Calculus and
Linear Algebra
Preface
Why This Material Matters for Data Science and Machine Learning
Disclaimers
A Final Word
I Review
1
Functions and Linear Equations
1.1
Lecture Content
1.2
Lecture Notes
1.2.1
Common Functions and Their Graphs
1.2.2
Finding the Domain of a Function
1.2.3
Solving Systems Using the Elimination Method
1.2.4
Revenue, Cost, and Profit Functions
1.3
Examples
1.3.1
Applications of Linear Equations
1.3.2
Systems of Linear Equations
1.4
Practice Problems
1.5
Self-Assessment
1.6
Lesson Checklist
II Matrix Algebra
2
Introduction to Matrices
2.1
Lecture Content
2.2
Lecture Notes
2.2.1
Augmented Matrix
2.2.2
Elementary Row Operations
2.2.3
Gauss-Jordan Elimination Steps
2.3
Examples
2.4
Practice Problems
2.5
Self-Assessment
2.6
Lesson Checklist
3
Matrix Operations
3.1
Lecture Content
3.2
Lecture Notes
3.2.1
Matrix Notation:
\(A_{m \times n}\)
3.2.2
Notation for Matrix Entries:
\(a_{ij}\)
3.2.3
Matrix Addition and Subtraction
3.2.4
Scalar Multiplication
3.2.5
Matrix Multiplication
3.2.6
Matrix Transpose
3.2.7
\(a_{ij}\)
Notation for Operations
3.3
Examples
3.3.1
Addition and Subtraction
3.3.2
Multiplication
3.4
Practice Problems
3.5
Self-Assessment
3.6
Lesson Checklist
4
Inverses and Determinants
4.1
Lecture Content
4.2
Lecture Notes
4.2.1
Determinant of a
\(2 \times 2\)
Matrix
4.2.2
Determinant of a
\(3 \times 3\)
Matrix
4.2.3
Determinant of Higher-Order Matrices
4.2.4
Inverse of a
\(2 \times 2\)
Matrix
4.2.5
Inverse of Higher-Order Matrices (Steps)
4.3
Examples
4.4
Practice Problems
4.5
Self-Assessment
4.6
Lesson Checklist
III Linear Algebra
5
Vector Projections
5.1
Lecture Content
5.2
Lecture Notes
5.2.1
Vector Operations
5.2.2
Standard Inner Product (Dot Product)
5.2.3
Euclidean Norm of a Vector
5.2.4
Cosine of the Angle Between Two Vectors
5.2.5
Vector Projection Formula
5.2.6
Geometric Interpretations
5.3
Examples
5.4
Practice Problems
5.5
Self-Assessment
5.6
Lesson Checklist
6
Basis and Orthonormal Basis
6.1
Lecture Content
6.2
Lecture Notes
6.2.1
Linear Independence
6.2.2
Span
6.2.3
Standard Basis
6.2.4
Gram-Schmidt Orthonormalization
6.2.5
Coordinates Relative to an Ordered Basis
6.2.6
Change of Basis Matrix in
\(\mathbb{R}^2\)
6.3
Practice Problems
6.4
Self-Assessment
6.5
Lesson Checklist
7
The Rotation Matrix
7.1
Lecture Material
7.2
Lecture Notes
7.2.1
Rotation Matrix in
\(\mathbb{R}^2\)
7.2.2
Geometric Interpretation: Left Multiplication as Rotation
7.2.3
Stretching and Compression
7.2.4
Reflection Matrices
7.2.5
Summary
7.3
Practice Problems
7.4
Self-Assessment
7.5
Lesson Checklist
8
Eigenvalues and Eigenvectors
8.1
Lecture Content
8.2
Lecture Notes
8.2.1
Eigenvalues and Eigenvectors
8.2.2
Finding Eigenvalues (for
\(2 \times 2\)
or
\(3 \times 3\)
matrices)
8.2.3
Finding Eigenvectors
8.2.4
Diagonalization (for
\(2 \times 2\)
matrices with two distinct eigenvalues)
8.3
Examples
8.4
Practice Problems
8.5
Self-Assessment
8.6
Lesson Checklist
IV Differential Calculus
9
Exponential and Logarithmic Functions
9.1
Lecture Content
9.2
Lecture Notes
9.2.1
Rules for Exponents
9.2.2
Rules for Logarithms
9.2.3
Finding an Exponential Equation from Two Points
9.2.4
Solving Exponential and Logarithmic Equations
9.2.5
Applications and Formulas
9.3
Examples
9.3.1
Quadratic Functions
9.3.2
Exponent Rules
9.3.3
Logarithm Rules
9.4
Practice Problems
9.5
Self Assessment
9.6
Lesson Checklist
10
Limits and Continuity
10.1
Lecture Content
10.2
Lecture Notes
10.2.1
Estimate a Limit Using a Table of Values
10.2.2
Use Factoring to Evaluate Limits
10.2.3
One-Sided Limits and Vertical Asymptotes
10.2.4
Dominant Term Analysis for Limits at Infinity
10.2.5
Continuity of Piecewise Functions
10.3
Examples
10.3.1
Limits
10.3.2
One Sided Limits
10.3.3
One Sided Limits
10.4
Practice Problems
10.5
Self Assessment
10.6
Lesson Checklist
11
