Applications of derivatives
Now that we can find derivatives quickly, we can use them. Derivatives tell us how fast things change, where a function is largest or smallest, and what its graph looks like.
In this chapter, you will solve problems about changing quantities, estimate values, sketch graphs, and find the best answer to real problems. At the end, derivatives also help us find hard limits.
This chapter covers the following topics:
Related rates
How to find the rate of change of one quantity from the rate of another, with circles, spheres, ladders, cars, and a cone-shaped tank.
Linear approximation and differentials
Using the tangent line to estimate values, the small-angle approximation, differentials, and errors in measurements.
Maximum and minimum values
Absolute and local extreme values, the Extreme Value Theorem, critical numbers, and the closed interval method.
The Mean Value Theorem
Rolle's Theorem, the Mean Value Theorem, and why a zero derivative means a constant function.
The shape of a graph
Increasing and decreasing functions, the first derivative test, concavity, inflection points, and the second derivative test.
Curve sketching
A step-by-step checklist for sketching a graph by hand, with three fully worked examples.
Optimization problems
How to find the largest or smallest value in a real problem: a fence, a box, a can, and the closest point on a curve.
L'Hôpital's rule
Using derivatives to find limits of the forms 0/0 and infinity/infinity, and other indeterminate forms.
Before you start
This chapter builds on the last two chapters. These pages may help: