Basic Engineering Mathematics (Term-Work)

flowchart TD
A[Start: What is a Determinant?] --> B{Key Idea: A Special Scalar Value<br>Calculated from a Square Matrix};
B --> C{Primary Interpretations};
C --> C1["Geometric: Scaling Factor<br>Area/Volume change by<br>linear transformation"];
C --> C2["Algebraic: Invertibility Test<br>detA = 0 → Matrix is singular<br>detA ≠ 0 → Matrix is invertible"];
C1 & C2 --> D[Methods of Computation];
subgraph D[Computation Methods by Matrix Size]
    D1["1x1 Matrix<br>det[a] = a"]
    D2["2x2 Matrix<br>det⎡a b⎤ = ad - bc<br>    ⎣c d⎦"]
    D3["3x3 Matrix (Sarrus' Rule)<br>+aei + bfg + cdh - ceg - bdi - afh"]
    D4["nxn Matrix (Laplace Expansion)<br>Choose row/column, sum of<br>a_ij * -1ⁱ⁺ʲ * detM_ij<br>M_ij = Minor matrix"]
end
D --> E[Important Properties];
subgraph E[Key Properties & Rules]
    E1["detI = 1"]
    E2["detAᵀ = detA"]
    E3["detAB = detA * detB"]
    E4["Swapping a row → flips sign"]
    E5["Proportional rows → det = 0"]
end
E --> F{Applications & Uses};
F --> G1["Solving Systems of Equations<br>Cramer's Rule"];
F --> G2["Finding Matrix Inverse<br>A⁻¹ = adjA / detA"];
F --> G3["Calculus: Jacobian Determinant<br>For change of variables in<br>multiple integrals"];
F --> G4["Geometry: Calculating Area/<br>Volume of parallelograms/<br>parallelepipeds"];
F --> G5["Eigenvalue Problems<br>Characteristic equation: detA - λI = 0"];
G1 & G2 & G3 & G4 & G5 --> H[End: Determinant as a Fundamental<br>Tool in Linear Algebra];

flowchart TD
A[Start: Properties of Determinants] --> B["detA is a scalar value<br>calculated from a square matrix A"];
B --> C{Core Properties};
subgraph C[Fundamental Rules]
    C1["detI = 1<br>Determinant of Identity matrix"]
    C2["detAᵀ = detA<br>Transpose doesn't change determinant"]
    C3["Triangular Matrix Diagonal Included<br>detA = product of diagonal elements"]
end
C --> D{Effect of Row/Column Operations};
subgraph D[Elementary Operations]
    D1["Row/Column Switching<br>Swaps sign of determinant: det → -det"]
    D2["Row/Column Scaling by k<br>Scales determinant by k: det → k⋅det"]
    D3["Row/Column Addition<br>Adding a multiple of one row to another<br>Does NOT change determinant: det unchanged"]
end
D --> E{Multiplicative Property & Invertibility};
subgraph E[Product and Inversion]
    E1["detAB = detA ⋅ detB<br>Determinant of product equals product of determinants"]
    E2["detA⁻¹ = 1 / detA<br>if A is invertible"]
    E3["detAⁿ = detAⁿ"]
end
E1 --> F{"If detA = 0?"};
F -- Yes --> G["Matrix A is Singular<br>Not Invertible"];
F -- No --> H["Matrix A is Nonsingular<br>Invertible"];
G --> I["Rows/Columns are Linearly Dependent"];
H --> J["Rows/Columns are Linearly Independent"];
D & E --> K{Special Cases & Other Properties};
subgraph K[Additional Important Properties]
    K1["Proportional Rows/Columns<br>If two rows/columns are proportional, then detA = 0"]
    K2["Zero Row/Column<br>If a row or column is all zeros, then detA = 0"]
    K3["Linearity Property<br>det is linear in each row/column separately"]
    K4["Block Triangular Matrix<br>det = detblock₁₁ ⋅ detblock₂₂ ⋅ ... ⋅ detblockₙₙ"]
    K5["Scalar Multiplication of Matrix<br>detkA = kⁿ ⋅ detA<br>for an n×n matrix A"]
end
K --> L{Applications of Properties};
subgraph L[Uses of These Properties]
    L1["Simplify Determinant Calculation<br>Use row operations to create zeros<br> or triangular form"]
    L2["Proving Matrix Invertibility"]
    L3["Solving Systems of Equations Cramer's Rule"]
    L4["Finding Eigenvalues Solve detA - λI = 0"]
    L5["Calculating Area/Volume Scale Factor<br> Absolute value of det gives area/volume change"]
end
L --> M[End: Properties Simplify Computation and Theory];

