credit not given for both MATH 052 and MATH 054. Co-requisite: MATH 021.
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School Search
Mathematics
Our Belief We believe that the success of a student depends on three factors:
1. The amount of interest the child puts into his education. 2. The amount of interest the parent puts into the child's education. 3. The learning environment created by the school.
Philosophy
Our expectation is that every student at August Martin High School will take four years of mathematics. All students are encouraged to take the most challenging mathematics courses at which they can function and succeed.
To meet the demands of the College Preparatory initiative and to help students transition to college and technical careers, our goal is to help students master the Sequential Mathematics curriculum. Those students who require Bridge to Algebra before entering Mathematics A, will be exposed to a variety of teaching strategies. All students will receive supplementary supportive assistance when needed.
Bilingual courses will be offered in Bridge to Algebra and Mathematics A.
Mathematical Placement
1. Entering freshman who score below the 26th percentile on the Queens Math Test, DRP or the CAT, will be programmed for MG1.
2. Those scoring within the 25th and 50th percentile on the Queens Math Test, DRP or the CAT, will be programmed in M$A M$A is an extended period program running 7or 8 periods per week.
3. Entering freshman scoring above the 50th percentile on the Queens Math Test, DRP or the CAT, will be programmed for M$1.
4. Students who pass Bridge to Algebra MG1 and MG2 will be programmed for M$A.
Visit Math Links at Library Website
Students are welcome to visit Math Assistant Principal regarding any inquiry about mathematics class. Students should come only during their lunch period unless there is an emergency. Students are discouraged from cutting their classes in order to see the Math Assistant Principal. You must get a pass from your class teacher.
Math Sequence
Mathematics A (M#A, M#B, M#C, M#D) – Prentice Hall A four term sequence Math A Regents Course. Each course is a double period class. M#A begins the sequence in the first term and so on. The Math A course includes logic,algebra, coordinate & analytical geometry, probability, systems of equation, trigonometry of right triangle, statistics, locus and construction.
Mathematics A (M&1, M&2, M&3 or M#1, M#2, M#3)
A three term sequence Math A Regents Course. The sequence M&1, M&2 and M&3 is an accelerated program where each course is a single period class. The sequence M#1, M#2 and M#3 is an accelerated program with extra support. Each course is a double period class. Mathematics A (M$A, M$B, M$C & M$D)
A four term sequence Math A Regents Course. Each course is a single period class.M$A begins the sequence in the first term and so on. M$D is given in theLast term and culminates in the Regents Math A Examination. Each class is the prerequisite for the next class. The Math A course includes logic, algebra, coordinate & analytical geometry, probability, systems of equation, trigonometry of right triangle, statistics, locus & construction. Mathematics B (*M$E, M$F, M$G & M$H- four terms) (**M$4, M$5, M$6- three terms)
Prerequisite: Passed the Math A Regents Exam *A four term sequence Math B Regents Course. M$E is given in the first term and so on. M$H is given in the last term and culminates in the Regents Math B Examination.
**A three term sequence Math B Regents Course. M$4 is given in the first term and so on. M$6 is given in the last term and culminates in the Regents Math B Examination. Each class is the prerequisite for the next class. The Math B course includes relations and functions, exponential functions, logarithmic functions, trigonometry, complex numbers, transformational geometry, probability, sequences, binomial theorem and statistics.
Intermediate Algebra (MG3 & MG4 or MGS)
An enriched course which contains algebra and geometry including matrices, determinants, sequences, trigonometry and complex fractions.
Students must have attained a passing score in the Course 1 Regents Exam or Math A Regents Exam along with the approval of the Assistant Principal-Mathematics.
Pre-Calculus (MEP1 & MEP2)
A course for those students planning to take Advanced Placement Calculus. The courseincludes topics such as sequences, series, methods of convergence for series, analytic geometry of parametric equations, polar graphs, Theory of Equations, matrices & limits.
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Could you suggest me a good TI calculator?
Could you suggest me a good TI calculator?
i have decided to buy a good TI calculator. The cost isn't really a factor.. since almost all of their calculators come under 349 USD.
My area of work is mostly calculus-based physics and analytical geometry [coordinate geometry]. In mathematics, i'd also be needing features for basic basic statistics and probability computation.
Advanced algebraic support and data visualization is a must in addition to vector algebra and vector calculus. From what I saw at their site, one of the Graphing calculators would be a good option with me, because I deal with a lot of function plots and data visualization. And since programming is one of my hobbies, I'd also need a programmable calculator which can be programmed with the TIGCC library.
And since i already have Mathematica with me and do most of my stuff with that.. the only reason i want a calculator is for portability [a laptop is gonna cost a lot more :P].. so the smaller the better.
Also.. is there any calculator which has some specific set of functions for chemistry based applications?
