Math 120 Spring 98

Quiz 7  Basic Calculus 2

Name:

Date:  Friday April 3, 1998
Time Begun:  Ron Buckmire

Time Ended:

Topic covered: Determining Convergence/Divergence of Improper Integrals by Comparison

The idea behind this quiz is to give you more practice determining whether a particular improper integral will converge or diverge without evaluating it, but by comparing it to an improper integral you already know something about.

Instructions:

1. Once you open the quiz, you have 90 minutes to complete it.

2. You may use the book or any of your class notes, and you may use a calculator. You must work alone.

3. If you use your own paper, please staple it to the quiz before coming to class. If you don’t have a stapler, buy one.

4. After completing the quiz, sign the pledge below stating on your honor that you have adhered to these rules.

5. Relax and enjoy...

6. This quiz is due on Monday, April 6, in class. NO LATE QUIZZES WILL BE ACCEPTED.

Pledge: I, , pledge my honor as a human being and Occidental student, that I have followed all the rules above to the letter and in spirit.

Math 120 Spring 1998 Quiz 7

SHOW ALL YOUR WORK

Consider the function f(z) = \frac{1}{\sqrt{z} + z} = \frac{1}{z^{1/2} + z} . We want to try and determine whether

I = ∫_{1}^{∞} \frac{1}{\sqrt{z} + z} dz and J = ∫_{0}^{1} \frac{1}{\sqrt{z} + z} dz converge or diverge.

(a) (4 points) Use the rules developed in class to say whether each of the following integrals converges or diverges:

∫_{0}^{1} \frac{dz}{2\sqrt{z}}  ∫_{1}^{∞} \frac{dz}{2\sqrt{z}}  ∫_{0}^{1} \frac{dz}{z}  ∫_{1}^{∞} \frac{dz}{z}

(b) (1 point) For z > 1 is \frac{1}{\sqrt{z} + z} > \frac{1}{z} or \frac{1}{\sqrt{z} + z} < \frac{1}{z}?

For z > 1 is \frac{1}{\sqrt{z} + z} > \frac{1}{\sqrt{z}} or \frac{1}{\sqrt{z} + z} < \frac{1}{\sqrt{z}}?

(c) (2 points) Does I = ∫_{1}^{∞} \frac{dz}{\sqrt{z} + z} converge or diverge? Support your answer by explaining

how the inequalities in part (b) and your answers to part (a) allow you to determine whether I converges or diverges without actually evaluating it. If you like, you can use a graph to support your explanation.

(d) (1 point) For 0 < z < 1 is \frac{1}{\sqrt{z} + z} > \frac{1}{z} or \frac{1}{\sqrt{z} + z} < \frac{1}{z}?

For 0 < z < 1 is \frac{1}{\sqrt{z} + z} > \frac{1}{\sqrt{z} + \sqrt{z}} or \frac{1}{\sqrt{z} + z} < \frac{1}{\sqrt{z} + \sqrt{z}}?

(e) (2 points) Does J = ∫_{0}^{1} \frac{dz}{\sqrt{z} + z} converge or diverge? Support your answer by explaining

how the inequalities in part (d) and your answers to part (a) allow you to determine whether I converges or diverges without actually evaluating it. If you like, you can use a graph to support your explanation.