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