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Give brief but clear answers to the questions below. Click on this link to send me an email with your answers. Make sure to type 'Calculus Assignment' and the date in the subject line.

For Thursday, September 2

1. The examples at the bottom of page 372 give the left and right approximations to the area under the curve y=x². Which is an over estimate, which is an underestimate, and why?
2. How do we obtain more accurate approximations to the area under the curve?
3. What notation allows us to write long sums more compactly?
4. What are the areas that you feel strongest in from Calculus 1? What areas will you need to work to improve this semester?

For Friday, September 3

1. Explain how the definite integral comes about as a limit.
2. What nice properties does the definite integral have, similar to some nice properties of the derivative?
3. Why does the sign of an integral change if we reverse the limits of integration?

For Tuesday September 7

1. Why is the calculation in the middle of page 395 (Where f(t)=t, a=0 and g(t) is the integral from 0 to x of f(x) ) valid?
2. In the statement of FTC2 (page 398), why does the theorem stipulate that we may choose F to be any antiderivative of f?
3. How is taking the area under a curve related to the antiderivative of that curve?

For Thursday, September 9

1. What are the differences between the power rules for derivatives and antiderivatives. (If you explain the rules, you should see that you are doing an 'opposite' operation in three places)
2. Why do all of our antiderivatives in Table 1 end in +C
3. What is the simplification process used for Example 5 on page 407. (Hint: They didn't really do 'long division').

For Friday, September 10

1. In Example 1 on page 415, why is it essential to have x³ as part of the antiderivative?
2. In Example 7 on page 418, how did the limits of 0 and 4 get changed to the limits 1 and 9?
3. Pages 419-420 give a discussion of symmetry, particularly even and odd functions. What kind of function is the product of two even functions? Of two odd functions? Of an even and an odd function?

For Tuesday September 14

1. How does this section extend the ideas of section 5.1?
2. Why is formula 3 (page 440) stated in terms of absolute values?
3. Look at the first four exercises on page 442. For each, determine whether integrating vertically or horizontally will be easier and why. (Use the discussion of example 6 if you're stuck on this one).

For Thursday, September 16

1. How does the determination of volumes lend itself to integration in a manner similar to the determination of areas? (In computing areas, we multiplied heights by infinitely small widths. Here, what are we multiplying by the widths?)
2. What is a washer? When are they used in computing volumes instead of solid disks? (Try and come up with a one or two sentence answer to the second question)
3. How is the side length s of the cross section of the pyramid in example 8 (page 451) determined?

For Friday, September 17

1. Give an explanation of the dimensions of the arbitrary cylindrical shell shown on page 456. (An explanation in your own words will help with both visualization of the object and understanding of settin up this type of problem.)
2. Why might Example 4 (page 458) be difficult to compute using disks/washers?
3. There is an exam in a week!!!! What would be most helpful for you in preparation for the test?

For Tuesday September 21

1. Units will be important to our discussion of work. What is a Newton? (hint: it's not a cookie) What is a joule?
2. Both examples 2 and 3 involve integrals to determine work. For each of these examples, what are the units on f(x) and dx
3. Recalculate example 4 if the cable were 25, 50, and 150 feet long. Can you make a statement on the amount of work required for a cable of arbitrary length?

You may download a copy of the practice exam here. We will go over this on Thursday. Here is a solution to the Work problem from that practice exam.

For Thursday, September 23

1. Review the Mean Value Theorem (Section 4.2 of Stewart, or Sections 26-27 of Gordon). How does it compare to the theorem given on page 465?
2. How is the mean value theorem like an 'infinite average'? (Think of taking the average of 3, 5, 10, 100, 10000 numbers)
3. Argue that a continuous function f with average value zero has at least one value c such that f(c)=0 .

For Friday, September 24

1. Relax! See me (early) if you have any questions.
2. Get a good night's sleep.
3. Arrive awake and in good physical condition (fed, caffeinated, or both, if those will help) to do your best on the exam. :)

For Tuesday September 28

1. Familiarize yourself with formulas 1 and 2 on page 476. How do they come about as a consequence of the product rule?
2. What similarities are there between Examples 2 and 5 from section 7.1?
3. Comment: The test was easier/about as hard/harder than I expected it would be.

For Thursday, September 30

1. Review your trigonometric identities from Calculus 1 (See Appendix D if you need further assistance). Which is the most important of the trigonometric identities, from which most others are derived?
2. Answer the question above Figure 1 on page 483.
3. Formula 1 on page 486 has a very slick verification in the explanation that follows. Read through it. What other algebraic technique does this remind you of?