Problem Set 4 — Derivatives and Such

Purpose

This problem set develops your understanding of derivatives and several related ideas. It addresses the following learning outcomes:

• Compute limits and derivatives of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions (outcome 1)
• Compute definite and indefinite integrals of algebraic, trigonometric, inverse trigonometric, exponential, logarithmic, and piecewise defined functions (outcome 2)
• Determine the continuity and differentiability of a function at a point and on a set (outcome 3)
• Use appropriate modern technology to explore calculus concepts (outcome 7).

Background

This problem set uses ideas from sections 3.1 through 3.3 of our textbook. We discussed this material in classes between September 21 and October 1. This problem set also draws on material we discussed on some of those dates but that is not in chapter 3 of the book, such as antiderivatives or derivatives in Mathematica.

Activity

Solve the following problems:

Question 1

Use the limit definition of the derivative to find the derivative of

After finding the derivative using the limit definition, check your answer using Mathematica.

Question 2

Use differentiation rules to find the derivatives of the following functions. Then confirm your answers by finding the same derivatives with Mathematica.

1. \(f(t) = \frac{t^3}{3} + 2t^2 - t + 5\)
2. \(s(t) = \frac{t^2 - 1}{6t}\)
3. \(f(z) = (z^3 + 3z^2)(z - 1)\)

Question 3

Part A

Use the antiderivative power rule and antiderivative constant rule that we deduced in class on September 25 (see “Antiderivatives” in the class notes from that day) to find antiderivatives of

1. \(3x^2\)
2. \(4x\)

Part B

(This part exceeds the level of mastery I expect for this course because it asks you to derive an antiderivative rule for yourself.)

Is there a “sum” antiderivative rule, i.e., a rule that tells you what the antiderivative of a sum is, i.e., an antiderivative of an expression of the form \(f(x) + g(x)\)? If so, state it in English and/or as an equation, and give an English justification for it. Then use it and the results from Part A to find the antiderivative of \(3x^2 + 4x\).

Question 4

Consider the function

The function \(f(x)\) is the derivative of some function, call it \(G(x)\), although we don’t know antiderivative rules that would let us find \(G(x)\) yet. However, you can still learn something about \(G(x)\). For example…

Part A

Where is \(G(x)\) differentiable?

Part B

Where is \(G(x)\) continuous?

Part C

(This part goes beyond what I expect for mastering calculus concepts, because it gets into some of the subtleties of logic.)

What, if anything, can you tell about whether \(G(x)\) is continuous at \(x = 3\)?

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.