flowchart TD
A[Start: Cramer's Rule] --> B["A method to solve a system of<br>n linear equations with n variables"];
B --> C{Prerequisites};
subgraph C[Conditions for Application]
    C1["System must be Square<br>Number of equations = Number of unknowns"]
    C2["Coefficient Matrix A must be Invertible<br>i.e., detA ≠ 0"]
end
C --> D["Given System: A⋅X = B<br>Where:
<br>A = n×n Coefficient Matrix
<br>X = n×1 Variable Matrix x₁, x₂,...,xₙ
<br>B = n×1 Constant Matrix"];
D --> E["Calculate Determinant of A, denoted detA"];
E --> F{Is detA = 0?};
F -- No, detA ≠ 0 --> G["Proceed with Cramer's Rule"];
F -- Yes, detA = 0 --> H["Stop. Cramer's Rule fails.<br>The system has either no solution or infinitely many solutions."];
G --> I["For each variable xᵢ"];
I --> J["Form Matrix Aᵢ by replacing<br>the i-th column of A with the constant matrix B"];
J --> K["Calculate determinant of Aᵢ, denoted detAᵢ"];
K --> L["Solution for xᵢ = detAᵢ / detA"];
L --> M["Repeat for i = 1 to n<br>to find all variables"];
M --> N["Solution Vector X = x₁, x₂, ..., xₙ"];
N --> O{Advantages & Disadvantages};
subgraph O[Analysis]
    O1["Advantages:<br>- Straightforward formula<br>- Useful for theoretical purposes<br>- Good for small systems n=2,3"]
    O2["Disadvantages:<br>- Computationally inefficient for n>3<br>- Requires detA ≠ 0<br>- Not practical for large systems"]
end
O --> P[End: A Elegant but Limited Tool];

flowchart TD
A[Start: What is a Matrix?] --> B["A rectangular array of numbers<br> arranged in rows and columns."]
B --> C{Primary Classification:<br> By Dimensions & Structure}
subgraph C[Types of Matrices]
    direction TB
    C1["Row Matrix<br>1 x n"]
    C2["Column Matrix<br>m x 1"]
    C3["Square Matrix<br>n x n"]
    C4["Rectangular Matrix<br>m x n, m≠n"]
    C5["Diagonal Matrix"]
    C6["Identity Matrix I"]
    C7["Zero / Null Matrix"]
    C8["Symmetric Matrix<br>A = Aᵀ"]
    C9["Singular Matrix<br>det = 0"]
end
C --> D[Matrix Algebra & Operations]
subgraph D[Key Operations]
    D1[Addition & Subtraction<br>Element-wise, requires same dimension]
    D2[Scalar Multiplication<br>Multiply every element by a scalar]
    D3[Matrix Multiplication<br>Rows of A × Columns of B<br>Requires: colsA = rowsB]
    D4[Transpose Aᵀ<br>Flip rows and columns]
    D5["Inverse A⁻¹<br>Exists only for square,<br>non-singular detA ≠ 0 matrices"]
    D6["Determinant detA or A<br>Scalar value for square matrices only"]
end
D --> E{Key Properties of Operations}
subgraph E[Properties]
    E1["Associative: A+B+C = A+B+C<br>AB C = A BC"]
    E2["Commutative: A+B = B+A<br>But AB ≠ BA generally"]
    E3["Distributive: A B+C = AB + AC"]
    E4["Transpose of Product: ABᵀ = BᵀAᵀ"]
end
E --> F{Applications & Real-World Uses}
F --> G1["Computer Graphics:<br>3D Transformations Rotation, Scaling, Translation"]
F --> G2["Data Science:<br>Dataset Representation, Linear Regression, PCA"]
F --> G3["Physics & Engineering:<br>Solving Systems of Linear Equations, Circuit Analysis"]
F --> G4["Economics & Operations Research:<br>Input-Output Models, Linear Programming"]
F --> G5["Computer Science:<br>Graph Theory Adjacency Matrix, Search Algorithms"]
F --> G6["Cryptography:<br>Encoding and Decoding Messages"]
G1 & G2 & G3 & G4 & G5 & G6 --> H[End: Matrices as the Language of Linear Algebra]

