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.
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
- 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
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