I would strongly suggest the TI-89. It can simplify algebraic expressions, do basic (and a few not-so-basic) integrals, and some other useful stuff. It won't substitute for your physics knowledge. But my good ol' 89 saved me on my PhD qualifier (no seriously, I would have failed if not for its diff eq solver), so I'd recommend it to practically anyone in the physical sciences.
In regards to vector algebra, it can do dot and cross products. It can allegedly also do coordinate transformations, though I've never dared to actually try this. It can also do vector calculus...sort of. You'll need to turn double, triple, line, and surface integrals into iterated integrals by hand. But once you have it in this form, you can plug it into the 89 and get an expression. Not sure what you mean by data visualization, but it can graph continuous functions and sequences. As for programming, I don't know what the TIGCC library is, but the 89 has some sort of programming language. I've seen people write some pretty impressive programs for the 89. It can even play Super Mario. Not that you want to use your calculator for games, but if it can do this, then I'm sure it can do some pretty useful mathematical applications.
thanks a lot for the reply.. even i was thinking of the TI-89 since it's quite a popular model. But.. what do you think about the Voyage 200 or the TI-92 series? The only reason i didn't want to go for it was that it might be bulky and as i said.. portability was a major concern in my case. It's photographs hardly give me an idea of what it's real life size is..
I do know that the Voyage 200 qualifies as a computer and is not allowed in SAT exams and stuff.. but the exams that i have to give in india don't allow any calculator at all.. so it doesn't really matter to me.
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Past Course Descriptions
Course Listings - Fall 2007
MATHEMATICS DEPARTMENT
COURSE DESCRIPTIONS
Fall 2007
MATH 112 The Language of Mathematics (4 credits) Barhorst, Garry
Prerequisite: Math Placement Level 22 or higher
This is an introduction to mathematics at the beginning college level. MATH 112 will explore topics in contemporary mathematics with a problem-solving approach.
The class meetings will include lectures, problem-solving sessions, and group work. The final grade will be based on quizzes, exams, a project, and/or a comprehensive final. This course is not intended to prepare students for further courses in mathematics. Mathematical-reasoning intensive.
MATH 118 Mathematics for Elementary and Middle School Teachers (4 credits) Staff Prerequisite: Math Placement Level 22 or higher
Study of number systems, number theory, patterns, functions, measurement, algebra, logic, probability, and statistics with a special emphasis on the processes of mathematics: problem solving, reasoning, communicating mathematically, and making connections within mathematics and between mathematics and other areas. Open only to students intending to major in education. Every year. Mathematical-reasoning intensive.
Prerequisite: Math Placement Level 24 or higher
This is a standard pre‑calculus mathematics course that explores the functions common to the study of calculus. Examination of polynomial, rational, exponential, logarithmic, and trigonometric functions will be done using algebraic, numeric, and graphical techniques. Applications of these functions in formulating and solving real-world problems will also be discussed.
The final grade in the course will be based on homework, quizzes, tests, and a comprehensive final exam. Students are required to have a TI-83, TI-84, or TI-86 graphing calculator for use in class and for homework assignments. Mathematical-reasoning intensive.
MATH 127 Introductory Statistics (4 credits) Andrews, Douglas
Prerequisites: Math Placement Level 23 or higher
A study of statistics as the science of using data to glean insight into real-world problems. Includes graphical and numerical methods for describing and summarizing data, sampling procedures and experimental design, inferences about the real-world processes that underlie the data, and student projects for collecting and analyzing data. Open to non-majors only.
Note: A student may receive credit for only one of the following statistics courses: MATH 127, MATH 227, PSYC 107, or MGT 210. Mathematical-reasoning intensive.
MATH 131 Essentials of Calculus (4 credits) Staff
Prerequisite: MATH 120 or Math Placement Level 25
This one semester calculus course is an introduction to the techniques and applications of differential and integral calculus. The applications come primarily from the bio-sciences and do not involve any trigonometric models. The final grade in the course will be based on homework, quizzes, tests, and a comprehensive final exam.
Students are required to have a TI-83, TI-84, or TI-86 graphing calculator for use in class and for homework assignments. Mathematical-reasoning intensive.
Notes: 1. Students may not receive credit for both MATH 131 and MATH 201
2. MATH 131 does not satisfy the prerequisite for MATH 202.
3. Take MATH 131 only if you are POSITIVE that you will take only one semester of calculus at Wittenberg. Otherwise, you should take MATH 201.
MATH 201 Calculus I (4 credits) Higgins, William and Parker, Adam
Prerequisite: MATH 120 or Math Placement Level 25
Calculus is the mathematical tool used to analyze changes in physical quantities. This is the first course in the standard calculus sequence. It develops the notion of "derivative", which is used for studying rates of change, and then introduces the concept of "definite integral", which is related to area problems. The overall approach will emphasize the concepts of calculus using graphical, numerical, and symbolic methods.
The two-semester calculus sequence, MATH 201/202, is required for all students majoring or minoring in mathematics, computer science, physics, or chemistry. MATH 201 and MATH 202 can also count as Asupporting science@ courses for the BA and BS programs in Biology, Geology, and Biochemistry/Molecular Biology. Students who are sure they will take only one semester of calculus may be better served in the single-semester introduction to calculus, MATH 131: AEssentials of Calculus@. Talk with your advisor or with any math professor for advice on which calculus course is most appropriate for you.Depending on the instructor, the final grade in the course could be based on homework, quizzes, tests, and a comprehensive final exam. Mathematical-reasoning intensive.
NOTE: Students may not receive credit for both MATH 131 and MATH 201.
MATH 202 Calculus II (4 credits) Stickney, Alan
Prerequisite: MATH 201
This is the second course in Wittenberg=s three semester calculus sequence. MATH 202 is primarily concerned with integration and power series representations of functions. Topics covered include indefinite and definite integrals, the Fundamental Theorem of Calculus, integration techniques, approximations of definite integrals, improper integrals, applications of integrals, power series, Taylor=s Series, geometric series, and convergence tests for series.
The final grade in the course will be based on quizzes, tests, and a comprehensive final exam. Mathematical-reasoning intensive.
MATH 205 Applied Matrix Algebra (4 credits) Higgins, William
Prerequisites: MATH 201
A course in matrix algebra and discrete mathematical modeling which considers the formulation of mathematical models, together with analysis of the models and interpretation of the results. Primary emphasis is on those modeling techniques which utilize matrix methods. Such methods are now in wide use in areas such as economic input‑output models, population growth models, Markov chains, linear programming, computer graphics, regression, numerical approximation, and linear codes.
Students in this course are required to have a TI-83, TI-84, or TI-86 graphing calculator for use in class, for homework, and for tests. A TI-89, TI-92, or Voyage 200 is also acceptable. The final grade in the course is based on homework, quizzes, tests, and a comprehensive final exam.
This course is a prerequisite for MATH 360 (Linear Algebra), and should be taken by all sophomore mathematics majors. Mathematical-reasoning intensive.
The final grade in this course is based on quizzes, tests, a computer project, and a comprehensive final exam. Mathematical-reasoning intensive.
MATH 221 Foundations of Geometry (4 credits) Stickney, Alan
Prerequisite: MATH 210
A rigorous study of Euclidean and non-Euclidean geometry from an axiomatic point of view. Special attention is given to the concepts of definition, theorem, and proof. The mathematics is studied in an historical context.
This course is primarily intended for junior/senior mathematics majors and minors, and should be of particular interest to those planning to teach mathematics at a pre‑college level. The course is WRITING INTENSIVE. Mathematical-reasoning intensive.
MATH 227 Data Analysis (4 credits) Andrews, Douglas
Prerequisite: MATH 131 or MATH 201 This introductory statistics course is designed not only for students majoring or minoring in math, but for any student who would benefit from a more substantial introduction to the field - especially prospective teachers of mathematics or statistics, as well as students considering careers as statisticians or actuaries. Students will learn general principles and techniques for summarizing and organizing data effectively, and will explore the connections between how the data was collected and the scope of conclusions that can be drawn from the data. Also emphasized are the logic and techniques of formal statistical inference, with greater focus on the mathematical underpinnings of these basic statistical procedures than is found in other introductory statistics courses. Software for probability and data analysis is used daily.
Note: A student may not receive credit for more than one of the following: MATH 127, MATH 227, PSYC 107, or MGT 210. Mathematical-reasoning intensive.
MATH 328 Mathematical Statistics (4 credits) Andrews, Douglas
Prerequisite: MATH 228
Essential for anyone interested in a career in statistics or actuarial science, this course extends the ideas of Univariate Probability (MATH 228) to probability of several variables, which is then used to explore the distribution theory underlying the most commonly used statistical methods. Mathematical-reasoning intensive.
MATH 360 Linear Algebra (4 credits) Stickney, Alan
Prerequisites: MATH 205 and MATH 210
Introduction to abstract vector spaces. Topics include Euclidean spaces, function spaces, linear systems, linear independence and basis, linear transformations and their matrices. Students are required to have a TI-83, TI-84, or TI-86 graphing calculator for use in class, for homework, and on tests. A TI-89, TI-92, or Voyage 200 is also acceptable.
The final grade in the course is based on written assignments, quizzes, tests, and a comprehensive final exam. WRITING INTENSIVE. Mathematical-reasoning intensive.
MATH 370 Real Analysis (4 credits) Higgins, William
Prerequisite: MATH 210
Through a rigorous approach to the usual topics of one‑dimensional calculus ‑ limits, continuity, differentiation, integration, and infinite series ‑ this course offers a deeper understanding of the ideas encountered in calculus. The course has two important goals for its students: the development of an accurate intuitive feeling for analysis and of skill at proving theorems in this area. The final grade in this course is based upon written assignments, tests, and a comprehensive final exam.