flowchart TD
A[Start: Matrix] --> B["A rectangular array of numbers,<br> symbols, or expressions arranged in rows and columns"];
B --> C{Classification of Matrices};
subgraph C[Based on Dimensions & Structure]
    C1["Row Matrix 1 x n"]
    C2["Column Matrix m x 1"]
    C3["Square Matrix n x n"]
    C4["Rectangular Matrix m x n m ≠ n"]
    C5["Zero Matrix All elements are 0"]
    C6["Diagonal Matrix<br>Square matrix with<br> non-zero elements only on main diagonal"]
    C7["Identity Matrix I<br>Diagonal matrix with 1's on diagonal"]
    C8["Symmetric Matrix A = Aᵀ"]
    C9["Skew-Symmetric Matrix A = -Aᵀ"]
    C10["Triangular Matrix Upper & Lower"]
    C11["Orthogonal Matrix Aᵀ = A⁻¹"]
    C12["Singular Matrix detA = 0"]
    C13["Non-Singular Matrix detA ≠ 0"]
end
C --> D{Matrix Algebra};
subgraph D[Operations on Matrices]
    D1["Addition & Subtraction<br>Element-wise, requires same dimensions"]
    D2["Scalar Multiplication<br>Multiply every element by a scalar"]
    D3["Matrix Multiplication<br>#Columns in A = #Rows in B<br> to form AB"]
    D4["Transpose Aᵀ or A'<br>Flip rows and columns"]
    D5["Inverse A⁻¹<br>Exists only for square, non-singular matrices<br> A A⁻¹ = I"]
    D6["Determinant detA or A<br>Scalar value for square matrices only"]
    D7["Trace Sum of diagonal elements"]
end
D --> E{Properties of Operations};
subgraph E[Key Algebraic Properties]
    E1["Addition: Commutative A+B=B+A, Associative"]
    E2["Multiplication: Associative, Distributive, NOT Commutative"]
    E3["Transpose: ABᵀ = BᵀAᵀ, A+Bᵀ = Aᵀ+Bᵀ"]
    E4["Inverse: AB⁻¹ = B⁻¹A⁻¹, Aᵀ⁻¹ = A⁻¹ᵀ"]
    E5["Determinant: detAB = detA detB, detAᵀ = detA"]
end
E --> F{Special Matrix Operations};
subgraph F[Advanced Concepts]
    F1["Matrix Decomposition<br>LU, QR, Eigen, SVD decompositions"]
    F2["Rank of Matrix Maximum number of<br> linearly independent rows/columns"]
    F3["Eigenvalues and Eigenvectors<br>Ax = λx"]
    F4["Matrix Calculus<br>Derivatives with respect to vectors/matrices"]
end
F --> G{Applications of Matrices};
subgraph G[Real-World Applications]
    G1["Computer Graphics<br>Transformations rotation, scaling, translation"]
    G2["Physics & Engineering<br>Solving systems of linear equations, quantum mechanics"]
    G3["Economics & Operations Research<br>Input-output models, linear programming"]
    G4["Data Science & Statistics<br>Datasets, covariance matrices, machine learning"]
    G5["Computer Science<br>Graph theory adjacency matrices, search algorithms"]
    G6["Cryptography<br>Encoding and decoding messages"]
end
G --> H[End: Matrices as Fundamental Building Blocks];

