Fall 2009 Math 104

October, 2009

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 10/2 at 9am

Section 6.1: Review Integration
Section 6.2: Integration by Parts

• To read: Skim through section 6.1 (it's just examples), read all of Section 6.2, and read the writing guide all the way through.
• Be sure to understand: In Section 6.2, pay particular attention to where the rule for integration by parts comes from, and to the basics of how it's used. The examples illustrate just about every possible variant, but for now pay particular attention to Examples 1-3.

E-mail Subject Line: Math 104 Your Name 10/2

1. Integration by parts attempts to undo one of the techniques of differentiation. Which one is it?
2. Pick values for u and dv in the integral int( x * exp(x), x). Use parts to find an antiderivative for x * exp(x).

Reminders:

• Begin PS 5.
• PS 5 is your second group assignment. I encourage (but do not require) that you switch group partners- the more people you've worked with, the more comfortable you will feel in the class. Also, just because someone is a good friend does not mean you have the same sort of schedule or the same approach to homework.
• Remember not to divvy up the problems amongst you--your group should discuss every problem. Whether you choose to each begin each problem on your own before meeting, or to work through them from scratch as a group will depend on how you feel you learn best.
• Meanwhile, of course, continue to work on your project (yes, it's a busy week!) Make every effort to find times 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.
• Aim to have finished the mathematics of the project no later than Sunday evening, as clearly formulating your response (in the form of a letter, remember) will take several days!

Due Monday 10/5 at 9am

Section 6.2: Integration by Parts
Guide to Writing Mathematics
Checklist

• To read: Re-read Section 6.2, this time paying particular attention to Examples 2.4 and 2.5. Be sure to read the marginal remarks! Also, read through all of Section 6.5.

And finally, read the writing guide -- this will help you figure out the differences between writing for homework and writing in response to a different situation. 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.

I have also linked to the writing checklist, rather than printing them out, because some groups will only need one to include with the finished product, while other groups like to have extra copies to make notes on as they proof-read rough drafts. This way, your group can have however many you need.

E-mail Subject Line: Math 104 Your Name 10/5

Each integral can be evaluated using u-substitution or integration by parts. Which technique would you use in each case? You do not need to evaluate the integral, but explain your choice.

1. int( x*cos(x), x)
2. int(x*cos(x2),x)

Reminders:

• 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.
• I would be happy to skim through a draft of your letter and give you suggestion, but I will need to have a draft no later than Wednesday before noon (perhaps earlier, depending on how my week's schedule is shaping up).

Due Wednesday 10/7 at 9am

Section 6.5: Integration Tables and Computer Algebra Systems
Section 6.6: Improper Integrals

• To read: Through the section Improper Integrals with a Discontinuous Integrand (through the middle of page 550).
• Be sure to understand: Definitions 6.1 and 6.2, and what is meant by convergent and divergent integrals.

E-mail Subject Line: Math 104 Your Name 10/7

1. Graphically, how would you interpret the question of whether or not an improper integral with a discontinuous integrand converges?
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:

• Continue taking advantage of the tutoring hours and my office hours, if you've been before; if you've never been, consider trying it for the first time.
• Bring unresolved problems on PS 5 to class.
• Once again, one member of your group for your homework will be the primary author for the week- put a star next to that person's name. Unless all members of your group were primary authors on the first problem set, the primary author should be someone who was not primary author before.

Due Friday 10/9 at 9am

Section 6.6: Improper Integrals

• To read: Read up to the subsection on A Comparison Test.

E-mail Subject Line: Math 104 Your Name 10/9

1. What are the two ways in which an integral may be improper?
2. Graphically, how would you interpret the question of whether or not an improper integral with infinite limits of integration converges?

Reminders:

• Remember that a blank copy of the check-list should be attached to the front of your project by paper clip when you turn it in.
• Begin PS 6.

Due Monday 10/12 at 9am

Fall Break!

Due Wednesday 10/14 at 9am

Section 6.6: Improper Integrals?

• To read: Finish the Section.

E-mail Subject Line: Math 104 Your Name 10/14

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?

