Problem Set 8—Graph Sketching and Optimization

Purpose

This lesson reinforces your understanding of how to use derivatives to find maximum and minimum values, and other important points, in the graph of a function, and of how to use the ability to find maxima and minima to solve optimization problems.

Background

This exercise builds on material in sections 4.1 through 4.3 of our textbook, but draws most directly on sections 4.4 (“Concavity and Curve Sketching”) and 4.5 (“Applied Optimization”). We covered (or will cover) this latter material in classes between approximately October 29 and November 5.

Activity

Solve each of the following problems:

Problem 1

Section 4.4, exercise 94 (sketch the graph of a function given information about it, its derivative, and its second derivative at various values of x; see the text for details).

Problem 2

A variation on section 4.4, exercise 88: using the steps of the graphing procedure given in the textbook, determine what range of x values the curve y = (x³ + x - 2) / (x - x²) should be graphed over in order to see all local maxima, minima, asymptotes, increasing and decreasing intervals, and inflection points.

Problem 3

Section 4.5, exercise 8 (find the shortest fence that can enclose 216m² and also split the area in half).

Problem 4

Section 4.5, exercise 60a (show that, when coughing, the human trachea contracts by an amount that maximizes the speed of the expelled air).

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.