MATH101, Calculus

Undergraduate course, Lancaster University, Department of Mathematics & Statistics, 2018

1 Sequences

Induction

Application to sums of squares and cubes

Series as a limit of partial sums

Geometric series, and sum of telescoping series

2. Functions of a real variable

Polynomials

Rational functions

Continuity and limits of a function of a real variable

Exponential and hyperbolic functions defined via series

Compositions and inverses

The natural logarithm function

3. Differentiation

Definition of the derivative as limit of tangent

Differentiable functions are continuous

Product and chain rules

Taylor series

Maxima and minima as stationary points

4. Complex numbers

Argand diagram

Complex exponentials

De Moivre’s Theorem

Roots of unity

Second order differential equations with linear constant coefficients

General solutions, and Particular Integrals in trig and polynomial cases

5. Integration

Indefinite Integration

Definite integration

Fundamental Theorem of Calculus

Integration by parts

Integration by substitution