Fall 2007 Math 104

October, 2007

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

All section and page numbers refer to sections from Ostebee/Zorn, Volume 2, Edition 2.

Due Monday 10/1 at 9am

Section 8.1 Integration by Parts(continued)

• Be sure to understand: Theorem 1. Be warned that Examples 8 and 9 can be a little slippery.

E-mail Subject Line: Math 104 Your Name 10/1

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 * sin(x), x). Use parts to find an antiderivative for x * sin(x).

Due Wednesday 10/3 at 9am

Section 8.1 Integration by Parts
Guide to Writing Mathematics

E-mail Subject Line: Math 104 Your Name 10/3

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:

• 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.
• The math for project 1 should be finished by Wednesday evening at the latest, so that you can spend plenty of time working on writing your letter to Jill.

Due Friday 10/5 at 9am (this is really the reading for next Wednesday's class)

Section 9.1 Taylor Polynomials

• To read: Read the section (you may skip the section 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 10/5

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

Reminder:

• Begin PS 6.
• I urge you to bring me a draft of your project for some suggestions.

Due Monday 10/08 at 9am

Fall Break!

Due Wednesday 10/10 at 9am

Section 9.1 Taylor Polynomials

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

(If you didn't do the reading questions for last Friday, do them now.)

Reminder:

• Come to my office hours, tutoring hours!
• Bring questions on PS 6 (which won't be due until the beginning of class Friday).
• Everyone in your group should be proof-reading your project and making constructive comments. Remember that a blank copy of the check-list should be attached to your project when you turn it in.

Due Friday 10/12 at 9am

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 10/12

What is the point of Theorem 2? Explain in your own words.
Reminders:
• 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 today -- it will be due next Thursday, as usual.

Due Monday 10/15 at 9am

Section 9.2 Taylor's Theorem: Accuracy Guarantees for Taylor Polynomials

E-mail Subject Line: Math 104 Your Name 10/15

Let f(x)=sqrt(x).
1. Find P3(x) for f at the base point x0=64.
2. What can you say about the error committed by using P3(x) as an approx for sqrt(x) on the interval [50,80]?
Reminders:
• 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 10/17 at 9am

Section 10.1: Improper Integrals: Ideas and Definitions

• Be sure to understand: The section Convergence and divergence: Formal definitions and Examples 1 - 5.

E-mail Subject Line: Math 104 Your Name 10/17

1. What are the two ways in which an integral may be improper?
2. Explain why int( 1/x2, x=1..infinity) is improper. Does the integral converge or diverge?
3. Explain why int( 1/x2, x=0..1) is improper. Does the integral converge or diverge?

Reminders:

• Exam 2 is Thursday 10/25
• Bring questions on PS 7 to class.

Due Friday 10/19 at 9am

Section 10.1: Improper Integrals: Ideas and Definitions

• To read: Re-read this section; fit it all together and make sense of the subtleties.

Reminders:

• Begin PS 8

Due Monday 10/22 at 9am

Section 10.2: Detecting Convergence, Estimating Limits

• Be sure to understand: The statements of Theorems 1 and 2 and Example 4.

E-mail Subject Line: Math 104 Your Name 10/22

1. If 0 < f(x) < g(x) and int( g(x), x=1. . infty) converges, will int(f(x), x=1. .infty) converge or diverge? Why?
2. There are two types of errors that arise in Example 4 for approximating int( 1/(x5 +1), x=1..infty). What are the two types?

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 10/26 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 10/26 at 9am

Section 10.2: Detecting Convergence, Estimating Limits

• Be sure to understand: Example 5.

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

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

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

Due Monday 10/29 at 9am

Section 10.2: Detecting Convergence, Estimating Limits

Reminders:

• Monday is the deadline to receive 90% on the antidifferentiation exam. If you haven't passed it yet, don't let this opportunity go by without the attempt -- several, if necessary!
• Begin PS 9

Due Wednesday 10/31 at 9am

Section 4.2 More on Limits: Limits Involving Infinity and l'Hopital's Rule
Section 11.1 Sequences and Their Limits

• To read: The section l'Hopital's rule: finding limits by differentiation that begins on page S-19 and all of Section 11.1.
For Thursday, read Section 11.2 through Example 4. This can be tough going, but really work at it.
• Be sure to understand: The statement of l'Hopital's rule and the section Terminology and basic examples in Section 11.1.

E-mail Subject Line: Math 104 Your Name 10/31

1. Does l'Hopital's Rule apply to lim(x -> infty) x2 / ex ? Why or why not?
2. Does l'Hopital's Rule apply to lim(x -> infty) x2 / sin(x) ? Why or why not?
3. Does the following sequence converge or diverge? Be sure to explain your answer.
1, 3, 5, 7, 9, 11, 13, . . .
4. Find a symbolic expression for the general term ak of the sequence
1, 2, 4, 8, 16, 32, . . .

Reminders:

• Take advantage of tutoring hours
• Bring questions on 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 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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