Problem Set 8 — Applications of Derivatives 2

Purpose

This problem set develops your ability to apply derivatives to solving extreme value problems and related problems involving the Mean Value Theorem or the shapes of graphs. It addresses the following learning outcomes:

• Compute limits and derivatives of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions (outcome 1)
• Use the derivative of a function to determine the properties of the graph of the function and use the graph of a function to estimate its derivative (outcome 4)
• Solve problems in a range of mathematical applications using the derivative or the integral (outcome 5).

Background

This problem set draws mainly on sections 4.3 through 4.5 of our textbook. We discussed this material in classes between October 28 and November 6.

Activity

Solve the following problems:

Question 1

(Based on exercises 142 and 143 in section 4.3 of our textbook.)

The production of gold between 1848 and 1888 in the California gold rush can be modeled by

where \(t\) is the number of years since the start of the gold rush (so \(0 \le t \le 40\)) and \(G(t)\) is the amount of gold produced, in millions of ounces. See the textbook for a graph of gold production versus time.

Part A

According to the model, when did the absolute maximum production of gold occur during the gold rush?

Part B

According to the model, when did the absolute minimum production occur?

Question 2

(Based on exercise 190 in section 4.4 of our textbook.)

Part A

At 10:17 a.m., you are traveling 55 mph when you pass a police car that is stopped on the freeway. You pass a second stopped police car at 10:53 a.m., when you are also traveling 55 mph. The second police car is located 39 miles from the first one. If the speed limit is 60 mph, can the police cite you for speeding? Why or why not?

Part B

After being cited for speeding in Part A, you appeal the ticket on the grounds that your car is the new model with the high tech teleporter drive, which allows the car to move instantaneously from one place to another. Assuming the car really does have such a device, should the court dismiss your ticket? Why or why not?

Question 3

Sketch graphs of functions \(f(x)\) and \(g(x)\) that have the following features over the interval \([-3,3]\).

Part A

(Exercise 216 in section 4.5 of our textbook.)

• \(f(x) > 0\) everywhere in the interval
• \(f^\prime(x) > 0\) for \(x < 0\)
• \(f^\prime(x) = 0\) for \(0 < x < 1\)
• \(f^\prime(x) > 0\) for \(x > 1\)

Part B

(Exercise 218 in section 4.5 of our textbook.)

• \(g\) has a local maximum at \(x = 0\)
• \(g\) has local minima at \(x = -2\) and \(x = 2\)
• \(g^{\prime\prime}(x) > 0\) for \(x < -1\)
• \(g^{\prime\prime}(x) < 0\) for \(-1 < x < 1\)
• \(g^{\prime\prime}(x) > 0\) for \(x > 1\)

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.