This course is intended only for junior and senior mathematics majors or minors. Others will be enrolled only with the permission of the instructor. WRITING INTENSIVE. Mathematical-reasoning intensive.
MATH 460 Senior Seminar (2 credits) Shelburne, Brian
This is a capstone course for mathematics majors. Its purpose is to let participants think about and reflect on what mathematics is and to tie together their years of studying mathematics at Wittenberg. The structure of the course will be taken from the book Journey Through Genius by W. Dunham which covers the story of mathematics from the 5th century B.C.E. up to the 20th century C.E. by looking at some of the famous problems, theorems, and "colorful" mathematical characters who worked on them. The course is a seminar where participants are expected to research areas of interest in mathematics and present their findings to the rest of the seminar. The grade will be based on class discussions and presentations. Mathematical-reasoning intensive.
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As a cornerstone of mathematical science, the importance of modern algebra and discrete structures to many areas of science and technology is apparent and growing–with extensive use in computing science, physics, chemistry, and data communications as well as in areas of mathematics such as combinatorics.
An Elementary Introduction to Statistical Learning Theory is an excellent book for courses on statistical learning theory, pattern recognition, and machine learning at the upper-undergraduate and graduate levels. It also serves as an introductory reference for researchers and practitioners in the fields of engineering, computer science, philosophy, and cognitive science that would like to further their knowledge of the topic.
Help learners in grades 1-8 get it 'write' with practical strategies to help them write and understand mathematics content. This resource is designed in an easy-to-use format providing detailed strategies, graphic organizers, and activities with classroom examples by grade ranges. Specific suggestions for differentiating instruction are included with every strategy for various levels of readers and learning styles.
Addressing the direction of high school mathematics in the 21st century, this resource builds on the ideas of NCTM s Principles and Standards for School Mathematics and focuses on how high school mathematics can better prepare students for future success.
Reasoning and sense making are at the heart of the high school curriculum. Discover the components of mathematical reasoning and sense making in grades 9-12.
Math for Moms and Dads: A dictionary of terms and concepts…just for parents
Kids are struggling with math in school, on tests, and with homework. Parents feel stressed, helpless, and math-phobic. They struggle to encourage and assist on the very subject they are least prepared to manage: MATH.
Broken down using straightforward, simple language, this guide offers parents who are easily intimidated by math instructive and handy concepts to use when helping their students with homework or studying for a big test.
Parents banish math phobia once and for all by facing math head-on in Math for Moms and Dads. Frequently, the issue isn't "how to," it's actually "what do they want me to do?"
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A+ National Pre-apprenticeship Maths and Literacy for Hospitality by Andrew Spencer
Pre-apprenticeship Maths and Literacy for Hospitality is a write-in workbook that helps to prepare students seeking to gain a Hospitality Apprenticeship. It combines practical, real-world scenarios and terminology specifically relevant to the Hospitality industry, and provides students with the mathematical skills they need to confidently pursue a career in the Hospitality trade. Mirroring the format of current apprenticeship entry assessments, Pre-apprenticeship Maths and Literacy for Hospitality includes hundreds of questions to improve students' potential of gaining a successful assessment outcome of 75-80% and above. This workbook will therefore help to increase students' eligibility to obtain a Hospitality Apprenticeship. Pre-apprenticeship Maths and Literacy for Hospitality also supports and consolidates concepts that students studying VET (Vocational Educational Training) may use, as a number of VCE VET programs are also approved pre-apprenticeships. This workbook is also a valuable resource for older students aiming to revisit basic literacy and maths in their preparation to re-enter the workforce at the apprenticeship level.
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* Foodservice managers need a firm understanding and mastery of the principles of cost control in order to run a successful operation. Dopson & Hayes have created a comprehensive resource for both students and managers.
Running a hotel is a tough business, but it' also a most rewarding and stimulating one. This book covers what you need to know about the process of buying and running a hotel and shares the real life experiences of its hotelier author. It presents the realities of being your own boss.
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Books By Author Andrew Spencer
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Area Calculation
Area calculationis one of the most important chapters of Integral Calculus. It takes a real test of the skills of Integration and that too Definite Integral in particular. The chapters of Indefinite and Definite Integral are prerequisite to study the topic ofArea Calculationin Integral calculus.
This chapter is devoted to the application of definite integration forcalculating the areaof the regions bounded by specified or derived curves. Comprehensive material is provided with the consideration of wide variety of regions. Efforts are made to present the subject matter in the most lucid manner in order to involve the students actively and intellectually. A large number of solved examples, both of objective type and subjective type, are so chosen as to cover all the possible variations of question likely to be encountered by the students in different competitive examinations.
