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(Last modified: Friday, February 17, 2012, 3:32 PM )

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

I'll use WebWork and/or Maple syntax for some of the mathematical notation on this page. (Paying attention to how I type various expressions is a good way to absorb WebWork/Maple notation). I will not use it when I think it will make the questions too difficult to read.

Due Friday 1/27 at 8:30am

Course policies
Syllabus
Wheaton's Honor Code
Wheaton's Description of Plagiarism
Suggestions for reading a math text
Background questionnaire, on OnCourse
WeBWorK Orientation

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: All of each of the above weblinks, except for WeBWorK Orientation. (See below for guidance on WeBWorK Orientation). Complete the questionnaire in OnCourse, if you're registered for the course already. (If you're not yet, complete it once you are).

  Review all of Section 4.1 (skip inverse trig functions) and 4.5, and remind yourself of the main ideas in Sections 4.2 (skip induction), 4.3, and 4.4 (we'll return to those sections later).

  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 ¹⁄_x, are true, and make sure you know them all (again, excluding the inverse trig functions).

  Use this review as an opportunity to get acquainted with the authors' style, if this book is new to you. 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.

  As for WeBWorK Orientation: if you've used WeBWorK before, you do not have to do the orientation, but do log on before class on Friday to make sure you can. (If I don't already know you, just send me an e-mail explaining that you've used it before.)

  For those of you who haven't used it before, log in to WeBWorK by using your Wheaton ID as both your username and as your password (with a lower-case "w" in both cases). (If you're not registered yet but did give me your Wheaton ID, you won't be able to log until late afternoon of the day you gave it to me. In the meantime you can get started by using "guest" as both your username and password. The server may be slow, so be patient.

  Once you're logged in, click on WeBWorK Orientation. Read the discussion on the right, and then click on Problem 1 to get started. Get as far as you can with the orientation before class Friday, and finish it before getting started on WW 1.

  If you have trouble logging in to WeBWorK, e-mail me . Let me know exactly what you used for username and password -- cutting and pasting is the most exact way.

• Be sure to understand: The statements of both forms of the Fundamental Theorem of Calculus, and how they are used.

Reading questions:

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 ∫ cos(x) dx but not ∫ tan(x)?
2. The following questions relate to the antiderivatives of the two products xe^x2 and of xe^x.

   (a) Which differentiation rule would you use to verify that an antiderivative of xe^x2 is ½ e^x2?
   (b) Which differentiation rule would you use to verify that an antiderivative of xe^x is e^x(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 3x⁵+sin(x) 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, or even that one is a real number and the other a family of functions. Why is a definite integral a real number (what does it represent)? Why is an indefinite integral a family of functions (again, what does it represent)?

Submit answers through OnCourse

Reminders:

• Come to lab at 1 pm Thursday, in room Mars Science 1141.
• In these reading assignments, always give your best shot at answering the questions. I do not expect you to get them all right, and I do not take points off for getting them wrong -- only for not trying. These questions and answers serve multiple purposes - they help you to focus when you're reading (and over time, hopefully they will help you learn what sorts of questions you should be asking yourselves as you read), they give me feedback on what level of understanding the class as a whole is walking in with after reading, they allow us to focus on understanding rather than on mere transfer of information, and they allow me to give you credit for the effort you're putting into the reading.
• 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.
• The homework assignments are accessed through a link toward the bottom of the course's main web page.
• Begin WeBWorK Orientation, WW 1, and PS 1.

Please Note:

• Because these sections should be review, these questions are more conceptual than the reading questions will usually be.

Due Monday 1/30 at 8:30am

A Description of Calculus 2 Homework Assignments
Problem Set Guidelines
Section 0.4 Trigonometric and Inverse Trigonometric Functions
Section 2.8: Implicit Differentiation and Inverse Trigonometric Functions

• To read:All of the Problem Set Guidelines. 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 for those of you who haven't seen it before, don't worry - it's just 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 ^dy⁄_dx, and why in the end we would solve for ^dy⁄_dx.

