Spring 2006, Math 104

January and February, 2006

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

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

Due Friday 1/27 at 9am

• Pay attention to: all of it. Any questions? Please do ask me!

Section 5.1: Areas and Integrals
Section 5.2: The Area Function
Section 5.3: The Fundamental Theorem of Calculus

• To read: All of sections 5.1, 5.2, and 5.3. The main ideas in this section should be review. If you did not take Calc 1 at Wheaton, however, the presentation of the ideas may be quite different from what you're used to -- use this review as an opportunity to get acquainted with the authors' style. Adjusting to a new book can be difficult -- your immediate reaction may be negative, just because it's not what you're used to. Keep an open mind, and remember that one would expect Calculus in college to be different from Calc in high school.

If the main ideas in these sections are not review to you, please e-mail me or come speak to me immediately.

If the sections are largely review, but you have one or more questions, please come to my office hours. Office hours are a huge part of Calculus at Wheaton; I expect nearly every one will come see me at least every couple weeks.

• Be sure to understand: The statements of both forms of the Fundamental Theorem of Calculus.

E-mail Subject Line: Math 104 Your Name 1/27

1. Why do you think it makes sense to call
int(f(x),x=a..b)/(b-a)
the average value?

Notes:

• The above is written in Maple notation -- it's as good a way as any to write mathematical ideas without symbols, with the added benefit that it gets you used to some Maple notation. The above says the integral of f(x), from x=a to x=b, all divided by b-a.
• The text doesn't specifically address this question; the reason I'm asking this is because this is exactly the sort of question you should be learning to ask yourself (and attempting to answer) when you read.
2. Find the signed area between x^5 and the x-axis from x=1 to x=2.
3. If f(x) is continuous, must it have an antiderivative? If your answer is yes, does that mean there must be a nice formula (or any formula at all) for the antiderivative?

Reminders:

• In these assignments, you should always briefly explain how you arrived at your answers.
• Begin Problem Set 1, listed at the bottom of this course's web page. . The problem sets due each Tuesday reflect an entire week's worth of work, and you should be working on them throughout the week.

Due Monday 1/30 at 9am

Problem Set Guidelines
Section 3.4 Inverse Functions and Their Derivatives (Appendix S in your book, I believe)

• To read: You can skim the beginning of Section 3.4 (Appendix S), but carefully read the section Working with inverse trigonometric functions beginning on page S-8.
• Be sure to understand: The various uses of the words "identity" and "inverse". The derivatives of the inverse trig functions.

E-mail Subject Line: Math 104 Your Name 1/30

1. What is the domain of the function arccos(x)? Why is this the domain?
2. Explain how we can tell lines which are neither horizontal nor vertical have inverses.
3. Why do you think we are studying the inverse trig functions now?
4. Find one antiderivative of 1 / (1+x2).

Reminder:

• Come to lab at 1 pm Tuesday, in room A102.
• PS 1 (individual) is due Tuesday at 1pm. Make sure you read and follow the guidelines referred to above.
• If you have any last remaining questions on the problem set after coming to my office hours, bring them to class Monday.
• Make sure you've read all the stuff I handed out in class and all the material on the course web page.

Due Wednesday 2/1 at 9am

More on Antidifferentiation & Inverse Trig Functions

• To read: Re-read any sections we've read so far that have caused you difficulty. If you're feeling not-quite-ready for this class, this is your day to catch up.
• Be sure to understand: Everything

Reminder:

• Begin working on PS 2. This is a group assignment -- that means you must work with one other person (or in some cases, two other people). Start introducing yourself to people in the class, and try to find someone you think you can work well with.

These groups are not permanent -- you're welcome to work with different people different weeks.

Due Friday 2/3 at 9am

Section 5.4 Finding Antiderivatives; The Method of Substitution

• To read: All. Remember to pay attention to tables and graphs; in particular, remind yourself why all of the basic antiderivatives in the table on page 333 are true, and begin to know them all.
• Be sure to understand: Examples 6, 7, 9, and 13 illustrate specific important points, but you should be paying attention to and doing your best to understand all the examples.

E-mail Subject Line: Math 104 Your Name 2/3

1. Explain the fundamental difference between a definite integral and an indefinite integral. Please go deeper than saying one has limits of integration and one doesn't. The first is a real number -- why? what does it represent? Then think similarly about indefinite integrals.
2. Substitution attempts to undo one of the techniques of differentiation. Which one is it?
3. What are the three steps in the process of substitution?

Reminder:

• Remember the problem sets due each Tuesday reflect the whole week's worth of work, and should be worked on throughout the week.
• As I mentioned before, office hours are an important part of Calculus, so please don't hesitate to come to them! I only set aside 10-15 minutes in class on Mondays to answer questions on the week's problem set, so don't save all your questions for that time--come to office hours, and get more personal attention.

Due Monday 2/6 at 9am

Section 5.6 Approximating Sums; The Integral as a Limit

E-mail Subject Line: Math 104 Your Name 2/6

1. Give an example of a partition of the interval [0,3].
2. When approximating an integral, which would you expect to be more accurate, L10 or L100? Why?
3. Explain the idea of a Riemann sum in your own words.