We have learnt that the definite integral between two values of Independent variable represents the area of the curve bound by the curve, the axis of the independent variable. Further, as we can calculate the area under one curve and the area under another curve then we can calculate the area between two curves. Depending upon the nature of the curves, this area can have different shapes and thus the tool of definite integral can be employed to calculate the area of different shapes. As a matter of fact, you will realize that the standard formulae to calculate the areas of different shapes can be derived by definite integral by choosing the appropriate curves.
Area Calculation is important from the perspective of scoring high in IIT JEE as there are few fixed pattern on which a number Multiple Choice Questions are framed on this topic. You are expected to do all the questions based on this to remain competitive in IIT JEE examination. It is very important to master these concepts as this this helps in many of the problems in Physics in your preparation for IIT JEE, AIEEE, DCE, EAMCET and other engineering entrance examinations.
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7th Grade Math
For Students: The Important Files (documents and power points) are located at the bottom of the page
Overview of 7th Grade MathAtStudentsStudents will review basic geometry vocabulary and concepts. They will learn how to set up equations given geometric information to solve for a variable. They will also learn properties of circles-- pi, radius, diameter-- and how to calculate their area and circumference.
GRAPHING: Students will be introduced
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AGraphical Approach to College Algebraillustrates equ... MOREation, the associated inequality of that equation, and ending with applications. The text covers all of the topics typically caught in a college algebra course, but with an organization that fosters studentsrs" understanding of the interrelationships among graphs, equations, and inequalities. With theFifth Edition,the text continues to evolve as it addresses the changing needs of todayrs"s students. Included are additional components to build skills, address critical thinking, solve applications, and apply technology to support traditional algebraic solutions, while maintaining its unique table of contents and functions-based approach.AGraphical Approach to College Algebracontinues to incorporate an open design, with helpful features and careful explanations of topics. This Package Contains: A Graphical Approach to College Algebra,Fifth Edition, (agrave; la Carte edition) with MyMathLab/MyStatLab Student Access Kit
Books ŕ la Carte are unbound, three-hole-punch versions of the textbook. This lower cost option is easy to transport and comes with same access code or media that would be packaged with the bound book.
A Graphical Approach to College Algebra illustrates equation, the associated inequality of that equation, and ending with applications. The text covers all of the topics typically caught in a college algebra course, but with an organization that fosters students' understanding of the interrelationships among graphs, equations, and inequalities.
With the Fifth Edition, the text continues to evolve as it addresses the changing needs of today's students. Included are additional components to build skills, address critical thinking, solve applications, and apply technology to support traditional algebraic solutions, while maintaining its unique table of contents and functions-based approach. A Graphical Approach to College Algebra continues to incorporate an open design, with helpful features and careful explanations of topics.
5.6. Further Applications and Modeling with Exponential and Logarithmic Functions
6. Analytic Geometry
6.1. Circles and Parabolas
6.2. Ellipses and Hyperbolas
6.3. Summary of Conic Sections
6.4. Parametric Equations
7. Systems of Equations and Inequalities; Matrices
7.1. Systems of Equations
7.2. Solution of Linear Systems in Three Variables
7.3. Solution of Linear Systems by Row Transformations
7.4. Matrix Properties and Operations
7.5. Determinants and Cramer's Rule
7.6. Solution of Linear Systems by Matrix Inverses
7.7. Systems of Inequalities and Linear Programming
7.8. Partial Fractions
8. Further Topics in Algebra
8.1 Sequences and Series
8.2 Arithmetic Sequences and Series
8.3 Geometric Sequences and Series
8.4 Counting Theory
8.5 The Binomial Theorem
8.6 Mathematical Induction
8.7 Probability
R. Reference: Basic Algebraic Concepts
R.1. Review of Exponents and Polynomials
R.2. Review of Factoring
R.3. Review of Rational Expressions
R.4. Review of Negative and Rational Exponents
R.5. Review of Radicals
Appendix: Geometry Formulas
John Hornsby: When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics, education, or journalism. His ultimate decision was to become a teacher, but after twenty-five years of teaching at the high school and university levels and fifteen years of writing mathematics textbooks, all three of his goals have been realized; his love for teaching and for mathematics is evident in his passion for working with students and fellow teachers as well. His specific professional interests are recreational mathematics, mathematics history, and incorporating graphing calculators into the curriculum.
John's personal life is busy as he devotes time to his family (wife Gwen, and sons Chris, Jack, and Josh). He has been a rabid baseball fan all of his life. John's other hobbies include numismatics (the study of coins) and record collecting. He loves the music of the 1960s and has an extensive collection of the recorded works of Frankie Valli and the Four Seasons.
