Spring 2012 Math 104

March, 2012

Be sure to check back often, because assignments may change!

As you learn Maple, I'll often use Maple syntax for mathematical notation on this page.
Unless otherwise noted, all section and page numbers refer to sections from Calculus: Early Transcendental Functions, Smith and Minton, 3rd Edition.

Due Friday 3/2 at 8:30am

Section 6.6: Improper Integrals (Postponed to Monday 3/5)

• To read: Through the section Improper Integrals with a Discontinuous Integrand (through the middle of page 550).
• Be sure to understand: Definitions 6.6.1 and 6.6.2; what is meant by convergent and divergent integrals.

1. Rephrase the question "Does an improper integral with a discontinuous integrand converge?" as a question about the graph of the integrand.
2. Explain why int( 1x3, x=0..3) is improper.
3. Does int( 1x3, x=0..3) converge or diverge?
4. Does int(1x3, x=-2..3) converge or diverge?

Reminders:

• For those who are working on the exam re-take for extra credit, it is due Monday at 3pm.
• Begin WW 6 and PS 6.
• Continue to work on your project (the mathematics should be complete by Sunday evening). Find times to meet that work for all members of your group. Every one should be contributing equally on the mathematics. Do not figure one person will work on the math and another will do the writing -- for one thing, this is completely counter to the point, and for another, it never results in a good paper.
• I'll give you the antidifferentiation exam toward the end of class Friday

Due Monday 3/5 at 8:30am

Guide to Writing Mathematics
Checklist
Section 6.6: Improper Integrals (Carried over from Friday 3/2)

• To read: Through the section Improper Integrals with a Discontinuous Integrand (through the middle of page 550).

All of the writing guide and the checklist.

• Be sure to understand: Definitions 6.6.1 and 6.6.2; what is meant by convergent and divergent integrals.

1. Rephrase the question "Does an improper integral with a discontinuous integrand converge?" as a question about the graph of the integrand.
2. Explain why int( 1x3, x=0..3) is improper.
3. Does int( 1x3, x=0..3) converge or diverge?
4. Does int(1x3, x=-2..3) converge or diverge?

Reminders:

• Finish the mathematics of the project before class. If you run into difficulties, come to my office hours Monday.
• If you didn't pass the antidifferentiation exam on Friday, try to find a few times this week when you can spare half an hour to try it again.
• For those who are working on the exam re-take for extra credit, it is due Monday at 3pm.

Due Wednesday 3/7 at 8:30am

Section 6.6: Improper Integrals

• To read: In Section 6.6, read up to the subsection on A Comparison Test.
Really read the writing guide -- it will help you figure out the differences between writing for homework and writing to explain a result to a person who is not an instructor. Pay attention to the different aims in writing to share mathematical ideas you've developed (as in your project) versus writing to convince someone you understand ideas they already know (as in homework), and the different phrasing and techniques that result.

Also pay attention to the writing checklist that I use to grade, so you can see how many points each aspect can earn. Paperclip a clean copy to the front of your paper.

1. What are the two ways in which an integral may be improper?
2. Rephrase the question "Does an improper integral with infinite limits of integration converge?" as a question about the graph of the integrand over that infinite interval.

Reminders:

• Bring questions on WW 6 and PS 6 to class.
• Your project response should be written by your group as a whole. Whether you accomplish this by having one person get the rough ideas on paper, then the next begin to polish it, and pass it back and forth repeatedly, or whether write the entire paper together is up to you. What is not okay is having one member of the group do all the writing, just as it was not okay to have one member do all the mathematics. It is also not a good idea to split the writing up, because rarely does such a letter or paper flow well -- people either repeat themselves or leave out details, as well as the resulting style being choppy.

Due Friday 3/9 at 8:30am

Section 6.6: Improper Integrals

• To read: Finish the Section.

Suppose that 0 < f(x) < g(x).

1. If int(g(x), x=1. .infty) diverges, what (if anything) can you conclude about int( f(x), x=1. . infty), and why?
2. If int(f(x), x=1. .infty) converges, what (if anything) can you conclude about int( g(x), x=1. . infty), and why?