Reading questions:

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).

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Reminder:

• WeBWorK Orientation and WW 1 are both due Wednesday at 7pm, and PS 1 is due Thursday at 1pm. WeBWorK assignments are always to be done individually. Although Problem Sets will alternate between individual and group assignments, PS 1 is an individual assignment. Read and follow the problem set guidelines referred to above.
• If you're having difficulties with the homework, or just want to make sure you're on the right track, come see me! Once the Kollett Center is open, you can also go to the tutors there -- they are most useful for specific questions, however, as they have students from other classes wanting their attention as well. If you are having difficulty with the concepts themselves, or in general if you think you'd need more than a few minutes of their help at a time, you're better off coming to my office hours.
• If you didn't already, make sure you've read the introductory material on the course web page.

Due Wednesday 2/1 at 8:30am

Section 4.6 Integration by Substitution

• To read: All.
• 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.

Reading questions:

1. Substitution attempts to undo one of the techniques of differentiation. Which one is it?
2. Find one antiderivative of ^ex⁄ _{(1-(e^x)²)^1/2}

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Reminder:

• WW Orientation and WW 1 are due Wednesday at 7pm. PS 1 (individual) is due Thursday at 1pm. When writing your final draft of PS 1, follow the guidelines referred to above.
• If you have any remaining questions on WW 1 or PS 1 after coming to my office hours, bring them to class Wednesday.

Due Friday 2/3 at 8:30am

Section 4.2: Sums and Sigma Notation
Section 4.3: Area
Section 4.4: The Definite Integral
Section 4.7: Numerical Integration

• To read: In Section 4.2, review all except the section Mathematical Induction, which is optional. Review all of Sections 4.3 and 4.4 Read Section 4.7 up to the top of page 407 (the beginning of the section Simpson's Rule).
• Be sure to understand: The statements of Theorem 1 and Theorem 2 in Section 4.7.

Reading questions:

1. Give an example of a partition of the interval [0,3].
2. Explain the idea of a Riemann sum in your own words.
3. Why might we want or need to approximate an integral?
4. 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?

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Reminder:

• Begin working on WW 2 and PS 2. PS 2 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 -- I strongly encourage you to work with different people different weeks.

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

Due Monday 2/6 at 8:30am

Section 4.7: Numerical Integration

• To read: The section Error Bounds for Numerical Integration beginning on page 410.
  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.

Reading questions:

1. Explain in words what K represents in Theorem 7.1.
2. Consider $$_-3∫ ¹ x³dx. Is 5 a valid value for K in Theorem 7.1? Explain.

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Reminders:

• Remember both the WeBWorK assignments and Problem Sets reflect the whole week's worth of work, and should be worked on throughout the week.
• Office hours are an important part of Calculus for all students -- do not hesitate to take advantage of them.
• Also take advantage of the tutoring in the Kollett Center -- just remember to come to me if you find yourself wantng more than a couple minutes of attention at a time.
• Just a reminder: the aims of the reading assignments are to help learn to technical material, give me feedback on what level of understanding the class as a whole has before class starts, allow us to focus on understanding rather than on mere transfer of information, and to give you credit for the effort you're putting into the reading. I do not take points off for incorrect answers -- only for not trying.

Due Wednesday 2/8 at 8:30am

Problem Set Guidelines
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 Sections 5.2 and 5.3 next.)
• Be sure to understand: where the derivative comes in in the derivation of the arclength formula; Examples 4.1 and 4.2.

Reading questions:
Let f(x)=10sin(πx) and g(x)=5sin(πx).

1. Find the area of the region bounded by y=f(x) and y=g(x) between x=0 and x=1.
2. Set up (but do not evaluate) the integral that gives the length of the curve y=g(x) from x=0 to x=1.

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Reminders:

• If, after coming to office hours, you still have questions on WW 2 or PS 2, bring them to class on Wednesday.
• Your group will turn in one joint version of PS 2; the recopying should all be done by one person, the "primary author". Put a star next to the name of the "primary author".
• 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.
• I suggest that you switch partners with every group problem set. You should be the primary author roughly every other group problem set.

