Difference Equations to Differential Equations
An introduction to calculus
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A companion multivariable 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 email 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
 Complexvalued functions: motion in the plane
 Applet: Projectile motion
 The twobody 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 massspring systems
 The geometry of solutions: the phase plane
 Applet: Phase plane for a pendulum
 Power series solutions

