Fall 2005 Math 104

October, 2005

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 10/3 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 10/3

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 10/5 at 9am

Section 8.1 Integration by Parts
Section 9.1 Taylor Polynomials

• To read: re-read Section 8.1 for Wednesday. For Thursday, read Section 9.1 (you may skip the section Trigonometric polynomials: Another nice family).

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:

• 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.

Due Friday 10/7 at 9am

Section 9.1 Taylor Polynomials

• To read: Re-read this section, combining what we discussed Thursday with the book's discussion to resolve as many of your previous difficulties as possible.
• 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 10/7

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 10/10 at 9am

Fall Break!

Due Wednesday 10/12 at 9am

Section 9.1 Taylor Polynomials

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

Reminder:

• Come to my office hours, tutoring hours!
• Bring questions on PS 6.
• 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 10/14 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 10/14

What is the point of Theorem 2? Explain in your own words.
Reminders:
• I'll give you the antidifferentiation exam toward the end of class.
• Begin PS 7.

Due Monday 10/17 at 9am

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

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

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 know, 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 10/19 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 10/19

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 10/27
• Bring questions on PS 7 to class.

Due Friday 10/21 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 10/21

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:

• Begin PS 8

Due Monday 10/24 at 9am

Section 10.2: Detecting Convergence, Estimating Limits

• Be sure to understand: Example 5.

E-mail Subject Line: Math 104 Your Name 10/24

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)?

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 10/28 at 9am

Section 10.2: Detecting Convergence, Estimating Limits

Reminders:

• Begin PS 9

Due Monday 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.
• 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, . . .

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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