The Limit Definition of the Derivative
11.1
Lecture Content
11.2
Lecture Notes
11.2.1
Limit Definition of the Derivative
11.2.2
Derivative Rules for Common Functions
11.2.3
Sum, Difference, and Constant Multiple Rules
11.2.4
Tangent Line to a Function at a Point
11.3
Examples
11.3.1
Average ROC
11.3.2
Average ROC
11.3.3
Derivatives using Limits
11.3.4
Basic Derivative Rules
11.4
Practice Problems
11.5
Self Assessment
11.6
Lesson Checklist
V Derivative Formulas and Applications
12
The Product and Quotient Rules
12.1
Lecture Content
12.2
Lecture Notes
12.2.1
Product Rule for Two Functions
12.2.2
Product Rule for Three Functions
12.2.3
Quotient Rule
12.3
Exercises
12.3.1
Applications of Differentiation
12.3.2
The Product and Quotient Rules
12.4
Practice Problems
12.5
Self Assessment
12.6
Lesson Checklist
13
The Chain Rule and L’Hopital’s Rule
13.1
Lecture Content
13.2
Lecture Notes
13.2.1
Derivative of a Composition of Functions (Chain Rule)
13.2.2
L’Hôpital’s Rule
13.3
Examples
13.3.1
The Chain Rule
13.3.2
Derivatives of Logs and Exponentials
13.4
Practice Problems
13.5
Self Assessment
13.6
Lesson Checklist
14
Implicit Differentiation
14.1
Lecture Content
14.2
Lecture Notes
14.2.1
Higher-Order Derivatives
14.2.2
Implicit Differentiation
14.3
Examples
14.4
Practice Problems
14.5
Self Assessment
14.6
Lesson Checklist
15
Graphical Analysis
15.1
Lecture Content
15.2
Lecture Notes
15.2.1
Finding Critical Numbers
15.2.2
Classifying Maximums, Minimums, and Inflection Points
15.2.3
Extreme Value Theorem (EVT)
15.2.4
Intervals of Increase, Decrease, and Concavity
15.3
Examples
15.3.1
Maximum and Minimum Values
15.3.2
Applied Problems
15.3.3
Increasing and Decreasing
15.3.4
Curve Sketching
15.4
Practice Problems
15.5
Self Assessment
15.6
Lesson Checklist
VI Integral Calculus
16
Antiderivatives
16.1
Lecture Content
16.2
Lecture Notes
16.2.1
Basic Antiderivatives and Rules
16.2.2
Integration by Substitution
16.2.3
Integration by Parts
16.3
Examples
16.3.1
Antiderivatives
16.3.2
U-Substitution
16.3.3
Integration by Parts
16.4
Practice Problems
16.5
Self Assessment
16.6
Lesson Checklist
17
Riemann Sums
17.1
Lecture Content
17.2
Lecture Notes
17.2.1
Estimating Area Using
\(n\)
Rectangles
17.2.2
Properties of Summation Notation
17.2.3
Summation Formulas
17.2.4
Limit Definition of the Definite Integral
17.3
Examples
17.4
Practice Problems
17.5
Self Assessment
17.6
Lesson Checklist
18
The Fundamental Theorem of Calculus
18.1
Lecture Content
18.2
Lecture Notes
18.2.1
Fundamental Theorem of Calculus – Part I
18.2.2
Fundamental Theorem of Calculus – Part II
18.2.3
Properties of Definite Integrals
18.2.4
Fundamental Theorem of Calculus
18.3
Practice Problems
18.4
Self Assessment
18.5
Lesson Checklist
VII Additional (Important) Topics
19
Area between Curves
19.1
Lecture Content
19.2
Lecture Notes
19.2.1
Definite Integrals as Net Area
19.2.2
Simple Improper Integrals
19.2.3
Area Between Two Curves
19.3
Examples
19.4
Practice Problems
19.5
Self Assessment
19.6
Lesson Checklist
20
Constrained Optimization
20.1
Lecture Content
21
Lecture Notes
21.0.1
Sketching the Feasible Region
21.0.2
Maximizing the Objective Function
21.1
Example
21.2
Practice Problems
21.3
Self Assessment
21.4
Lesson Checklist
22
Partial Derivatives
22.1
Lecture Content
23
Lecture Notes
23.0.1
Evaluation of Functions of Two Variables
23.0.2
Finding Partial Derivatives
23.0.3
Clairaut’s Theorem (Equality of Mixed Partials)
23.0.4
D-Test for Classifying Critical Points
23.1
Example
23.1.1
Functions of Several Variables
23.1.2
Partial Derivatives
23.1.3
Maximum and Minimum Values
23.2
Practice Problems
23.3
Self Assessment
23.4
Lesson Checklist
Calculus and Linear Algebra
Lesson 22
Partial Derivatives
22.1
Lecture Content
Video: Functions of Several Variables
Video: Partial Derivatives
Video: Maximums and Minimums