Marge Lial has always beenGary Rockswold has been teaching mathematics for 33 years at all levels from seventh grade to graduate school, including junior high and high school students, talented youth, vocational, undergraduate, and graduate students, and adult education classes. He is currently employed at Minnesota State University, Mankato, where he is a full professor of mathematics. He graduated with majors in mathematics and physics from St. Olaf College in Northfield, Minnesota, where he was elected to Phi Beta Kappa. He received his Ph.D. in applied mathematics from Iowa State University. He has an interdisciplinary background and has also taught physical science, astronomy, and computer science. Outside of mathematics, he enjoys spending time with his lovely wife and two children.
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A Friendly Introduction to Number Theory, Coursesmart eTextbook, 4th Edition
Description
For one-semester undergraduate courses in Elementary Number Theory.
A Friendly Introduction to Number Theory, Fourth Edition is designed to introduce students to the overall themes and methodology of mathematics through the detailed study of one particular facet—number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.
Table of Contents
Preface
Flowchart of Chapter Dependencies
Introduction
1. What Is Number Theory?
2. Pythagorean Triples
3. Pythagorean Triples and the Unit Circle
4. Sums of Higher Powers and Fermat's Last Theorem
5. Divisibility and the Greatest Common Divisor
6. Linear Equations and the Greatest Common Divisor
7. Factorization and the Fundamental Theorem of Arithmetic
8. Congruences
9. Congruences, Powers, and Fermat's Little Theorem
10. Congruences, Powers, and Euler's Formula
11. Euler's Phi Function and the Chinese Remainder Theorem
12. Prime Numbers
13. Counting Primes
14. Mersenne Primes
15. Mersenne Primes and Perfect Numbers
16. Powers Modulo m and Successive Squaring
17. Computing kth Roots Modulo m
18. Powers, Roots, and "Unbreakable" Codes
19. Primality Testing and Carmichael Numbers
20. Squares Modulo p
21. Is -1 a Square Modulo p? Is 2?
22. Quadratic Reciprocity
23. Proof of Quadratic Reciprocity
24. Which Primes Are Sums of Two Squares?
25. Which Numbers Are Sums of Two Squares?
26. As Easy as One, Two, Three
27. Euler's Phi Function and Sums of Divisors
28. Powers Modulo p and Primitive Roots
29. Primitive Roots and Indices
30. The Equation X4 + Y4 = Z4
31. Square–Triangular Numbers Revisited
32. Pell's Equation
33. Diophantine Approximation
34. Diophantine Approximation and Pell's Equation
35. Number Theory and Imaginary Numbers
36. The Gaussian Integers and Unique Factorization
37. Irrational Numbers and Transcendental Numbers
38. Binomial Coefficients and Pascal's Triangle
39. Fibonacci's Rabbits and Linear Recurrence Sequences
40. Oh, What a Beautiful Function
41. Cubic Curves and Elliptic Curves
42. Elliptic Curves with Few Rational Points
43. Points on Elliptic Curves Modulo p
44. Torsion Collections Modulo p and Bad Primes
45. Defect Bounds and Modularity Patterns
46. Elliptic Curves and Fermat's Last Theorem
Further Reading
Index
*47. The Topsy-Turvey World of Continued Fractions [online]
*48. Continued Fractions, Square Roots, and Pell's Equation [online]
*49. Generating Functions [online]
*50. Sums of Powers [online]
*A. Factorization of Small Composite Integers [online]
*B. A List of Primes [online]
*These chapters are available online
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Designed to inform readers about the formal development of Euclidean geometry and to prepare prospective high school mathematics instructors to teach Euclidean geometry, this text closely follows Euclid's classic, Elements. The text augments Euclid's statements with appropriate historical commentary and many exercises — more than 1,000 practice exercises provide readers with hands-on experience in solving geometrical problems. In addition to providing a historical perspective on plane geometry, this text covers non-Euclidean geometries, allowing students to cultivate an appreciation of axiomatic systems. Additional topics include circles and regular polygons, projective geometry, symmetries, inversions, knots and links, graphs, surfaces, and informal topology. This republication of a popular text is substantially less expensive than prior editions and offers a new Preface by the author.
Table of Contents for Geometry from Euclid to Knots
Preface to the Dover Edition
Preface
1. Other Geometries: A Computational Introduction
2. The Neutral Geometry of the Triangle
3. Nonneutral Euclidean Geometry
4. Circles and Regular Polygons
5. Toward Projective Geometry
6. Planar Symmetries
7. Inversions
8. Symmetry in Space
9. Informal Topology
10. Graphs
11. Surfaces
12. Knots and Links
Appendix A: A Brief Introduction to the Geometer's Sketchpad
Appendix B: Summary of Propositions
Appendix C: George D. Birkhoff's Axiomatization of Euclidean Geometry
Appendix D: The University of Chicago School Mathematics Project's Geometrical Axioms
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Hello experts! Are there any online resources to learn about the basics of algebra free quiz? I didn't really get the chance to cover the entire syllabus as yet. This is probably why I encounter problems while solving questions.
