Chapter Review

Much of this chapter has been devoted to developing the body of computational formulas that make calculus a CALCULUS, i.e., a system of calculation. Most of us don't calculate just for the fun of it — we calculate to solve problems. Thus our development of calculational tools has been embedded in the process of solving meaningful problems.

We have now developed all the general rules for differentiation:

• Constant Multiple Rule
• Sum Rule
• General Power Rule
• Product Rule
• Inverse Function Rule
• Chain Rule

The last four were new in this chapter.

Early in the chapter we introduced second derivatives and considered the relationships between the graph of a function and the values of its first two derivatives. This led us to discover the importance of points at which the first derivative is zero — critical points — and at which the second derivative is zero — inflection points (usually). This brought us to the problem of finding numeric solutions of equations in one variable. Modern electronic tools solve this problem for us, but differential calculus is the key to understanding how they do it. In particular, we saw that local linearity leads to an elegant way to solve an equation quickly: Newton's Method.

To calculate critical points, to use Newton's Method, or to carry out many other problem-solving procedures, one must have formulas for computing derivatives. The focus of the rest of the chapter was on developing those formulas. For example, in the context of computing the rate of change of U.S. energy consumption, we developed the Product Rule and gave an interpretation of its components.

Next we turned to a mathematical analysis of the physical observation that light rays follow a path which minimizes travel time. In particular, analysis of the angle of reflection led us naturally to the most important derivative formula of all, the Chain Rule. When combined with the formulas known already, plus a direct calculation of the derivative of the square root function, the Chain Rule provided the tool we needed for optimizing the reflection travel-time function.

At the end of the chapter we examined the meaning of differential as a free­standing object. We found that we could interpret the differential of a dependent variable to mean rise to a tangent line, as opposed to difference of a dependent variable, which is the rise to a point on the graph. For small values of the run, the tangent line and the curve are close together, so differential and difference approximate each other. We used that approximation to estimate the volume of the earth's crust — modeled as a difference of the volumes of two concentric spheres — and we saw that the corresponding differential was easier to calculate.