# Difference Equations to Differential Equations

## An Introduction to Calculus

By Dan Sloughter, Furman University

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

Calculus: Areas and tangents

Java applet: Area of a circle

NEW: Dart version: Area of a unit circle

Java applet: Tangent line for a parabola

NEW: Dart version: Tangent line for a parabola

Sequences

The sum of a sequence

Difference equations

Nonlinear difference equations

### Chapter 2: Functions and their properties

Functions and their graphs

Trigonometric functions

Java applet: Square wave approximation

NEW: Dart version: Square wave approximation

Java applet: Sound wave approximation

NEW: Dart version: Sound wave approximation

Limits and the notion of continuity

Continuous functions

Some consequences of continuity

### Chapter 3: Best affine approximations

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

Newton’s method

Java applet: Newton’s method

NEW: Dart version: Newton’s method

Rolle’s Theorem and the Mean Value Theorem

Finding maximum and minimum values

The geometry of graphs

### Chapter 4: Integration

The definite integral

Numerical approximations of definite integrals

Java applet: Numerical integration rules

NEW: Dart version: Numerical integration rules

The fundamental theorem of calculus

Java applet: Cumulative area

NEW: Dart version: Cumulative area

Using the fundamental theorem

Java applet: Change of variable

NEW: Dart version: Change of variable

More techniques of integration

Improper integrals

More on area

Distance, position, and the length of curves

### Chapter 5: Polynomial approximations and Taylor series

Polynomial approximations

Java applet: Taylor polynomials

NEW: Dart version: Taylor polynomial approximations

Taylor’s theorem

Infinite series revisited

Infinite series: the comparison test

Infinite series: the ratio test

Infinite series: absolute convergence

Power series

Taylor series

Some limit calculations

### Chapter 6: More transcendental functions

The exponential function

The natural logarithm function

Models of growth and decay

Integration of rational functions

Inverse trigonometric functions

Trigonometric substitutions

Hyperbolic 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

The two-body problem

### Chapter 8: Differential equations

Numerical solutions of differential equations

Java applet: Euler’s method

NEW: Dart version: Euler’s method

Separation of variables

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

NEW: Dart version: Phase plane for a pendulum

Power series solutions

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.

Dan Sloughter, Department of Mathematics, Furman University, Greenville, SC, 29613