LECTURE NOTES OF WILLIAM CHEN
# FIRST YEAR CALCULUS

### Chapter 1 : THE NUMBER SYSTEM >>

### Chapter 2 : FUNCTIONS >>

### Chapter 3 : INTRODUCTION TO DERIVATIVES >>

### Chapter 4 : SOME SPECIAL FUNCTIONS >>

### Chapter 5 : APPLICATIONS OF DERIVATIVES >>

### Chapter 6 : LIMITS OF FUNCTIONS >>

### Chapter 7 : CONTINUITY >>

### Chapter 8 : DIFFERENTIATION >>

### Chapter 9 : THE DEFINITE INTEGRAL >>

### Chapter 10 : TECHNIQUES OF INTEGRATION >>

### Chapter 11 : NUMERICAL INTEGRATION >>

### Chapter 12 : APPLICATIONS OF INTEGRATION >>

### Chapter 13 : IMPROPER INTEGRALS >>

### Chapter 14 : ORDINARY DIFFERENTIAL EQUATIONS >>

### Chapter 15 : FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS >>

### Chapter 16 : SECOND ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS >>

### Chapter 17 : FUNCTIONS OF TWO VARIABLES >>

### Chapter 18 : INTERPOLATION AND APPROXIMATION >>

### Chapter 19 : SEQUENCES >>

### Chapter 20 : SERIES >>

### Chapter 21 : POWER SERIES >>

### Chapter 22 : THE BINOMIAL THEOREM >>

This set of notes has been compiled over a period of more than 30 years. Some chapters were used in various forms and on many occasions between 1981 and 1990 by the author at Imperial College, University of London. The remaining chapters were written in Sydney.

The material has been organized in such a way to create a single volume suitable for use as an introduction to elementary calculus.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

- The Real Numbers
- The Natural Numbers
- Completeness of the Real Numbers
- Further Discussion on the Real Numbers
- The Complex Numbers
- Polar Coordinates
- Finding Roots
- Analytic Geometry

- Introduction
- Composition of Functions
- Real Valued Functions
- One-to-One and Onto Functions
- One-to-One and Onto Real Valued Functions

- Introduction
- Stationary Points and Second Derivatives
- Curve Sketching
- Linearization of Error and Approximation of Derivative
- Resolving Indeterminate Limits
- Implicit Differentiation

- Exponential Functions
- The Exponential and Logarithmic Functions
- Derivatives of the Inverse Trigonometric Functions
- Rates of Growth of some Special Functions

- Kinematics on a Line
- Cost and Revenue Analysis
- Modelling with Maxima and Minima
- Global Maxima and Minima
- Newton's Method

- Introduction
- Further Techniques
- One Sided Limits
- Infinite Limits
- Limits at Infinity

- Introduction
- Continuity in Intervals
- Continuity in Closed Intervals
- An Application to Numerical Mathematics
- An Application to Inequalities

- Elementary Results on Derivatives
- Two Important Results on Derivatives
- Consequences of the Mean Value Theorem

- Finite Sums
- An Example
- The Riemann Integral
- Antiderivatives
- Fundamental Theorems of the Integral Calculus
- Average Values of Functions
- Further Discussion

- Integration by Substitution
- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitutions
- Completing Squares
- Partial Fractions

- Introduction
- The Trapezium Rule
- The Midpoint Rule
- Simpson's Rule
- Truncation Errors
- Richardson Extrapolation

- Areas on the Plane
- Volumes of Solids
- Application to Modelling in Science
- Application to Modelling in Economics
- Application to Probability Theory
- Separable Differential Equations
- Exponential Growth and Decay

- Introduction
- Unbounded Integrands
- Unbounded Intervals

- Introduction
- How Ordinary Differential Equations Arise
- Some Modelling Problems

- Introduction
- Separable Variable Type
- The Homogeneous Equation
- The Linear Equation
- Application to a Problem in Physics

- Introduction
- The Homogeneous Case
- An Analogy
- The Non-Homogeneous Case
- The Method of Undetermined Coefficients
- Lifting the Trial Functions
- Further Examples
- A More Systematic Approach for Particular Integrals
- Initial Conditions
- Summary
- Application to Problems in Physics

- Introduction
- Partial Derivatives
- The Differential
- Directional Derivatives
- The Total Derivative
- Change of Variables
- Tangent Planes and Normals
- Stationary Points
- An Application to Ordinary Differential Equations

- Exact Fitting
- Approximate Fitting
- Minimax Approximation
- Least Squares Approximation

- Introduction
- Special Results for Real Sequences
- Recurrence Relations
- Further Discussion

- Introduction
- Some Well Known Series
- Series of Non-Negative Terms
- Conditional Convergence
- Absolute Convergence
- Relationship with Integrals
- Further Discussion

- Introduction
- Taylor Series
- Application to Differential Equations
- Further Discussion

- Finite Binomial Expansions
- Infinite Binomial Expansions