flowchart TD
A[Start: Matrix Properties] --> B["A matrix is a rectangular array<br>of numbers, symbols, or expressions"];
B --> C{Classification by Dimensions & Structure};
subgraph C[Types of Matrices]
    C1["Row & Column Matrix"]
    C2["Square & Rectangular Matrix"]
    C3["Diagonal & Identity Matrix I"]
    C4["Zero or Null Matrix"]
    C5["Symmetric A = Aᵀ & Skew-Symmetric A = -Aᵀ"]
    C6["Triangular Upper & Lower"]
    C7["Orthogonal AᵀA = I"]
    C8["Singular det=0 & Nonsingular det≠0"]
end
C --> D{Matrix Algebra Properties};
subgraph D[Operation-Specific Properties]
    D1["Addition/Subtraction<br>- Commutative: A+B = B+A<br>- Associative: A+B+C = A+B+C<br>- Additive Identity: A + 0 = A<br>- Additive Inverse: A + -A = 0"]
    D2["Scalar Multiplication<br>- Associative: c d A = c d A<br>- Distributive: c A+B = cA + cB<br>-
c+d A = cA + dA"]
    D3["Matrix Multiplication<br>- NOT Commutative: AB ≠ BA general<br>- Associative: AB C = A BC<br>- Distributive: A B+C = AB + AC<br>-
B+C A = BA + CA<br>- Multiplicative Identity: AI = IA = A"]
    D4["Transpose<br>- Aᵀᵀ = A<br>- A+Bᵀ = Aᵀ + Bᵀ<br>- cAᵀ = cAᵀ<br>- ABᵀ = BᵀAᵀ crucial"]
    D5["Inverse exists if detA ≠ 0<br>- A A⁻¹ = A⁻¹A = I<br>- AB⁻¹ = B⁻¹A⁻¹<br>- Aᵀ⁻¹ = A⁻¹ᵀ<br>- Aⁿ⁻¹ = A⁻¹ⁿ"]
    D6["Trace Sum of diagonal elements<br>- trA+B = trA + trB<br>- trcA = c trA<br>- trAB = trBA"]
end
D --> E{Special Properties & Theorems};
subgraph E[Key Results]
    E1["Determinant Properties<br>- detI = 1<br>- detAᵀ = detA<br>- detAB = detA detB<br>- detA⁻¹ = 1/detA<br>- detcA = cⁿ detA for n×n matrix"]
    E2["Rank Properties<br>- ρA = ρAᵀ<br>- ρAB ≤ min ρA, ρB<br>- ρA+B ≤ ρA + ρB"]
    E3["Eigenvalue & Eigenvector Properties<br>- Sum of eigenvalues = trA<br>- Product of eigenvalues = detA"]
end
E --> F{Applications of Properties};
subgraph F[Real-World Significance]
    F1["Solving Linear Systems<br>Consistency, solutions"]
    F2["Matrix Decompositions<br>LU, QR, SVD for numerical stability"]
    F3["Vector Spaces & Linear Transformations<br>Basis, dimension, rank-nullity theorem"]
    F4["Computer Graphics & Physics<br>Transformations, quantum mechanics"]
    F5["Data Science & Optimization<br>PCA, least squares solutions"]
end
F --> G[End: Properties Define Matrix Behavior];

flowchart TD
A[Start: Matrix Inversion] --> B["Find A⁻¹ for a square matrix A<br>such that A × A⁻¹ = I"];
B --> C{Prerequisite Check};
subgraph C[Is Inversion Possible?]
    C1["Matrix must be Square n×n"]
    C2["Matrix must be Nonsingular<br>i.e., detA ≠ 0"]
end
C1 & C2 --> D{Choose an Inversion Method};
subgraph D[Methods for Finding A⁻¹]
    D1["Elementary Row Operations<br>Gauss-Jordan Elimination"]
    D2["Adjugate Matrix Method<br>A⁻¹ = adjA / detA"]
    D3["Decomposition Methods<br>LU Decomposition, Cholesky"]
    D4["Special Case Methods<br>For 2x2, 3x3, orthogonal, etc."]
end
D1 --> E["Augment matrix A with Identity Matrix: A | I"];
E --> F["Perform Gauss-Jordan Elimination<br>Apply row ops to reduce A to I"];
F --> G["The right block becomes A⁻¹<br>Result: I | A⁻¹"];
D2 --> H["Calculate the Determinant of A, detA"];
H --> I["Find the Matrix of Minors"];
I --> J["Create Matrix of Cofactors<br>Apply checkerboard pattern of signs +/ -"];
J --> K["Transpose to get the Adjugate Matrix adjA"];
K --> L["A⁻¹ = 1 / detA × adjA"];
D3 --> M["Factorize A into simpler matrices<br>e.g., A = L × U<br>Where L is lower triangular, U is upper triangular"];
M --> N["Solve for A⁻¹ by solving systems<br>using forward/back substitution"];
D4 --> O["For 2x2 Matrix:<br>a b⁻¹ = 1/ad-bc × d -b<br>c d    -c  a"];
G & L & N & O --> P["Verify Result: A × A⁻¹ = I<br>to avoid calculation errors"];
P --> Q{Applications of Matrix Inverse};
subgraph Q[Why Invert a Matrix?]
    Q1["Solve Systems of Linear Equations<br>A⋅x = b → x = A⁻¹⋅b"]
    Q2["Compute Division in Matrix Algebra"]
    Q3["Find Basis for Vector Space"]
    Q4["Coordinate Changes & Transformations"]
end
Q --> R[End: Inverse is a Fundamental Tool];

