Difference Equations to Differential Equations
An introduction to calculus
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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.
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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
- 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
Copyright © 1997, 2000 by
Dan Sloughter.
This work comes with ABSOLUTELY NO WARRANTY.
This work is licensed under a Creative Commons
License.
Last modified: Monday 30 June 2008 17:30 EDT
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