Dan Sloughter, Department of Mathematics, Furman University, Greenville, SC 29613
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
- Applet: Area of a circle
- Applet: 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
- Applet: Square wave approximation
- Applet: Sound wave approximation
- Limits and the notion of continuity
- Continuous functions
- Some consequences of continuity
Chapter 3: Best affine approximations
- Best affine approximations
- Applet: 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
- Applet: 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
- Applet: Numerical integration rules
- The fundamental theorem of calculus
- Applet: Cumulative area
- Using the fundamental theorem
- Applet: 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 of definite integrals
- Applet: Taylor polynomials
- 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
- Applet: Projectile motion
- The two-body problem
Chapter 8: Differential equations
- Numerical solutions of differential equations
- Applet: 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
- Applet: Phase plane for a pendulum
- Power series solutions
Answers for selected problems are available here.
A concatenated version (single download) is available here. This file is approximately 3.4 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.