I'm know little in algebra free quiz. But, it's quite hard to explain it. I may help you answer it but since the solution is complex, I doubt you will really understand the whole process of solving it so it's recommended that you really have to ask someone to explain it to you in person to make the explaining clearer. Good thing is that there's this software that can help you with your problems. It's called Algebra Buster and it's an amazing piece of program because it does not only show the answer but it also shows the process of solving it. How cool is that? problem. It's almost like a tutor is teaching it to you. I have been using it for four weeks and so far, haven't come across any problem that Algebra Buster can't solve. I have learnt so much from it!
I remember having difficulties with fractional exponents, geometry and exponent rules. Algebra Buster is a really great piece of math software. I have used it through several math classes - Pre Algebra, Remedial Algebra and Pre Algebra. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly recommended.
Don't worry my friend
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Mathematics
Introduction
Mathematics is a science that involves abstract concepts and language. Students develop their mathematical thinking gradually through personal experiences and exchanges with peers. Their learning is based on situations that are often drawn from everyday life. In elementary school, students take part in learning situations that allow them to use objects, manipulatives, references and various tools and instruments. The activities and tasks suggested encourage them to reflect, manipulate, explore, construct, simulate, discuss, structure and practise, thereby allowing them to assimilate concepts, processes and strategies1 that are useful in mathematics. Students must also call on their intuition, sense of observation, manual skills as well as their ability to express themselves, reflect and analyze. By making connections, visualizing mathematical objects in different ways and organizing these objects in their minds, students gradually develop their understanding of abstract mathematical concepts. With time, they acquire mathematical knowledge and skills, which they learn to use effectively in order to function in society.
In secondary school, learning continues in the same vein. It is centred on the fundamental aims of mathematical activity: interpreting reality, generalizing, predicting and making decisions. These aims reflect the major questions that have led human beings to construct mathematical culture and knowledge through the ages. They are therefore meaningful and make it possible for students to build a set of tools that will allow them to communicate appropriately using mathematical language, to reason effectively by making connections between mathematical concepts and processes, and to solve situational problems. Emphasis is placed on technological tools, as these not only foster the emergence and understanding of mathematical concepts and processes, but also enable students to deal more effectively with various situations. Using a variety of mathematical concepts and strategies appropriately provides keys to understanding everyday reality. Combined with learning activities, everyday situations promote the development of mathematical skills and attitudes that allow students to mobilize, consolidate and broaden their mathematical knowledge. In Cycle Two, students continue to develop their mathematical thinking, which is essential in pursuing more advanced studies.
This document provides additional information on the knowledge and skills students must acquire in each year of secondary school with respect to arithmetic, algebra, geometry, statistics and probability. It is designed to help teachers with their lesson planning and to facilitate the transition between elementary and secondary school and from one secondary cycle to another. A separate section has been designed for each of the above-mentioned branches, as well as for discrete mathematics and analytic geometry. Each section consists of an introduction that provides an overview of the learning that was acquired in elementary school and that is to be acquired in the two cycles of secondary school, as well as content tables that outline, for every year of secondary school, the knowledge to be developed and actions to be carried out in order for students to fully assimilate the concepts presented. A column is devoted specifically to learning acquired in elementary school.2 Where applicable, the cells corresponding to Secondary IV and V have been subdivided to present the knowledge and actions associated with each of the options that students may choose based on their interests, aptitudes and training needs: Cultural, Social and Technical option (CST), Technical and Scientific option (TS) and Science option (S).
Information concerning learning acquired in elementary school was taken from the Mathematics program and the document Progression of Learning in Elementary School - Mathematics, to indicate its relevance as a prerequisite and to define the limits of the elementary school program. Please note that there are no sections on vocabulary or symbols for at the secondary level, these are introduced gradually as needed.
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2. From this window you will
see the topics for the course, the first of which is the orientation to
the tutorial program.
3. Open the Orientation
topic and work through all the exercises until you feel comfortable using
the system. The exercises will teach you how to form the math symbols
you need to input your answers and so forth. Every topic has a fairly
useful Help screen if you need immediate help with math or detailed help
with using the program.
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(2 terms, 2 credits) This course consists of a review of the principles and ideas of algebra and the extension of these concepts. Some of the topics include equations, formulas, inequalities, variations, graphs, linear relationships, systems of equations, functions, powers and roots, exponents and logarithms, trigonometry and polynomials. Materials: graphing calculator, notebook, and folder.
This course prepares the student for college or AP Calculus. Students will examine real numbers, polynomial and rational functions, exponential and logarithmic functions, trigonometric and circular function. Topics include vectors, systems of equations and inequalities, matrices, and polar coordinates.
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There is a choice of three courses in mathematics each tailored to challenge students of the appropriate initial ability and give them the necessary skills for further study. All our courses aim to enable students to access and appreciate the power and usefulness of mathematics. They also aim to give students an enjoyment of the subject and a desire to produce elegant solutions in their own work. We will give students insight into the ongoing nature of learning, in particular encouraging them to make appropriate use of new technologies as they become available. Students will be encouraged to develop independent working techniques whilst being supported in the process of acquiring academic maturity.
