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经典教材
Difference Equations to Differential Equations
(An Introduction to Calculus)
by Dan Sloughter, Furman University
2009
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
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