flowchart TD
A[Start: What is a Vector?] --> B{Vector Fundamental Concepts};
subgraph B[Core Properties]
    direction LR
    B1[Magnitude<br>Length of the arrow]
    B2[Direction<br>Angle or bearing]
end
B --> C{Representation};
C --> C1["Cartesian Coordinates<br>(x, y, z)"];
C --> C2["Polar Coordinates<br>(r, θ)"];
C --> C3["Unit Vectors<br>î, ĵ, k̂"];
C1 & C2 & C3 --> D[Key Vector Operations];
subgraph D[Operations]
    D1[Addition & Subtraction<br>Tip-to-Tail or Parallelogram Method]
    D2[Scalar Multiplication<br>Scaling magnitude,<br>flipping direction]
    D3[Dot Product<br>Result: Scalar<br>Measures Projection]
    D4[Cross Product<br>Result: Vector<br>Perpendicular to both inputs]
end
D --> E{Applications in Physics & Engineering};
E --> F1["Mechanics<br>Force, Velocity, Acceleration"];
E --> F2["Electromagnetism<br>Electric & Magnetic Fields"];
E --> F3["Computer Graphics<br>3D Modeling, Lighting"];
E --> F4["Navigation<br>Heading, Wind Correction"];
F1 & F2 & F3 & F4 --> G[End: Vector as a Fundamental Tool];

flowchart TD
A[Start: What is a Function?] --> B["A relation where each input (x)<br>has exactly one output (y)<br>f: X → Y"];
B --> C{Ways to Represent a Function};
subgraph C[Representation]
    direction LR
    C1["Algebraically<br>f(x) = x² + 1"]
    C2["Graphically<br>Plot on Coordinate Plane"]
    C3["Numerically<br>Table of Values"]
    C4["Verbally<br>Description in Words"]
end
C --> D{Classify by Behavior & Properties};
subgraph D1[Key Properties]
    D11["Injective (One-to-One)<br>f(a) = f(b) ⇒ a = b"]
    D12["Surjective (Onto)<br>Range = Codomain"]
    D13["Bijective<br>Both 1-1 and Onto<br>(Has an inverse)"]
end
subgraph D2[Symmetry]
    D21["Even Function<br>f(-x) = f(x)<br>Symmetric about y-axis"]
    D22["Odd Function<br>f(-x) = -f(x)<br>Symmetric about origin"]
end
subgraph D3[Rates of Change]
    D31["Increasing<br>x₁ < x₂ ⇒ f(x₁) < f(x₂)"]
    D32["Decreasing<br>x₁ < x₂ ⇒ f(x₁) > f(x₂)"]
    D33["Constant<br>f(x) = c"]
end
D --> E{Common Types of Functions};
subgraph E[Function Types]
    E1["Polynomial<br>f(x) = aₙxⁿ + ... + a₁x + a₀"]
    E2["Rational<br>f(x) = P(x)/Q(x)"]
    E3["Exponential<br>f(x) = a ⋅ bˣ"]
    E4["Logarithmic<br>f(x) = logₐ(x)"]
    E5["Trigonometric<br>sin(x), cos(x), tan(x)"]
    E6["Piecewise<br>Defined by different rules<br>on different intervals"]
end
E --> F[Operations on Functions];
subgraph F[Operations]
    F1["Arithmetic<br>f+g, f-g, f⋅g, f/g"]
    F2["Composition<br>(f∘g)(x) = f(g(x))"]
    F3["Inversion<br>f⁻¹(x) exists only if f is bijective<br>f(f⁻¹(x)) = x"]
end
F --> G{Applications of Functions};
G --> H1["Physics: Describe motion<br>Position, velocity, acceleration<br>as functions of time"]
G --> H2["Economics & Finance:<br>Cost, revenue, profit functions,<br>Compound interest"]
G --> H3["Computer Science:<br>Functions as subroutines,<br>Input-Output processing"]
G --> H4["Engineering: Signal processing,<br>Electrical waveforms"]
G --> H5["Everyday Life: Modeling trends,<br>Converting units, Temperature scales"]
H1 & H2 & H3 & H4 & H5 --> I[End: Functions as a Fundamental Concept];

