Free Calculus Notes and Worked Examples

Teaching

I have a license to teach mathematics at the secondary level in the state of Minnesota.

Minnesota State Teaching License

As a calculus instructor, I prepared these class notes as a review of the chapters in Stewart's Single Variable Calculus, Early Transcendentals . These notes include worked examples and study materials for limits, derivatives, integrals, applications, sequences, series, parametric equations, polar coordinates, vectors, and introductory differential equations.

Use the sections below to find notes by topic. Each unit can be expanded or collapsed to make the page easier to browse.

Limits and Continuity

Foundational topics on tangent and velocity problems, limits, continuity, and limits at infinity.

Class Notes
2.1 Tangent and Velocity Problems
2.2 Limit of a Function
2.3 Calculating Limits Using the Limit Laws
2.4 Precise Definition of a Limit
2.5 Continuity
2.6 Limits at Infinity, Horizontal Asymptotes
Derivatives

Core derivative concepts and derivative rules.

Class Notes
2.7 Derivatives and Rates of Change
2.8 Derivative as a Function
3.1 Derivatives of Polynomials and Exponential Functions
3.2 Product and Quotient Rules
3.3 Derivatives of Trigonometric Functions
3.4 Chain Rule
3.5 Implicit Differentiation
3.6 Derivatives of Logarithmic Functions
Applications of Derivatives

Related rates, linear approximation, optimization, curve sketching, and Newton's method.

Class Notes
3.9 Related Rates
3.10 Linear Approximations and Differentials
4.1 Maximum and Minimum Values
4.2 The Mean Value Theorem
4.3 How Derivatives Affect the Shape of a Graph
4.4 Intermediate Forms & L'Hospital's Rule
4.5 Curve Sketching
4.7 Optimization Problems
4.8 Newton's Method
4.9 Antiderivatives
Integrals

Riemann sums, definite integrals, the Fundamental Theorem of Calculus, and substitution.

Class Notes
5.1 Riemann Sums and Distances
5.2 Definite Integral
5.3 Fundamental Theorem of Calculus
5.4 Indefinite Integrals and the Net Change Theorem
5.5 Substitution Rule
Applications of Integration

Areas, volumes, average value, arc length, surface area, and physical applications.

Class Notes
6.1 Areas Between Curves
6.2 Volumes
6.3 Volumes by Cylindrical Shells
6.5 Average Value of a Function
8.1 Arc Length
8.2 Surface Revolution
8.3 Physics Engineering
Techniques of Integration

Integration by parts, trigonometric integrals, trigonometric substitution, integration strategies, approximation, and improper integrals.

Class Notes
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitution
7.4 Integration by Partial Fractions
7.5 Integration Strategies
7.7 Approximate Integration
7.8 Improper Integrals
Introductory Differential Equations

Differential equation modeling, direction fields, Euler's method, separable equations, growth models, linear equations, and predator-prey systems.

Class Notes
9.1 Modeling Differential Equations
9.2 Direction Fields and Euler Method
9.3 Separable Equations
9.4 Population Growth Models
9.5 Linear Equations
9.6 Predator Prey
Parametric Equations, Polar Coordinates, and Conic Sections

Parametric curves, polar coordinates, polar area and length, and conic sections.

Class Notes
10.1 Parametric Equation Curves
10.2 Parametric Curves
10.3 Polar Coordinates
10.4 Polar Area Lengths
10.5 Conic Sections
Sequences and Series

Sequences, series, convergence tests, power series, and Taylor/Maclaurin series.

Class Notes
11.1 Sequences
11.2 Series
11.3 Integral Test, Sum Estimates
11.4 Comparison Test
11.5 Alternating Test
11.6 Absolute, Ratio, and Root Tests
11.7 Testing Series Convergence Strategy
11.8 Power Series
11.9 Functions as Power Series
11.10 Taylor and Maclaurin Series
Vectors and Three-Dimensional Geometry

Three-dimensional coordinates, vectors, dot products, cross products, lines, and planes.

Class Notes
12.1 3D Coordinates
12.2 Vectors
12.3 Dot Product
12.4 Cross Product
12.5 Equations for Lines and Planes