Problem Set 7—Extreme Values

Purpose

This lesson reinforces your understanding of how to use derivatives to find maxima and minima of functions.

Background

This exercise is mainly based on material in sections 4.1 through 4.3 (“Extreme Values of Functions,” “The Mean Value Theorem,” and “Monotonic Functions and the First Derivative Test” respectively) of our textbook. We covered (or will cover) this material in classes between approximately October 20 and October 28.

(Note that the extra credit part of the exercise is based on earlier material, and is not connected to extreme values or the Mean Value Theorem. It’s just a neat thing I realized you can do with the material we’ve been studying.)

Activity

Solve each of the following problems:

Problem 1

Section 4.1, exercise 50 (find extreme values of y = x³ - 2x + 4).

Problem 2

Section 4.1, exercise 76 (find the peak value of an alternating current; see textbook for details).

Problem 3

Section 4.2, exercise 4 (find the value(s) for c that satisfy the Mean Value Theorem for y = √(x-1)).

Problem 4

Section 4.2, exercise 50 (speeding trucker). Explain your answer in terms of the Mean Value Theorem.

Problem 5

Section 4.3, exercise 20 (find increasing and decreasing intervals and extreme values for g(t) = -3t² + 9t + 5).

Extra Credit

The arctangent, or inverse tangent, function, written tan^-1x, is the function that yields an angle y such that tany = x. By convention, y is always between -π/2 and π/2.

For up to 2 points extra credit, show that d/dx (tan^-1x) = 1/(1+x²). Use only differentiation rules and methods that we have studied so far in this course (and standard trigonometric identities and rules of algebra). Hint: if y = tan^-1x, then tany = x. Begin with this latter equation, and derive an equation for dy/dx.

For one more point extra credit, use the same general tactic you used to differentiate the inverse tangent function to find a derivative of one other inverse trigonometric function of your choosing.

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.