Reminder:

• Attach a blank copy of the check-list by paperclip to the front of your project when you turn it in.
• Begin WW 7 and PS 7 (group). Remember: groups should consist of 2 or 3 people; think about working with new people; don't split the problems; switch primary authors; put a star next to the primary author's name.

3/10 through 3/18:

Spring Break!

Due Monday 3/19 at 8:30am

Section 6.6: Improper Integrals

• To read: Re-Read the Section, paying particular attention to the portion we haven't covered yet.

Reminder:

• If you haven't passed the antidifferentiation exam yet, make the time to try it again.

Due Wednesday 3/21 at 8:30am

Section 3.2: Indeterminate Forms and l'Hôpital's Rule
Section 8.1 Sequences of Real Numbers

• To read: All of Section 3.2. Through Example 1.13 in Section 8.1.
• Be sure to understand: In Section 8.1, why neither 00 nor are automatically 1.

1. Does l'Hôpital's Rule apply to lim(x -> infty) x2 ex ? Why or why not?
2. Does l'Hôpital's Rule apply to lim(x -> infty) x2 sin(x) ? Why or why not?
3. Find a symbolic expression for the general term ak of the sequence
{0, 3, 6, 9, 12, 15, . . . }
4. Does the following sequence converge or diverge? (You may assume that the obvious pattern that you see is in fact the pattern the sequence will continue to follow!) Be sure to explain your answer.
{1, 3, 5, 7, 9, 11, 13, . . .}
5. In Section 8.1, Example 1.10, what two methods of demonstrating that the sequence is increasing were used? Which did you prefer, in that particular situation?
6. Give an example of a sequence (other than the one in Example 1.13) that is bounded and monotonic. (Keep it simple. You may find it helpful to think of a function with a horizontal asymptote at infinity.)
7. Give an example of a sequence that converges even though it is not both bounded and monotonic. (Again keep it simple.)

Reminders:

• Exam 2 will be Thursday 3/29
• Bring questions on WW 7 and PS 7 to class.

Due Friday 3/23 at 8:30am

Section 8.2: Infinite Series

• Be sure to understand: The distinction between a sum and an infinite series; that an infinite series is in fact a limit of a sequence; geometric series; the nuances of the kth term test.

1. There are (at least) two sequences associated with every series. What are they?
2. Does the geometric series (14)k converge or diverge? Why?
3. What does the kth Term Test tell you about each of the following series? Explain.
1. sin(k)
2. 1k

Reminders:

• Get an earlystrong start on WW 8. Once again, the entire assignment is on WeBWorK, so that you can get instant feedback before the exam. The questions on the study guide assume that you have already worked through WW 8.

Due Monday 3/26 at 8:30am

Section 8.2: Infinite Series

• Be sure to understand: All

Reminders:

• Monday is the deadline for receiving full credit on the antidifferentiation exam.

Wednesday 3/28 at 8:30am

Questions for Exam 2

To read: Once again, re-read the Honor Code, Wheaton's description of plagiarism, and the portion in the course policies that applies to the Honor Code, paying particular attention to how it all applies to exam situations.

Reminder:

• Get most of your questions answered before Wednesday, in office hours.
• As before, you may have an 8 1/2 x 11 handwritten front-only sheet of notes, and you may begin the exam at 12:30.

Due Friday 3/30 at 8:30am

Section 8.3: The Integral Test and Comparison Tests

• To read: Through Example 3.4.
• Be sure to understand: Examples 3.1, 3.3.

1. Explain in a couple of sentences (in your own words, of course) why the integral test makes sense. That is, explain the big ideas behind the integral test.

Reminders:

• Begin WW 9 and PS 9. PS 9 is a group assignment - remember the usual guidelines: variety in group make-up; 2-3 people; don't divvy; switch and label primary authors.

Here ends the reading for March
Next, go to the reading for April and May!

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 1306
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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