Problem Set 9 — Applications of Derivatives 3

Purpose

This problem set develops your understanding of applications of derivatives, including extreme values and L'Hôpital's Rule. It also exercises limits at infinity, and applications of Mathematica to solving mathematical problems. It addresses the following learning outcomes:

• Compute limits and derivatives of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions (outcome 1)
• Solve problems in a range of mathematical applications using the derivative or the integral (outcome 5)
• Use appropriate modern technology to explore calculus concepts (outcome 7).

Background

This problem set draws mainly on sections 4.3 (extreme values), 4.6 (limits at infinity), and 4.8 (L'Hôpital's Rule) of our textbook. We talked about those sections in classes between October 28 and November 2, between November 9 and 11, and on November 13, respectively.

This problem set also asks you to use Mathematica in a fairly open-ended way to solve an extreme value problem. We talked about Mathematica as a tool for extreme value problems between October 29 and November 2; the class notes from those days demonstrate the Mathematica features you’ll need for this problem set and discuss approaching them from a perspective of a set of tools you can combine as you wish in order to solve a problem rather than as isolated commands you use when directed to do so in order to answer a single question.

Activity

Solve the following problems:

Question 1

(Loosely based on question 138 in section 4.3 of our textbook.)

Use Mathematica to find the absolute minimum and maximum value of the function

on the interval \(-6 \le x \le 0\).

Question 2

(Based on exercise 264 in section 4.6 of our textbook.)

Find

\[\lim_{x \to \infty} \frac{3x^3 - 2x}{x^2+2x+8}\]

Question 3

(This question exceeds the mastery I expect for the “limits and derivatives” outcome, because it asks you to prove a result about limits, which is beyond the normal scope of this course.)

Use the formal definition of a limit at infinity to prove that

Question 4

(Exercise 370 in section 4.8 of our textbook.)

Evaluate the limit

\[\lim_{x \to \frac{\pi}{2}} \frac{\cos x}{\frac{\pi}{2} - x}\]

Follow-Up

I will grade this exercise during one of your weekly individual meetings with me. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.