Fall 2009, Math 104

September, 2009

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

As you get more familiar with Maple, I'll use Maple syntax for mathematical notation on this page.
All section and page numbers refer to sections from Calculus: Early Transcendental Functions, Smith and Minton, 3rd edition, unless otherwise noted.

Due Friday 9/4 at 9am

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

Also:

Section 4.1: Antidifferentiation
Section 4.2: Sums and Sigma Notation
Section 4.3: Area
Section 4.4: The Definite Integral
Section 4.5: The Fundamental Theorem of Calculus

• To read: Review all of Section 4.1. Remember to always pay attention to tables and graphs; in particular, remind yourself why all the basic antiderivatives (excluding the inverse trig functions) in the table on page 347, as well as the antiderivative of 1x, are true, and make sure you know them all. For Sections 4.2, 4.3, and 4.4, just really quickly refresh yourself on the main ideas, but don't get hung up on the particulars - we'll be returning to these sections later. Finally, review all of Section 4.5.

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.

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, and how they are used.

E-mail Subject Line: Math 104 Your Name 9/4

1. In Section 4.1, the text states that we do not yet have formulae for the indefinite integrals of several elementary functions, including ln(x), tan(x), and sec(x). Why not? For instance, why do we know $\int cos\left(x\right) dx$ but not $\int tan\left(x\right)$?
2. The following questions relate to the antiderivatives of the two products xex2 and of xex.
(a) Which differentiation rule would you use to verify that an antiderivative of xex2 is ½ ex2?
(b) Which differentiation rule would you use to verify that an antiderivative of xex is ex(x-1)?
(c) Why do your answers to (a) and (b) make it unlikely that we will find a general product rule for antidifferentiation?
3. Find the signed area between x5 and the x-axis from x=1 to x=2.
4. 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?
5. 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? The second is a family of functions - why? Again, what does it represent?

Reminders:

• Always briefly explain how you arrived at your answers. Use shortened notation like int(x^2) to denote mathematical symbols. As you learn Maple, you can use Maple notation.
• 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.

• Because I believe these sections are review, these questions are perhaps a bit more difficult than the reading questions will usually be.
• I have set it up so that you will get automatic notification that I got your reading assignment, but this is not a perfect system. First of all, you won't get the message unless my mail system is active -- that is, unless I'm logged in and either on-campus or connected by VPN. That means that if you send it at night, you might not get this message until the next morning when I arrive on campus. Also, the notification process 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 9/7 at 9am

Labor Day vacation!

Due Wednesday 9/9 at 9am

Problem Set Guidelines
Section 0.4 Trigonometric and Inverse Trigonometric Functions
Section 2.8: Implicit Differentiation and Inverse Trigonometric Functions

• To read: In Section 0.4: Read the subsection Inverse Trigonometric Functions, beginning on page 41. (I am assuming you're comfortable with trig functions themselves.) In Section 2.8: Skim the first portion, on implicit differentiation. I won't talk about it much in class, but if you haven't seen it before, don't worry - it's just revisiting the chain rule. Then read the subsection Derivatives of Inverse Trigonometric Functions.
• Be sure to understand: Why differentiating sin(y)=x leads to a result involving dydx, and why in the end we would solve for dydx.

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

1. Why do you think mathematicians often prefer to use arcsin(x) (or arccos(x), etc) rather than sin-1(x)?
2. What is the domain of the function arccos(x)? Why is this the domain?
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 9/11 at 9am

Section 4.6 Integration by Substitution

• Be sure to understand: Examples 3, 5, 7, 9, and 10 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 9/11

1. Substitution attempts to undo one of the techniques of differentiation. Which one is it?
2. In brief, what are the steps in the process of substitution?

Reminder:

• Begin working on PS 2. This is a group assignment. Groups should consist of two or three people; never one, never four. 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.

• Your group may not divvy up the problems amongst you (see course policies - Honor Code).

Due Monday 9/14 at 9am

Section 4.2: Sums and Sigma Notation
Section 4.3: Area
Section 4.4: The Definite Integral

• To read: In Section 4.2, read all except the section Mathematical Induction, which is optional. Read all of Section 4.3. In Section 4.4, only read through the bottom of page 370.

E-mail Subject Line: Math 104 Your Name 9/14

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, a midpoint sum with 10 subintervals or a midpoint sum with 100? Why?
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 9/16 at 9am

Problem Set Guidelines
Section 4.4: The Definite Integral
Section 4.7: Numerical Integration

• To read: Finish Section 4.4. Read Section 4.7 up to the top of page 407 (the beginning of the section Simpon's Rule).
• Be sure to understand: The statements of Theorem 1 and Theorem 2

E-mail Subject Line: Math 104 Your Name 9/16

1. Why would we ever want to approximate an integral?
2. If a function f(x) is concave up, does the Trapezoidal Rule produce an over- or an under-estimate of the signed area between f(x) and the x-axis?

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 of who the "primary author" was this time by putting a star next to that person's name. Next time, switch.
• 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 9/18 at 9am

Section 4.7: Numerical Integration

• To read: The section Error Bounds for Numerical Integration. Optional: The section on Simpson's Rule, and any subsequent discussion of Simpson's Rule.
• Be sure to understand: The statement of Theorem 7.1, and Examples 7.10 and 7.11.

E-mail Subject Line: Math 104 Your Name 9/18

1. Explain in words what K represents in Theorem 7.1.
2. Consider -31 x3dx.
(a)Find values for K
(b) How many subdivisions does the midpoint method require to approximate this integral with error less than 0.0001?

Reminders:

• Begin PS 3.
• Exam 1 will be Thursday 9/24.

Due Monday 9/21 at 9am

Section 5.1: Area Between Curves
Section 5.4: Arclength

• To read: All of Section 5.1. Then leap ahead to 5.4 and read up to the section Surface Area. (We will be omitting Surface Area; if you'd like to read about it, please do. We will cover Section 5.2 after the exam but will be omitting Section 5.3)
• Be sure to understand: where the derivative comes in in the derivation of the arclength formula; Examples 4.1 and 4.2.

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

Due Wednesday 9/23 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 9/25 at 9am

Section 5.2: Volume: Slicing, Disks and Washers

• To read: To beginning of the section The Method of Washers, bottom of page 448. Remember to pay attention to graphs, labels, and remarks, as well as examples.
• Be sure to understand: Where formulas 2.2 and 2.3 come from, how they're different and how they're the same (not just symboically, but in their derivation).

E-mail Subject Line: Math 104 Your Name 9/25

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 Monday 9/28 at 9am

Section 5.2: Volume: Slicing, Disks and Washers

• To read: Finish the section
• Be sure to understand: Make sure you understand the ideas, so that you don't have to memorize a different formula for every different situation.

E-mail Subject Line: Math 104 Your Name 9/28

Let R be the region bounded by the parabolas y=2(x-3)2-8 and y=(x-3)2-4.

Note: As in your reading, this phrasing means that you look for the region that is completely enclosed by only these two curves.
1. As best you can, describe the shape of the solid formed when R is rotated about the x-axis.
2. Describe the solid formed when R is rotated about the y-axis.

Due Wednesday 9/30 at 9am

Project 1

Reminders:

• Remember to take advantage of the tutoring hours, as well as my office hours.
• Bring unresolved questions on PS 4.

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 101A
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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