Mathematics Batch 02 - Analysis & Calculus - Programming Framework Analysis
This document presents analysis and calculus processes analyzed using the Programming Framework methodology. Each process is represented as a computational flowchart with standardized color coding: Red for triggers/inputs, Yellow for structures/objects, Green for processing/operations, Blue for intermediates/states, and Violet for products/outputs. Yellow nodes use black text for optimal readability, while all other colors use white text.
1. Limit Calculation Process
graph TD
A1[Function f of x] --> B1[Point of Interest]
C1[Approach Direction] --> D1[Limit Analysis]
E1[Indeterminate Forms] --> F1[L'Hopital Rule]
B1 --> G1[Direct Substitution]
D1 --> H1[Left Hand Limit]
F1 --> I1[Right Hand Limit]
G1 --> J1[Limit Evaluation]
H1 --> K1[Limit from Below]
I1 --> L1[Limit from Above]
J1 --> M1[Limit Exists Question]
K1 --> L1
L1 --> N1[Limit Comparison]
M1 --> O1[Limit Value]
N1 --> P1[One Sided Limits]
O1 --> Q1[Limit Calculation Process]
P1 --> R1[Limit Validation]
Q1 --> S1[Limit Verification]
R1 --> T1[Limit Calculation Result]
S1 --> U1[Limit Calculation Validation]
T1 --> V1[Limit Calculation Parameters]
U1 --> W1[Limit Calculation Output]
V1 --> X1[Limit Calculation Analysis]
W1 --> Y1[Limit Calculation Final Result]
X1 --> Z1[Limit Calculation Analysis Complete]
style A1 fill:#ff6b6b,color:#fff
style C1 fill:#ff6b6b,color:#fff
style E1 fill:#ff6b6b,color:#fff
style B1 fill:#ffd43b,color:#000
style D1 fill:#ffd43b,color:#000
style F1 fill:#ffd43b,color:#000
style G1 fill:#ffd43b,color:#000
style H1 fill:#ffd43b,color:#000
style I1 fill:#ffd43b,color:#000
style J1 fill:#ffd43b,color:#000
style K1 fill:#ffd43b,color:#000
style L1 fill:#ffd43b,color:#000
style M1 fill:#ffd43b,color:#000
style N1 fill:#ffd43b,color:#000
style O1 fill:#ffd43b,color:#000
style P1 fill:#ffd43b,color:#000
style Q1 fill:#ffd43b,color:#000
style R1 fill:#ffd43b,color:#000
style S1 fill:#ffd43b,color:#000
style T1 fill:#ffd43b,color:#000
style U1 fill:#ffd43b,color:#000
style V1 fill:#ffd43b,color:#000
style W1 fill:#ffd43b,color:#000
style X1 fill:#ffd43b,color:#000
style Y1 fill:#ffd43b,color:#000
style Z1 fill:#ffd43b,color:#000
style M1 fill:#51cf66,color:#fff
style N1 fill:#51cf66,color:#fff
style O1 fill:#51cf66,color:#fff
style P1 fill:#51cf66,color:#fff
style Q1 fill:#51cf66,color:#fff
style R1 fill:#51cf66,color:#fff
style S1 fill:#51cf66,color:#fff
style T1 fill:#51cf66,color:#fff
style U1 fill:#51cf66,color:#fff
style V1 fill:#51cf66,color:#fff
style W1 fill:#51cf66,color:#fff
style X1 fill:#51cf66,color:#fff
style Y1 fill:#51cf66,color:#fff
style Z1 fill:#51cf66,color:#fff
style Z1 fill:#b197fc,color:#fff
Triggers & Inputs
Limit Methods
Limit Operations
Intermediates
Products
Figure 1. Limit Calculation Process. This analysis process visualization demonstrates limit evaluation and convergence analysis. The flowchart shows function inputs and approach directions, limit methods and evaluation strategies, limit operations and comparisons, intermediate results, and final limit calculation outputs.