Reminders:
• Come to my office hours, tutoring hours!
• PS 6 will be due at the beginning of class Friday. (PS 7 will be due next Thursday as usual).
• Bring questions on PS 6 to class on Thursday.

Due Friday 10/16 at 9am

(From Ostebee and Zorn handout) Section 9.1 Taylor Polynomials

• To read: Read the section (you may skip the subsection Trigonometric polynomials: Another nice family).
• Be sure to understand: The statement of Theorem 1, Example 7, and the definition of the Taylor polynomial.

E-mail Subject Line: Math 104 Your Name 10/16

Explain the basic idea of the Taylor polynomial for a function f(x) at x=x0 in your own words.

Reminder:

• PS 6 is due at the beginning of class Friday.
• I'll give you the antidifferentiation exam toward the end of class.
• Begin PS 7. This is another group problem set; just to remind you, groups should consist of 2 or 3 people; think about working with different people; don't divvy up the problems; switch primary authors; put a star next to the primary author's name.

Due Monday 10/19 at 9am

(From Ostebee and Zorn handout) Section 9.1 Taylor Polynomials

• To read: Re-read this section yet again, trying to fit everything together.

Reminder:

• Mid-semester reminder -- office hours really are an important part of this course. If you haven't come in for questions yet, make a point of stopping in this week, whether it's to address a problem you wish you'd dealt with weeks ago, something that's confusing you now, a subtle point you'd like resolved, or to discuss how what we're learning connects to something you've learned in another class.

Due Wednesday 10/21 at 9am

(From Ostebee and Zorn handout) Section 9.2: Taylor's Theorem: Accuracy Guarantees for Taylor Polynomials

• To read: All, but you can skip the section Proving Taylor's theorem.
• Be sure to understand: The statement of Theorem 2 and Examples 2 and 3.

E-mail Subject Line: Math 104 Your Name 10/21

1. What is the point of Theorem 2? Explain in your own words.

Reminders:

• Exam 2 is Thursday 10/29
• Bring questions on PS 7 to class.

Due Friday 10/23 at 9am

(From Ostebee and Zorn handout)Section 9.2 Taylor's Theorem: Accuracy Guarantees for Taylor Polynomials
Section 8.1 Sequences of Real Numbers

• To read: Re-read Section 9.2 (from the hand-out), and read through bottom of page 615 in Section 8.1 (from your usual text)

E-mail Subject Line: Math 104 Your Name 10/23

1. Let f(x)=sqrt(x).
(a)Find P3(x) for f at the base point x0=64.
(b) What can you say about the error committed by using P3(x) as an approximation for sqrt(x) on the interval [50,80]?
2. Find a symbolic expression for the general term ak of the sequence
{0, 3, 6, 9, 12, 15, . . . }
3. 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, . . .}
Reminders:
• Begin PS 8

Due Monday 10/26 at 9am

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

• To read: All of Section 3.2, then finish Section 8.1.
• Be sure to understand:Why neither 00 nor are automatically 1.

E-mail Subject Line: Math 104 Your Name 10/26

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. 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?

Reminders:

• Monday is the deadline for receiving full credit on the antidifferentiation exam. You really need antidifferentiation to come easily to you for the exam Thursday!

Wednesday 10/28 at 9am

Questions for Exam 2

Reminder:

• Don't wait to bring all your questions your class on Wednesday -- we might not have time to get through them all. Bring some to my office hours and/or the tutoring hours.
• PS 8 will not be turned in, but will be covered on this exam.
• 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 10/30 at 9am

Section 8.2 Infinite Series

• To read: Through Example 2.5
• Be sure to understand: The distinction between a sum and an infinite series, and that an infinite series is in fact a limit of a sequence; geometric series; the nuances of the kth term test.

E-mail Subject Line: Math 104 Your Name 10/30

1. There are (at least) two sequences associated with every series. What are they?
2. Does the geometric series sum((1/4)k,k=0..infinity) converge or diverge? Why?
Reminders:
• Begin PS 9

Here ends the reading for October
Next, go to the reading for November!

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 101A
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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