Math 121 Winter 2010, Midterm 2 practice

Instructions

Closed book, closed notes, except for one 8.5”-by-11” (or A4) sheet of paper, okay to use both sides. You may be required to turn in your note sheet with the exam, so write your name on it.

50 minutes are allowed for this exam.

Clearly indicate your answer.

You must show all relevant work and justify your answers appropriately.

Partial credit will be given, but not without sufficient support.

No calculators that have a QWERTY-type keyboard are allowed, unless previously approved. The proctor's discretion is final.

This test covers Chapter 7.2 through Chapter 7.7, plus other previous material like integration by u-substitution.

Answers that violate simple upper or lower bounds may result in more point deductions.

There is a shortcut we learned about the average value of a sine or cosine; you may use that shortcut (but say you’re using it).

If you come across a problem that needs right-triangle calculations to compute something like cos(arcsin(theta)), you may stop there—you do not have to do those calculations. But, you should say that’s what you’re doing.

THIS PRACTICE TEST DOES NOT INCLUDE EVERY TOPIC THAT MIGHT BE ON THE REAL TEST!

A good way to study is to make a list of all the types of problems we’ve talked about (in class, worksheets, the textbook, homeworks, and projects if any) and make sure you know how to do each type, and understand the connections between them.

  1. Let f(x) = 5 sin(x) + 3 sin(2x). Find
  1. Find the RMS of f(x) = 5 sin(x) + 3 sin(2x) on the interval [0,2*pi].
  1. Suppose you want to integrate the following function via its data table:

x / 0 / 1 / 2 / 3
f(x) / 0.3 / 0.7 / 1.9 / 2.4

Can you use Simpson's method directly to do it? If so, do it. If not, give all the reasons why not.

  1. Use Simpson's method with n=4 to compute the integral from 0 to pi of sin(x). Show work.
  1. The following spreadsheet is trying to integrate f(x)=x^2 from 0 to 0.3 using the Trapezoid rule. Explain what, if anything, is wrong, and how to fix it. Both the formulas and the resulting numbers are given.

Col. A / Col. B / Col. C / Col. D
X / f(x)=x^2 / each trapez. / running sum / row 9
0 / 0 / 0 / 0 / row 10
0.1 / 0.01 / 0.05 / 0.05 / row 11
0.2 / 0.04 / 0.25 / 0.3 / row 12
0.3 / 0.09 / 0.65 / 0.95 / row 13
Col. A / Col. B / Col. C / Col. D
x / f(x)=x^2 / each trapez. / running sum / row 9
0 / =A10^2 / 0 / 0 / row 10
0.1 / =A11^2 / =AVERAGE(B10:B11)/(A11-A10) / =D10+C11 / row 11
0.2 / =A12^2 / =AVERAGE(B11:B12)/(A12-A11) / =D11+C12 / row 12
0.3 / =A13^2 / =AVERAGE(B12:B13)/(A13-A12) / =D12+C13 / row 13
  1. For each, circle True or False, and explain.

a)True/False: The Midpoint rule is always more accurate than the Trapezoid rule.

b)True/False: The RMS of a function can never be negative.

  1. What is the partial fraction expansion of (x+1)/[x*(x-1)] ?
  1. For each of these integrals, say (using a few words, like "IBP" or "U-sub", etc) which technique would be best suited to solving it. Hint: you can sort of check your answers by actually trying what you suggest, though that is not needed to get credit).
  2. Integral of x * e^(-x^2) dx
  3. Integral of sqrt(1-x^2) dx
  4. Integral of 1/(x^2-1) dx
  5. Integral from 0 to pi of sin(x)/x dx

A table of possibly useful formulas:

sin2(x) + cos2(x) = 1

tan2(x) + 1 = sec2(x)

sin(x) sin(y) = 0.5[cos(x-y) – cos(x+y)]

sin(2x) = 2 sin(x) cos(x)

d/dx tan(x) = sec2(x)

d/dx sec(x) = sec(x) tan(x)

integral of tan(t) = ln|sec(t)| + C

integral of sec(t) = ln |sec(t) + tan(t)| + C