Fall 2007, Math 104

September, 2007

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 8/31 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 8/31

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.
• Come to lab at 1 pm Thursday, in room A102.
• Begin Problem Set 1, listed at the bottom of this course's web page. . The problem sets due each Thursday reflect an entire week's worth of work, and you should be working on them throughout the week.

I have set it up so that you will get automatic notification that I got your reading assignment, but that notification will only work if your reading assignment had exactly the right heading. If you don't get the "message received" notice, don't panic, but it probably wouldn't hurt to e-mail me to check whether or not I got it.

Due Monday 9/3 at 9am

Labor Day vacation!

Due Wednesday 9/5 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 9/5

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:

• PS 1 (individual) is due Thursday 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 Wednesday.
• Make sure you've read all the stuff I handed out in class and all the material on the course web page.

Due Friday 9/7 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 9/7

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:

• 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 Monday 9/10 at 9am

Section 5.6 Approximating Sums; The Integral as a Limit

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

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

Reminders:

• Remember the problem sets due each Thursday 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 Wednesdays 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 Wednesday 9/12 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 9/12

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:

• The Kollett Center for Collaborative Learning has office hours held by student tutors. 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 classes, you simply need to wait until the Calc 1 students have their questions answered. These hours are posted around the science center, and are also available on the web.
• After coming to office hours and/or going to the Kollett Center for the tutoring hours, bring any still-unresolved questions on PS 2 to class on Wednesday.
• (Last reminder of the semester:) Come to lab Thursday 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 (a star by that person's name will do), 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 Friday 9/14 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 9/14

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).

Reminders:

• Begin PS 3.
• Exam 1 will be Thursday 9/20.

Due Monday 9/17 at 9am

Section 6.2 Error Bounds for Approximating Sums

• Be sure to understand: Example 7

E-mail Subject Line: Math 104 Your Name 9/17

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

Reminders:

• Begin studying now for the exam, if you haven't already!

Due Wednesday 9/19 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 3 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 Thursday.

Due Friday 9/21 at 9am

Section 7.1 Measurement and the Definite Integral; Arc Length

• Be sure to understand: The Fact on page 416, Example 5, the Fact on page 419, and Example 8.

E-mail Subject Line: Math 104 Your Name 9/21

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

1. 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.
2. Set up the integral that gives the length of the curve y=g(x) from x=-1 to x=3.

Reminder:

• Begin PS 4. This problem set looks shorter than usual, but beware -- the problems take thought. Get started early, so you have time to ponder them and put various ideas together.

Due Monday 9/24 at 9am

Section 7.2 Finding Volumes by Integration
Guide to Writing Mathematics

• Be sure to understand: The section Solids of revolution

E-mail Subject Line: Math 104 Your Name 9/24

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?

Due Wednesday 9/26 at 9am

Section 7.2 Finding Volumes by Integration

Reminders:

• Remember to take advantage of the tutoring hours, as well as my office hours.
• Bring unresolved questions on PS 4.
• Remember to come to the Johnson lecture When is the Pen Mightier than the Keyboard?, given by Andries Van Dam of Brown University -- 5:30 pm Wednesday afternoon.

Due Friday 9/28 at 9am

Continue working on Project 1

Here ends the reading for September
Go to the reading assignments for October!

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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