Chapter Review

We began this chapter by examining three different models for falling raindrops: Galileo's Model, which only accounted for the acceleration due to gravity, a model incorporating Stokes' law, which includes a term in the differential equation for air resistance proportional to the velocity, and the Velocity-Squared Model, which assumes air resistance proportional to the square of the velocity. For the first two models, we obtained symbolic solutions of the corresponding initial value problems.

However, we did not obtain symbolic solutions for the Velocity-Squared Model. Rather, we used Euler's Method, a tool for generating numerical and graphical solutions to initial value problems. Euler's Method, a natural extension of the concept of the differential, is a general tool that has nothing to do with falling raindrops. With a computer or graphing calculator at hand, we can solve an initial value problem, whether or not we know how to find a formula for the solution.

Next we turned to a physical problem as a prototype for problems whose solutions are periodic: the spring-mass system. This model led us to the study of sine and cosine functions, a major component of the description of these systems. We applied sines and cosines to model annual variations in nature — minutes of daylight and average temperature — and then to a model for the spring-mass motion.

Our first approach to solving the spring-mass problem — with acceleration negatively proportional to displacement — was to rewrite it as a system of two first order problems, using velocity as an intermediate variable. That turned our problem into one that we could solve numerically by Euler's Method. The numerical solution looked like a cosine, which fit with our calculation of derivatives of sine and cosine. Then we found a symbolic solution, which was indeed a cosine function, with period determined by the spring constant and mass, and amplitude determined by the initial displacement.

Having introduced two trigonometric functions, we moved on to other trigonometric functions (specifically, tangent and secant), as well as inverse trigonometric functions (specifically, arcsine and arctangent) to solve problems of right-angle trigonometry. In all cases, we found that we could calculate derivatives of the new functions from knowledge of the derivatives of sine and cosine, together with techniques from Chapter 4.

Finally, in Section 5.6 we consolidated all our derivative formulas from Chapters 2-5 and illustrated how they could be used to differentiate virtually any function defined by formulas whose components are algebraic, exponential, logarithmic, trigonometric, or inverse trigonometric functions. Furthermore, we saw that we could easily employ a graphing tool (computer or calculator) to check the results of such calculations.