Problem Set 4—Transcendental Function Miscellany

Purpose

This problem set reinforces your understanding of the inverse trigonometric and hyperbolic functions. It also develops your ability to use L’Hôpital’s rule to find limits.

Background

This exercise is based on material on inverse trigonometric functions in section 7.6 of our textbook, the basic definitions of the hyperbolic functions at the beginning of section 7.7, and on L’Hôpital’s rule in section 7.5. We discussed, or will discuss, this material in class on February 3 and 4.

Activity

Solve each of the following problems:

Problem 1

A boat is floating 10 miles west of a straight north-south coastline. At what angle from due north should the boat sail in order to reach a dock on the coastline 5 miles north of the boat’s position?

Problem 2

Exercise 2 in section 7.7 of our textbook (given that sinhx = 4/3, find the values of the other 5 hyperbolic functions).

Problem 3

Exercise 34 in section 7.6 of our textbook (differentiate tan^-1(lnx)).

Problem 4

Exercise 54 in section 7.6 of our textbook (evaluate a certain definite integral of 1 / (y√(9y²-1))).

Problem 5

Exercise 68 in section 7.6 of our textbook (find the indefinite integral of 1 / √(2x-x²)).

Problem 6

In class I claimed that compared to e^x, the function f(x) = x “might as well be zero.” There is a very specific sense in which this statement is literally true: as x approaches infinity, the limit of the ratio x / e^x approaches 0. Show that this claim about the limit of the ratio really is true. Then show that the same is true of the limits of the ratios x² / e^x and x³ / e^x. Is the general statement true of every positive integer power of x? Why or why not?

Problem 7

Analogous to problem 6, show that as x approaches infinity, the ratio lnx / x also approaches 0.

(In colloquial terms, problems 6 and 7 show that as x gets very large, any positive power of x “grows” towards being 0 times as big as e^x, while lnx grows towards being 0 times as big as x.)

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.