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Students see how calculus can explain the structure of a rainbow, guide a robot arm, or analyze the spread of AIDS. Each module starts with a concrete problem and moves on to provide a solution. The discussions are detailed, realistic, and pay careful attention to the process of mathematical modeling. Exercises, solutions, and references are provided.
Introduction: Resources for Calculus
Preface: Using these Modules
Derivatives
Arbitrating Disputes.
Maximizing quadratic polynomials finds a fair solution to a labor dispute.
Fitting Lines to Data.
Minimizing quadratic polynomials finds the regression line to a data set.
Somewhere within the Rainbow.
Minimizing trigonometric functions explains properties of the rainbow.
Three Optimization Problems in Computing.
Maximizing functions shows how to store and transmit data most efficiently.
Newton’s Method and Fractal Patterns.
Newton’s method generates fractal designs in the complex plane.
Differential Equations and Integrals
How Old is the Earth?
The exponential decay of rubidium is used to date very old rocks.
Falling Raindrops.
Four differential equation models describe how a raindrop falls.
Measuring Voting Power.
Integrals of polynomials give a measure of voting power.
How to Time a Radio.
Trigonometric integrals explain tuning a radio.
Volumes and Hypervolumes.
Integrals calculate the volumes of objects in three and higher dimensions.
Reliability and the Cost of Guarantees.
Probability integrals predict how long equipment will last.
Queueing Systems.
Differential equations and limits analyze waiting lines.
Moving a Planar Robot Arm.
Derivatives guide a robot arm along a curve.
Designer Curves.
Parametric curves are used in computer aided design of automobiles.
Modeling the AIDS Epidemic.
Differential equations model the spread of AIDS.
Speedy Sorting.
Integrals, limits and L’Hopital’s rule compare the speed of sorting algorithms.
Calculus of Several Variables.
Hydro-Turbine Optimization
Lagrange multipliers allocate water to hydroelectric turbines.
Portfolio Theory.
Lagrange multipliers choose a best stock-market portfolio