First Year Calculus

by WWL Chen

*This set of notes has been compiled
over a period of more than 25 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. All chapters have been in use
at Macquarie University since 1994.*

*The material has been organized in
such a way to create a single volume suitable for use in the calculus
half of the units MATH135, MATH136, MATH132 and MATH133 at Macquarie
University.*

*To read the notes, click the chapters
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

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

- 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

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

- 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

Chapter 22: THE BINOMIAL THEOREM

- Finite Binomial Expansions
- Infinite Binomial Expansions