Problem Set 4—Derivatives of Trigonometric Functions

Purpose

This lesson develops your understanding of the rules for differentiating trigonometric functions. In the process, it also reinforces rules for differentiating sums, products, quotients, etc.

Background

This exercise is mainly based on material in in section 3.5 (“Derivatives of Trigonometric Functions”) of our textbook. We covered (or will cover) this material in classes between approximately September 30 and October 3. The exercise also draws on section 3.3 of the textbook (“Differentiation Rules”), which we dicussed in class between September 22 and 29.

Activity

Solve each of the following problems:

Problem 1

Section 3.5, exercise 10 (Find dy/dx given that y = (sinx + cosx)secx).

Problem 2

(A slight variation on Section 3.5, exercise 2). Find dy/dx given that y = 3/x - 5sinx. Also find d²y/dx², and one antiderivative of y with respect to x.

Problem 3

Section 3.5, exercise 22 (Find ds/dt given that s = sint / (1-cost)).

Problem 4

Section 3.5, exercise 60a (derive the rule for the derivative of secx).

Problem 5

A slight modification of Section 3.5, exercise 63. Use R to graph y = cosx between x = -π and x = 2π. On the same graph, also plot y = ( sin(x+h) - sinx )/h for several small values of h (the text suggests some values, of which you may try some or all—but be sure to try both positive and negative values). Each value of h should yield a separate curve as x varies from -π to 2π. What does this exercise demonstrate? (You might find your plot easier to contemplate if you use different colors for different curves, e.g., one color for the cosx curve, and perhaps different colors for different absolute values of h in the other curves.)

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.