# Difference Equations to Differential Equations

## An Introduction to Calculus

By Dan Sloughter, Furman University

### Chapter 1: Sequences, limits, and difference equations

Java applet: Area of a circle

Java applet: Tangent line for a parabola

Nonlinear difference equations

### Chapter 2: Functions and their properties

Java applet: Square wave approximation

Java applet: Sound wave approximation

Limits and the notion of continuity

Some consequences of continuity

### Chapter 3: Best affine approximations

Java appplet: Affine approximations

Best affine approximations, derivatives, and rates of change

Differentiation of polynomials and rational functions

Differentiation of compositions of functions

Differentiation of trigonometric functions

Java applet: Newton’s method

Rolle’s Theorem and the Mean Value Theorem

Finding maximum and minimum values

### Chapter 4: Integration

Numerical approximations of definite integrals

Java applet: Numerical integration rules

The fundamental theorem of calculus

Java applet: Cumulative area

Java applet: Change of variable

More techniques of integration

Distance, position, and the length of curves

### Chapter 5: Polynomial approximations and Taylor series

Java applet: Taylor polynomials

Infinite series: the comparison test

Infinite series: the ratio test

Infinite series: absolute convergence

### Chapter 6: More transcendental functions

The natural logarithm function

Integration of rational functions

Inverse trigonometric functions

### Chapter 7: The complex plane

The algebra of complex numbers

The calculus of complex functions

Complex-valued functions: motion in the plane

Java applet: Projectile motion

### Chapter 8: Differential equations

Numerical solutions of differential equations

Java applet: Euler’s method

First order linear differential equations

Second order linear differential equations

Applications: pendulums and mass-spring systems

The geometry of solutions: the phase plane

Java applet: Phase plane for a pendulum

Answers for selected problems are available here.

A concatenated version (single download) is available here. The file is approximately 3.1 MB.

Send e-mail to Dan Sloughter to report any errors.

#### Copyright © 1997, 2000 by Dan Sloughter

This work is licensed under a Creative Commons License.