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Difference Equations to Differential Equations

An introduction to calculus

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Each section of the text is in Portable Document Format (PDF). PDF viewers are available here and here.

A PostScript version may be found here (the old homepage).

Difference Equations to Differential Equations was written with the help of Tex, DVIPS, xdvi, PDFTeX, XEmacs, nedit, XFig, epstopdf, pstoedit, Acrobat Reader® and Mathematica®.

A companion multi-variable calculus text, The Calculus of Functions of Several Variables, is available here.

For an alternative introduction to calculus, see Yet Another Calculus Text.

Answers for selected problems are available here.

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

Chapter 1: Sequences, limits, and difference equations

Calculus: areas and tangents
Applet: Area of a circle
Applet: Tangent line for a parabola
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   Tell A Friend