Problem Set 2—Derivatives and Limits

Purpose

This lesson develops your understanding of the limit definition of the derivative, and reinforces your understanding of limits.

Background

This exercise is mainly based on material in section 3.2 (“The Derivative as a Function”) of our textbook. We covered (or will cover) this material in classes between approximately September 11 and September 17. This exercise also draws, either directly or indirectly, on material from chapter 2 (“Limits and Continuity”) of the textbook.

One question asks you to use R to graph a function. Examples of drawing graphs in R have been presented in lectures, for example on September 2, September 3, or September 9. I will be happy to discuss and demonstrate this further if you want, so feel free to ask.

Activity

Solve each of the following problems:

Section 3.2, Exercise 10

(Find dv/dt given that v = t - 1/t)

Section 3.2, Exercise 14

(Differentiate k(x) = 1/(2+x) and use the derivative to find the slope of k’s graph at x = 2)

Section 3.2, Exercise 28

(Match a graph of an approximately cubic function to a graph of its derivative.)

Section 3.2, Exercise 8

(Find dr/ds given that r = s³ - 2s² + 3)

Section 2.6, Exercise 4b

(Find lim_x→-∞(π - 2/x². Test your answer by using R to graph the function over a suitable interval of x values.)

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.