LECTURE NOTES OF WILLIAM CHEN

# FIRST YEAR CALCULUS

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.

### Chapter 1 : THE NUMBER SYSTEM >>

• 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

### Chapter 2 : FUNCTIONS >>

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

### Chapter 3 : INTRODUCTION TO DERIVATIVES >>

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

### Chapter 4 : SOME SPECIAL FUNCTIONS >>

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

### Chapter 5 : APPLICATIONS OF DERIVATIVES >>

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

### Chapter 6 : LIMITS OF FUNCTIONS >>

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

### Chapter 7 : CONTINUITY >>

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

### Chapter 8 : DIFFERENTIATION >>

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

### Chapter 9 : THE DEFINITE INTEGRAL >>

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

### Chapter 10 : TECHNIQUES OF INTEGRATION >>

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

### Chapter 11 : NUMERICAL INTEGRATION >>

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

### Chapter 12 : APPLICATIONS OF INTEGRATION >>

• 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

### Chapter 13 : IMPROPER INTEGRALS >>

• Introduction
• Unbounded Integrands
• Unbounded Intervals

### Chapter 14 : ORDINARY DIFFERENTIAL EQUATIONS >>

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

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

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

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

• 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

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

• 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

### Chapter 18 : INTERPOLATION AND APPROXIMATION >>

• Exact Fitting
• Approximate Fitting
• Minimax Approximation
• Least Squares Approximation

### Chapter 19 : SEQUENCES >>

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

### Chapter 20 : SERIES >>

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

### Chapter 21 : POWER SERIES >>

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

### Chapter 22 : THE BINOMIAL THEOREM >>

• Finite Binomial Expansions
• Infinite Binomial Expansions