Contents

1 Techniques of integration

Integration by substitution

Integration by parts

Doing calculus using SageMath

Integrating powers of trigonometric functions

2 Limits

Revisiting limits

L’Hôpital’s rule

Evaluating limits using SageMath

3 Further concepts of differentiation

Implicit differentiation

Logarithmic differentiation

Related rates

4 Sequences

Introduction to sequences

Convergence of sequences: limit

Convergence of sequences: bounded and monotonic

5 More transcendental functions

Inverse trigonometric functions

Calculus with inverse trigonometric functions

Hyperbolic functions

Calculus with hyperbolic functions

Integrating powers of hyperbolic functions

6 Miscellany of algebra topics

Complex numbers

Factoring

Partial fraction decomposition

7 Further techniques of integration

Trigonometric substitutions

Integrating rational functions

Improper integrals

8 Further applications of integration

Probability

Arc length

9 Linear algebra

Solving linear systems: Gauss’s method

Transition to matrix notation

Gauss-Jordan reduction

Solving systems of linear equations using SageMath

10 Series

Summation notation

Convergence of series

Integral test

Comparison test

Ratio test

6 Systems of ODEs

Introduction to systems of ODEs

Linear systems of ODEs

Eigenmethod

Two dimensional systems and their vector fields

Second order systems and applications

11 Polynomials with infinitely-many terms

Power series

Taylor series

Calculus with power series

12 Useful formulas

Number sets

Exponentials and logarithms

Trigonometry

Hyperbolics