Spring 2012, Math 104

April and May, 2012

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

All section and page numbers refer to sections from Calculus: Early Transcendental Functions, Smith and Minton, 3rd Edition.

Due Monday 4/2 at 8:30am

Section 8.3: The Integral Test and Comparison Tests

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 several attempts!

Due Wednesday 4/4 at 8:30am

Project 2

• Be sure to understand: What is being asked of you.

Reminders:

• Bring questions on WW 9 and PS 9.

Due Friday 4/6 at 8:30am

Section 8.3: The Integral Test and Comparison Tests

• To read: Through Example 3.7
• Be sure to understand: Examples 3.5, 3.6, and 3.7.

1. Explain in a couple of sentences why the Comparison Test makes sense. That is, explain the idea behind it.
2. Show that the series 1(2+3j) converges.

Reminders:

• Continue to work on your project (the mathematics should be complete by Sunday evening). As usual, find times to meet that work for all members of your group. Every one should be contributing equally on the mathematics. Do not assign one or two people to work on the math and another to do the writing -- for one thing, this is completely counter to the point, and for another, it never results in a good paper.
• Begin WW 10 and PS 10.

Due Monday 4/9 at 8:30am

Guide to Writing Mathematics
Checklist
Section 8.3: The Integral Test and Comparison Tests

• To read: While I may not be covering the Limit Comparison Test in class, it is a useful test so please do read the rest of the section -- it resolves some of the difficulties we can run into with the Comparison Test. On the down side, it is not obvious how it can be used in approximating a series. Re-read the entire section, making sure you understand each example and looking for more nuance than you saw on the first read-through. Also, in class, I have covered using the Integral and Comparison Tests to help approximate convergent series in more detail than the text did. Read through your notes and in-class work related to this.

Consider the series 1(k4+5).

1. Show that the series converges.
2. Estimate the limit of the series within 0.01.
3. Is your estimate an over- or under- estimate?

Reminders:

• Finish the mathematics of the project before class. If you run into difficulties, come to my office hours Monday.

Due Wednesday 4/11 at 8:30am

Section 8.4: Alternating Series

• Be sure to understand: The statement of the Alternating Series test and how it differs from the kth Term Test. Examples 4.1, 4.2, 4.5, and 4.6.

1. Does sum( (-1)ksqrt(k), k=1..∞) converge or diverge?
2. How closely does S100 approximate the series sum((-1)k (1/k), k=1 .. ∞) ? Why?

Reminders:

• Bring questions on WW 10 and PS 10
• Your project response should be written by your group as a whole. Do not have one member of the group do all the writing. It is also not a good idea to split the writing up, because rarely does such a letter or paper flow well.

Due Friday 4/13 at 8:30am

Section 8.5: Absolute Convergence and the Ratio Test

• To read: Through the section The Ratio Test
• Be sure to understand: Examples 8.5.3 and Example 8.5.5. How to use the ratio test to determine convergence and divergence, and when the ratio test is inconclusive.

1. Give an example of a series that is conditionally convergent. Explain.
2. Give an example of a series that is absolutely convergent. Explain.

Reminders:

• Attach a blank copy of the check-list by paperclip to the front of your project when you turn it in.
• Begin WW 11 and PS 11 (group). As always, groups should consist of 2 or 3 people; think about working with new people; don't split the problems; switch primary authors; put a star next to the primary author's name.

Due Monday 4/16 at 8:30am

Section 8.5: Absolute Convergence and the Ratio Test

• To read: Re-read the section on the Ratio Test, and read the section Summary of Convergence Tests. Optional: the section The Root Test.
• Be sure to understand:

1. Explain in a couple of sentences why the Ratio Test makes sense.

Due Wednesday 4/18 at 8:30am

Section 8.6: Power Series

1. How do power series differ from the series we have looked at up to this point?
2. What is the interval of convergence of a power series? Why have we not discussed the interval of convergence before? Explain in your own words.
3. Suppose the power series bk(x-c)k has a non-trivial and non-infinite interval of convergence. Where will the midpoint of that interval be?
4. As you have read, we can differentiate and integrate a convergent power series term-by-term. Why is this not obvious?
5. Why might you want or need to use a power series?

Reminder:

• Bring questions on WW 11 and PS 11
• Exam 3 will be Thursday 4/26

Due Friday 4/20 at 8:30am

Section 8.7: Taylor Series

• Be sure to understand: This section can be tough (particularly if you didn't learn Taylor polynomials in Calc 1). Work hard at understanding it, and I'll be ready to answer your questions!

1. Give two good reasons for writing a known function ( such as cos(x) ) as a power series.

Reminders:

• Get an early start on WW 12, to allow you plenty of time to work on the study guide (which as usual assumes you have completed WW 12 first.)

Reminders:

• Monday is the last day you can receive 75% on the antidifferentiation exam. After that, it goes down to 50% until the last day of classes.

Due Monday 4/23 at 8:30am

Section 8.7: Taylor Series

• Be sure to understand: Continue to work hard at understanding this section.

Due Wednesday 4/25 at 8:30am

Questions for Exam 3

To read: And again, re-read the Honor Code, Wheaton's description of plagiarism, and the portion in the course policies that applies to the Honor Code, paying particular attention to how it all applies to exam situations.

Reminders:

• Get most of your questions answered before Wednesday, in office hours.
• As always, you may have handwritten notes on one side of a standard sheet of paper, and may begin the exam at 12:30.

Due Friday 4/27 at 8:30am

Section 8.8: Applications of Taylor Series

1. If you wanted to find esqrt(3) using a Taylor series, what would you use as a base point?

Reminder:

• Begin WW 13

Due Monday 4/30 at 8:30am

Section 9.4: Polar Coordinates

• Be sure to understand: Examples 4.2, 4.7, and 4.10

1. Why might we use polar coordinates rather than rectangular coordinates?
2. Give two different representations in polar coordinates of the point with rectangular coordinates (2sqrt(2), 2sqrt(2))
3. What is the relationship between the graphs y=sin(x) and y=sin(2x) in rectangular coordinates?
4. What is the relationship between the graphs r=sin(θ) and r=sin(2θ) in polar coordinates?

Due Wednesday 5/2 at 8:30am

Section 9.5: Calculus and Polar Coordinates

• To read: Begin on page 757 after the end of Example 5.3 and read through Example 5.7. (We'll focus on integration in polar coordinates. Differentiation in polar coordinates is usually covered in Multivariable Calculus.)
• Be sure to understand: Examples 5.3, 5.4, and 5.5

Consider the polar "rectangle" described by α ≤ θ ≤ β and 0 ≤ r ≤ R.

1. What is the shape of this "rectangle"?
2. What is the area of this "rectangle"?

Reminder:

• I'll answer questions on WW 13 either Thursday or Friday.

Due Friday 5/4 at 8:30am

The Big Picture

Reminder:

• Friday is the deadline to receive any credit on the antidifferentiation exam.

Here ends the reading for the Semester!

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 1306
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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