Problem Set 1—Functions and Inverse Functions

Purpose

This problem set consolidates your understanding of one-to-one functions and their inverses, and of how to find the derivative of a function’s inverse.

Background

This exercise is based on material in section 7.1 of our textbook. We discussed this material in class on January 21.

Activity

Solve each of the following problems:

Problem 1

Consider the function f(x) = x³ - 1.

1. Show that f is one-to-one
2. Find f^-1
3. Find the derivative of f^-1, using Theorem 1 from Chapter 7 of our textbook
4. Directly differentiate the inverse function you found in step 2 above, and verify that you get a function equivalent to the one found in step 3.

Problem 2

Exercise 44 in Section 7.1 of our textbook (find the slope of g^-1 at the origin, given that g(x) has slope 2 at the origin).

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 reading through it will help me know what to focus on in the rest of the meeting.

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.

My basic expectation in grading this exercise is that your solution will show that you understand how to solve each problem, although there may be arithmetic or copying mistakes, inefficient solution methods, incorrect or irrelevant statements incidental to the solution, or similar minor mistakes. If you understand how to solve all the problems and have no minor errors, I will consider the solution to be in between “what I expect” and “surprisingly beyond expectations.” I will consider solutions to be 3/4, 1/2, 1/4, or none of what I expect according to what (rough) fraction of the problems your solution shows understanding of, although I will raise grades slightly if it is clear by the end of your grading meeting that you have come to understand things you didn’t understand when you arrived at the meeting.