Problem Set 2—The Natural Logarithm

Purpose

This problem set consolidates your understanding of the natural logarithm and some of its uses with derivatives and antiderivatives.

Background

This exercise is based on material in section 7.2 of our textbook. We discussed (or will discuss) this material in class on January 22 and 27.

Activity

Solve each of the following problems:

Problem 1

Use algebraic properties of logarithms to simply

Problem 2

Show that if x < y, then the natural logarithm function is increasing at a slower rate at y than at x. (Less formally, this question asks you to show that the function lnx grows at ever slower rates as x gets larger; this in turn means that any real-world phenomenon—and there are some—with a logarithmic “cost” becomes a better and better “deal” as it gets bigger.)

Problem 3

Exercise 34 in Section 7.2 of our textbook (differentiate ln√( (x+1)⁵ / (x+2)²⁰ )).

Problem 4

Exercise 40 in Section 7.2 of our textbook (integrate 8rdr / (4r² - 5)).

Problem 5

Exercise 46 in Section 7.2 of our textbook (evaluate a definite integral of dx / ( 2x √(lnx) )).

Problem 6

Exercise 72 in Section 7.2 of our textbook (find the area between tanx and the x axis given -π/4 ≤ x ≤ π/3).

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.