Due Friday 2/10 at 8:30am

Section 5.2: Volume: Slicing, Disks and Washers

• To read: All. 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).

Reading questions:

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?
3. 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.

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Reminders:

• Begin WW 3 and PS 3.

Due Monday 2/13 at 8:30am

Section 5.2: Volume: Slicing, Disks and Washers

• To read: Re-read all.
  

No Reading Questions Today

Due Wednesday 2/15 at 8:30am

Section 5.3: Volumes By Cylindrical Shells

• To read: All.
  
• Be sure to understand: Examples 5.3.2 and 5.3.4.

Reading questions:

1. Give an example where you would use cylindrical shells rather than washers to find volume.
2. Let S be a solid formed by rotating about the y-axis.
   1. What is the variable of integration, if you're finding the volume using shells?
   2. What is the variable of integration, if you're finding the volume using washers?

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Reminders:

• Bring remaining questions on WW 3 and PS 3 to class.
• Exam 1 will be Thursday 2/23.

Due Friday 2/17 at 8:30am

Section 5.2 and Section 5.3: Volume

No Reading Questions Today

Reminder:

• Get an early and strong start on WW 4. There is no PS 4 for this week, so that you can get instant feedback on all the homework before the exam. The questions on the study guide assume that you have already worked through WW 4. (The study guide is not on WeBWorK, because you need to practice writing up solutions, as that's the format of an exam.)

Due Monday 2/20 at 8:30am

Section 6.1: Review Integration

• To read: All
• Be sure to understand: All - this is review, so you should follow it all.

No Reading Questions Today

Reminders:

• Spread your studying for the exam out over several days -- I know it's difficult for some people to study that way, but the material really does sink in better when you do.
• Come to my office hours to resolve any difficulties before the exam.

Due Wednesday 2/22 at 8:30am

Questions for Exam 1

Wheaton's Honor Code
Wheaton's Description of Plagiarism
Course policies

To read: Re-read the Honor Code, Wheaton's description of plagiarism, and the portion in the course policies that applies to the Honor Code. Pay particular attention to how all of this applies to exam situations.

No Reading Questions Today

Reminders:

• Keep coming to my office hours!
• Get questions on WW4 out of the way before class! This problem set should be finished before class, so you can be focusing on reviewing the semester to date.
• 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 2/24 at 8:30am

Section 6.2: Integration by Parts

• To read: All
• Be sure to understand: Where the rule for integration by parts comes from, and the basics of how it's used. Examples 1-3.

Reading Questions

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 ∫ xcos(x)dx. Use integration by parts to find an antiderivative for xcos(x).

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Reminders:

• Begin WW 5 and PS 5.
• PS 5 is your second group assignment. I encourage but do not require that you switch group partners- the more people you've worked with, the more comfortable you will feel in the class. Also, just because someone is a good friend does not mean you have the same sort of schedule or the same approach to homework.
• Remember not to split up the problems amongst you--your group should discuss every problem.

Due Monday 2/27 at 8:30am

Section 6.2: Integration by Parts
Section 6.5: Integration Tables and Computer Algebra Systems

• To read: Re-read Section 6.2. Read all of Section 6.5.
• Be sure to understand: Pay particular attention to Examples 2.4 and 2.5. Be sure to read the marginal remarks!

Reading questions:

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. ∫ xe^xdx
2. ∫ xe^x2dx

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Due Wednesday 2/29 at 8:30am

Project 1

• To read: Carefully read the letter you received in class
• Be sure to understand: What is being asked of you.

No Reading Questions Today

Reminders:

• Bring unresolved question on WW 5 and PS 5 to class.
• Once again, one member of your group for your homework will be the primary author for the week. Put a star next to that person's name. Unless all members of your group were primary authors on the first problem set, the primary author should be someone who was not primary author before.