Mathematics HL: The course will focus on developing the fundamental mathematical concepts in a structured and rigorous way. Our teaching will equip students with a wide range of skills and empower them to identify the most appropriate approach. The teaching will challenge students to routinely consider the limitations of their own work whilst encouraging them to consider abstract methods in order to produce general solutions.
Mathematics SL: The course will focus on preparing students with a solid basis for further study where some knowledge of advanced mathematics is required. It will also support the other parts of the IB course which require some use of mathematics. Mathematical Studies SL: The course will focus on mathematics that can be commonly applied to real-world scenarios. It will support the student's work in their other subjects and will make use of technology to access more technical methods where necessary.
In all courses we aim to give students a cultural, historical and personal perspective on mathematics by including study of inventions and situations that were instrumental in major mathematical breakthroughs. We will look at the work of many mathematicians with a variety of backgrounds considering the prevailing culture that may have been formative in their thinking. Students will also learn to appreciate the universal language of mathematics and the need for international standards in notation and measurement.
Course Content
Mathematics HL and SL
Following a short introductory course ensuring that all students have the correct basic skills all
Mathematics HL and SL candidates will undertake the following areas of study:
• Algebra
• Functions and Equations
• Circular Functions and Trigonometry
• Matrices
• Vectors
• Statistics and Probability
• Calculus
HL candidates would study more widely in each of these areas and also take an additional course in Discrete Mathematics.
Mathematical Studies SL
Mathematical Studies SL candidates will follow a course containing the following areas of study:
• Using the Graphical Display Calculator
• Number, Algebra and Sequences
• Sets, Logic and Probability
• Functions
• Geometry and Trigonometry
• Statistics
• Differential Calculus
• Maths of Finance
At each stage these students' work will place emphasis on the applications of their acquired knowledge to the real world.
Assessment
All courses are assessed by a combination of 80% final examinations and 20% coursework.
Mathematics HL
Students will sit three papers, including two 2 hour examinations covering the main content and a 1 hour paper covering the Discrete Mathematics element and submit a portfolio of their work.
Mathematics SL
Students will sit two 90 minute papers covering the main content and submit a portfolio of their work.
In both courses Paper 1 is undertaken without the aid of a calculator.
The portfolio comprises two pieces of work, one mathematical investigation and one mathematical modelling task.
Mathematical Studies SL
Students will sit two 90 minute papers covering the main content (a calculator is permitted for both) and submit a single 2000 word
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Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
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Operating Systems9/27/2007Two Tracks Track 1: based on NachosKernel AssignmentsCS 256/456 Dept. of Computer Science, University of Rochester Track 2: based on Xen/Linux Both tracks are group assignments (groups of 2) You are on your own t
Midterm 3: Practice Problems1. Find the first three elements of the Taylor series of the function f (x) = x3 ln(1 - x). 2. Find the radius of convergence of the seriesn=1(2n + 1)! n x (n!)23. Find the equation of the plane through the points
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Understand relations and functions and select, convert flexibly among, and use various representations for them.
Analyze functions of one variable by investigating local and global behavior, including slopes as rates of change, intercepts, and zeros.
Core Idea 2: Mathematical Reasoning
Employ forms of mathematical reasoning and proof appropriate to the solution of the problem, including deductive and inductive reasoning, making and testing conjectures, and using counterexamples and indirect proof.
Show mathematical reasoning in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.
Explain the logic inherent in a solution process.
Use induction to make conjectures and use deductive reasoning to prove conclusions.
Draw reasonable conclusions about a situation being modeled.
Core Idea 3: Algebraic Properties and Representations
Represent and analyze mathematical situations and structures using algebraic symbols.
Compare and contrast the properties of numbers and number systems including real numbers.
Understand the meaning of equivalent forms of expressions, equations, inequalities, or relations.
Write equivalent forms of equations, inequalities, and systems of equations and solve them.
Use symbolic algebra to represent and explain mathematical relationships.
Judge the meaning, utility, and reasonableness of results of symbolic manipulations.
Use symbolic expressions to represent relationships arising from various contexts.
Approximate and interpret rates of change from graphic and numeric data.
Use geometric models to gain insights into, and answer questions in, other areas of mathematics.
Core Idea 5: Data Analysis
Select and use appropriate statistical methods to analyze data and understand and apply basic concepts of probability.
Understand the relationship between two sets of data (bivariate), display such data in a scatterplot, and describe trends and shape of the plot including correlations (positive, negative, or no) and lines of best fit.
Make inferences based on the data and evaluate the validity of conclusions drawn.
Compute and interpret the expected value of random variables in simple cases.
Understand the concepts of conditional probability and independent events and compute the probability of a compound event.
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