Reminders:

• The Kollett Center for Cooperative Learning (KCCL) has office hours held by student tutors. As you can see, each of the Calculus classes has some hours that are listed as being priority for that class, but you're welcome to go to any of the Calculus hours. If you go to hours whose first priority is one of the Calc 1 or multivariable classes, you simply need to wait until those students have their questions answered.
• After coming to office hours and/or going to the KCCL for the tutoring hours, bring any still-unresolved questions on PS 2 to class on Monday.
• (Last reminder of the semester:) Come to lab Tuesday at 1pm, bring your completed problem set with you.
• Your group will turn in one joint version of PS 2; the recopying should all be done by one person, the "primary author". Make a note on it who the "primary author" was this time, and switch next time.
• Be sure that each member of your group has a photocopy of the problem set you turn in, both for studying purposes and for your records.

Due Wednesday 2/8 at 9am

Problem Set Guidelines
Section 6.1 Approximating Integrals Numerically

• Be sure to understand: The statements of Theorem 1 and Theorem 2

E-mail Subject Line: Math 104 Your Name 2/8

1. Why would we ever want to approximate an integral?
2. Let f(x)=x2 and I=int( f(x), x= -1. . 2). Does Theorem 1 apply to I? Explain.
3. Let f(x)=x2 and I=int( f(x), x= -1. . 2). Does Theorem 2 apply to I? Explain.

Reminders:

• Begin PS 3.

Due Friday 2/10 at 9am

Section 6.2 Error Bounds for Approximating Sums

• Be sure to understand: The statement of Theorem 3 and Example 6.

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

1. Explain in words what K1 is in Theorem 3.
2. Explain in words what K2 is in Theorem 3.
3. Find values for K1 and K2 for int( x3, x= -3. . 1).

Due Monday 2/13 at 9am

Section 6.2 Error Bounds for Approximating Sums
Section 7.1 Measurement and the Definite Integral; Arc Length

• Be sure to understand: Example 7 in Section 6.2; in Section 7.1, the facts on pages 416 and 419, and examples 5 and 8.

E-mail Subject Line: Math 104 Your Name 2/13

1. How many subdivisions does the trapezoid method require to approximate int( cos(x3), x = 0. . 1) with error less than 0.0001?

2. Let f(x)=sin(Pi*x/2)+10 and g(x)=3x/10+10.

(a) Set up the integral that determines the area of the region bounded by y=f(x) and y=g(x) between x=0 and x=5/3.
(b) Set up the integral that gives the length of the curve y=g(x) from x=-1 to x=3.

Reminders:

• Remember to take advantage of both my office hours and the tutoring hours in the KCCL.
• Bring unresolved questions to class Monday.
• Exam 1 will be Tuesday 2/21.

Due Wednesday 2/15 at 9am

Section 7.2 Finding Volumes by Integration

• Be sure to understand: The section Solids of revolution

E-mail Subject Line: Math 104 Your Name 2/15

1. Let R be the rectangle formed by the x-axis, the y-axis, and the lines y=1 and x=3. What is the shape of the solid formed when R is rotated about the x-axis?
2. Let T be the triangle formed by the lines y=x, x=1 and the x-axis. What is the shape of the solid formed when T is rotated about the x-axis?

Reminder:

• Begin PS 4.

Due Friday 2/17 at 9am

Section 7.2 Finding Volumes by Integration

Reminders:

• Remember to take advantage of the tutoring hours, as well as my office hours.
• Begin studying now for the exam, if you haven't already!

Due Monday 2/20 at 9am

Bring Questions for Exam 1

Reminders:

• Take advantage of my office hours as well as tutoring hours for help resolving any questions, whether big or small, subtle or major. Every semester it turns out someone is embarrassed to come to me for help, because they feel like they should have come earlier. Please don't worry about it -- better now than later, and better late than never.
• PS 4 will not be turned in, but it will be covered on the exam. Get questions on it out of the way before class!
• For the exam, you may have a "cheat sheet", consisting of handwritten notes on one side of an 8 1/2 x 11 (or smaller) piece of paper.
• You may begin taking the exam at 12:30pm Tuesday.

Due Wednesday 2/22 at 9am

Section 8.1 Integration by Parts

• 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 2/22

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).

Reminders:

• Begin PS 5. This is your second group assignment. While you're welcome to stick with the same partner(s) this time around, I encourage you to work with someone else instead.

Due Friday 2/24 at 9am

Section 8.1 Integration by Parts

E-mail Subject Line: Math 104 Your Name 2/24

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.

Due Monday 2/27 at 9am

Antidifferentiation

• To read: Review substitution and integration by parts
• Be sure to understand: Everything

Reminders:

• Bring questions on the hw. In the past, this problem set has caused some difficulty; while I've shortened it, you still should allow plenty of time to work on it (as you ALWAYS should be doing).

Here ends the reading for January and February
Go to the reading assignments for March!

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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