2. Derivative Calculation Process
graph TD
A2[Function f of x] --> B2[Differentiation Method]
C2[Point of Evaluation] --> D2[Derivative Analysis]
E2[Chain Rule] --> F2[Product Rule]
B2 --> G2[Power Rule]
D2 --> H2[Quotient Rule]
F2 --> I2[Implicit Differentiation]
G2 --> J2[Constant Rule]
H2 --> K2[Trigonometric Derivatives]
I2 --> L2[Logarithmic Derivatives]
J2 --> M2[Sum Rule]
K2 --> L2
L2 --> N2[Exponential Derivatives]
M2 --> O2[Difference Rule]
N2 --> P2[Inverse Function Derivatives]
O2 --> Q2[Derivative Calculation Process]
P2 --> R2[Derivative Validation]
Q2 --> S2[Derivative Verification]
R2 --> T2[Derivative Calculation Result]
S2 --> U2[Derivative Calculation Validation]
T2 --> V2[Derivative Calculation Parameters]
U2 --> W2[Derivative Calculation Output]
V2 --> X2[Derivative Calculation Analysis]
W2 --> Y2[Derivative Calculation Final Result]
X2 --> Z2[Derivative Calculation Analysis Complete]
style A2 fill:#ff6b6b,color:#fff
style C2 fill:#ff6b6b,color:#fff
style E2 fill:#ff6b6b,color:#fff
style B2 fill:#ffd43b,color:#000
style D2 fill:#ffd43b,color:#000
style F2 fill:#ffd43b,color:#000
style G2 fill:#ffd43b,color:#000
style H2 fill:#ffd43b,color:#000
style I2 fill:#ffd43b,color:#000
style J2 fill:#ffd43b,color:#000
style K2 fill:#ffd43b,color:#000
style L2 fill:#ffd43b,color:#000
style M2 fill:#ffd43b,color:#000
style N2 fill:#ffd43b,color:#000
style O2 fill:#ffd43b,color:#000
style P2 fill:#ffd43b,color:#000
style Q2 fill:#ffd43b,color:#000
style R2 fill:#ffd43b,color:#000
style S2 fill:#ffd43b,color:#000
style T2 fill:#ffd43b,color:#000
style U2 fill:#ffd43b,color:#000
style V2 fill:#ffd43b,color:#000
style W2 fill:#ffd43b,color:#000
style X2 fill:#ffd43b,color:#000
style Y2 fill:#ffd43b,color:#000
style Z2 fill:#ffd43b,color:#000
style M2 fill:#51cf66,color:#fff
style N2 fill:#51cf66,color:#fff
style O2 fill:#51cf66,color:#fff
style P2 fill:#51cf66,color:#fff
style Q2 fill:#51cf66,color:#fff
style R2 fill:#51cf66,color:#fff
style S2 fill:#51cf66,color:#fff
style T2 fill:#51cf66,color:#fff
style U2 fill:#51cf66,color:#fff
style V2 fill:#51cf66,color:#fff
style W2 fill:#51cf66,color:#fff
style X2 fill:#51cf66,color:#fff
style Y2 fill:#51cf66,color:#fff
style Z2 fill:#51cf66,color:#fff
style Z2 fill:#b197fc,color:#fff
Triggers & Inputs
Differentiation Methods
Derivative Operations
Intermediates
Products
Figure 2. Derivative Calculation Process. This analysis process visualization demonstrates differentiation techniques and rate of change analysis. The flowchart shows function inputs and differentiation methods, differentiation methods and rules, derivative operations and calculations, intermediate results, and final derivative calculation outputs.
3. Integral Calculation Process
graph TD
A3[Function f of x] --> B3[Integration Method]
C3[Integration Limits] --> D3[Integral Analysis]
E3[Substitution Method] --> F3[Integration by Parts]
B3 --> G3[Power Rule Integration]
D3 --> H3[Trigonometric Integration]
F3 --> I3[Partial Fractions]
G3 --> J3[Exponential Integration]
H3 --> I3
I3 --> K3[Logarithmic Integration]
J3 --> L3[Definite Integral]
K3 --> M3[Indefinite Integral]
L3 --> N3[Area Calculation]
M3 --> O3[Antiderivative]
N3 --> P3[Fundamental Theorem]
O3 --> Q3[Integral Calculation Process]
P3 --> R3[Integral Validation]
Q3 --> S3[Integral Verification]
R3 --> T3[Integral Calculation Result]
S3 --> U3[Integral Calculation Validation]
T3 --> V3[Integral Calculation Parameters]
U3 --> W3[Integral Calculation Output]
V3 --> X3[Integral Calculation Analysis]
W3 --> Y3[Integral Calculation Final Result]
X3 --> Z3[Integral Calculation Analysis Complete]
style A3 fill:#ff6b6b,color:#fff
style C3 fill:#ff6b6b,color:#fff
style E3 fill:#ff6b6b,color:#fff
style B3 fill:#ffd43b,color:#000
style D3 fill:#ffd43b,color:#000
style F3 fill:#ffd43b,color:#000
style G3 fill:#ffd43b,color:#000
style H3 fill:#ffd43b,color:#000
style I3 fill:#ffd43b,color:#000
style J3 fill:#ffd43b,color:#000
style K3 fill:#ffd43b,color:#000
style L3 fill:#ffd43b,color:#000
style M3 fill:#ffd43b,color:#000
style N3 fill:#ffd43b,color:#000
style O3 fill:#ffd43b,color:#000
style P3 fill:#ffd43b,color:#000
style Q3 fill:#ffd43b,color:#000
style R3 fill:#ffd43b,color:#000
style S3 fill:#ffd43b,color:#000
style T3 fill:#ffd43b,color:#000
style U3 fill:#ffd43b,color:#000
style V3 fill:#ffd43b,color:#000
style W3 fill:#ffd43b,color:#000
style X3 fill:#ffd43b,color:#000
style Y3 fill:#ffd43b,color:#000
style Z3 fill:#ffd43b,color:#000
style M3 fill:#51cf66,color:#fff
style N3 fill:#51cf66,color:#fff
style O3 fill:#51cf66,color:#fff
style P3 fill:#51cf66,color:#fff
style Q3 fill:#51cf66,color:#fff
style R3 fill:#51cf66,color:#fff
style S3 fill:#51cf66,color:#fff
style T3 fill:#51cf66,color:#fff
style U3 fill:#51cf66,color:#fff
style V3 fill:#51cf66,color:#fff
style W3 fill:#51cf66,color:#fff
style X3 fill:#51cf66,color:#fff
style Y3 fill:#51cf66,color:#fff
style Z3 fill:#51cf66,color:#fff
style Z3 fill:#b197fc,color:#fff
Triggers & Inputs
Integration Methods
Integral Operations
Intermediates
Products
Figure 3. Integral Calculation Process. This analysis process visualization demonstrates integration techniques and area calculation. The flowchart shows function inputs and integration methods, integration methods and techniques, integral operations and calculations, intermediate results, and final integral calculation outputs.