flowchart TD
A[Start: What is a Limit?] --> B["The value a function f(x)<br> approaches as x approaches<br> some value c.<br>limₓ→c f(x) = L"];
B --> C{Approach to c can be from};
C --> C1["Both Sides limₓ→c"]
C --> C2["Left-Hand Limit limₓ→c⁻"]
C --> C3["Right-Hand Limit limₓ→c⁺"]
C1 & C2 & C3 --> D{The Limit Exists IF};
D --> D1["limₓ→c⁻ f(x) = L"]
D --> D2["limₓ→c⁺ f(x) = L"]
D --> D3["AND L is a finite number"]
D1 & D2 & D3 --> D4["limₓ→c f(x) = L exists"]
D4 --> E{How to Find the Limit?<br>Evaluation Methods};
subgraph E[Methods of Evaluation]
    E1["Direct Substitution<br>Plug in x = c"]
    E2["Algebraic Manipulation<br>Factor & Cancel, Rationalize"]
    E3["Special Limits<br>limₓ→0 sinx/x = 1"]
end
E1 --> F1{Get an Answer?};
E2 --> F1;
E3 --> F1;
F1 -->|Yes, a finite number L| G["Success!<br>limₓ→c f(x) = L"];
F1 -->|No, get Indeterminate Form 0/0, ∞/∞, etc.| H["Use Advanced Techniques"];
subgraph H[Advanced Techniques for Indeterminate Forms]
    H1["More Algebra<br>Factorization, Rationalization"]
    H2["L'Hôpital's Rule<br>Take derivative of numerator and denominator"]
end
H --> I["Re-evaluate the limit"];
I --> F1;
E --> J["Limits at Infinity<br>limₓ→±∞ f(x)"];
J --> J1["Compare degrees of numerator<br>and denominator in rational functions"]
J --> J2["Use rules of growth rates<br>e.g., Exponential > Polynomial > Log"]
G --> K{Applications of Limits};
subgraph K[Where Limits Are Used]
    K1["Define Continuity<br>f is continuous at c if<br>limₓ→c f(x) = f(c)"]
    K2["Define the Derivative<br>f'(x) = limₕ→₀ f(x+h)-f(x)/h"]
    K3["Define Improper Integrals<br>∫ₐ∞ f(x dx = limₜ→∞ ∫ₐᵗ f(x dx"]
    K4["Analyze Asymptotes & Behavior<br>Horizontal & Vertical Asymptotes"]
end
K --> L[End: Limits are the Foundation of Calculus];

flowchart TD
    A[Start: Is fx continuous at x = c?] --> B{Three Conditions Must Hold}
    
    subgraph B[Continuity Checklist]
        direction TB
        B1["1. fc is defined<br>c is in the domain of f"]
        B2["2. limₓ→c fx exists<br>LHL = RHL = L"]
        B3["3. limₓ→c fx = fc<br>The limit equals the function value"]
    end
    
    B1 -->|Yes| B2
    B1 -->|No| C1["Condition FAILED<br>Discontinuity at x=c"]
    
    B2 -->|Yes, L exists| B3
    B2 -->|No, LHL ≠ RHL| C2["Condition FAILED<br>Limit does not exist"]
    
    B3 -->|Yes| D["All Conditions MET<br>f is CONTINUOUS at x = c"]
    B3 -->|No, L ≠ fc| C3["Condition FAILED<br>Removable Discontinuity"]
    
    C1 & C2 & C3 --> E["f is DISCONTINUOUS at x = c"]
    
    E --> F{Classify the Type of Discontinuity}
    
    subgraph F[Types of Discontinuities]
        F1["Removable Hole<br>lim exists but ≠ fc<br>or fc is undefined"]
        F2["Jump<br>LHL and RHL exist but are not equal"]
        F3["Infinite Asymptotic<br>Limit is ±∞"]
        F4["Oscillating<br>Limit does not exist due to<br>wild oscillation e.g., sin1/x"]
    end
    
    D --> G[Extended Analysis]
    
    subgraph G[Beyond a Single Point]
        G1["Continuity on an Interval<br>Continuous at every point in a, b"]
        G2["Types of Continuous Functions<br>Polynomial, Rational, Root, Trig,<br>Exp, Log functions are continuous<br>on their domains"]
        G3["The Intermediate Value Theorem IVT<br>If f is continuous on a, b,<br>it takes on every value<br>between fa and fb"]
    end
    
    G --> H[End: Continuity Ensures Predictable, Connected Behavior]