# MATH101, Calculus

Undergraduate course, Lancaster University, Department of Mathematics & Statistics, 2019

### 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