Table of Contents for Maple Demonstrations for Math 103, 104, 114 and 115

In this list, the demonstrations progress from the beginning of Math 103 through the end of Math 114 and 115. There is an indication next to each demo concerning to which courses it applies. Each demo should be loadable with either Maple 6, 7, or 8.

Functions and their properties

• Linear functions / plotting to determine the range of nonlinear functions
• Basic properties of exponential functions
• Power functions / use of solve and evalf
• Calculating inverse functions in Maple / logarithms
• Operations on functions (scaling of independent and dependent variables) / trigonometric functions

Limits and derivatives

• Limits - by graphing, numerical approximation, and symbolically
• Idea of the derivative, average rate of change as slope of secants
• Derivative as limit of average rate of change
• Definition of the derivative
• Maple's diff command, finding equations of tangent lines, graphical interpretations of derivatives
• Derivatives and linear approximations / second derivatives
• Bitangents (lines tangent to a curve at two different points) - connects to a sequence of problems in the Lab Manual
• Review of basic Maple (plot, solve, diff); illustrations of the chain rule and implicit differentiation
• Newton's method
• Using the derivative to solve max and min problems

Integrals

• Illustration of integrals as limits of Riemann sums
• More on integrals and Riemann sums - graphics and evaluation of integrals as limits
• Fundamental theorem of calculus illustrations, area between two curves
• Solids of revolution -- illustration via tubeplot and volume via integration (see also the first demo in the differential equations section below)
• Symbolic integration - definite and indefinite

Exponentials and logarithms

• Exponential functions -- finding the exponential function through two points, exponential growth, derivatives of exponential functions, exponential models from differential equations
• Logarithms - use to solve exponential equations, calculus of logarithms
• More exponential function -- compound interest
• Exponential (radioactive) decay

Sequences and series

• Numerical sequences and series; illustration of the integral test
• Taylor polynomials, starting from linear and quadratic approximations, illustration of how higher-degree approximations improve
• More examples of Taylor polynomials and series, complex exponential
• Using Taylor polynomials to approximate functions with given error bounds; integrals via Taylor series

Calculus in three dimensions

• Lines and planes in three dimensions, quadric surfaces
• Functions of two variables -- plotting, partial derivatives, classification of critical points
• Tangent planes and linear approximations (differentials) of functions of two variables
• Lagrange multipliers
• Double integrals in rectangular and polar coordinates (see also the demo on least squares under "Other topics" below)

Differential equations

• Using DEtools and dsolve on differential equations; calculating arclength
• First-order differential equations, plotting direction fields with DEplot
• Orthogonal trajectories
• Second-order differential equations; damped and forced oscillations

Other topics

• Plotting parametrically defined curves and curves in polar coordinates
• Permutations and combinations in Maple -- the Pennsylvania lottery
• The binomial distribution, Maple for mean, variance, etc of discrete probability distributions
• The birthday problem
• The normal distribution (and the statevalf library)
• Least-squares linear fits (Maple commands), double integrals
• Matrices in Maple -- row operations, Gauss elimination
• Inverse of I-A for small A via "geometric series"; the simplex method in Maple