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Math 106 — Second Midterm
November 2, 2012
Name:
Instructor: Section:
- Do not open this exam until you are told to do so.
- This exam has 9 pages including this cover AND IS DOUBLE SIDED. There are 6 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.
- Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out when you hand in the exam.
- Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions.
- Show an appropriate amount of work (including appropriate explanation). Include units in your answer where that is appropriate. Time is of course a consideration, but do not provide no work except when specified.
- You may use any previously permitted calculator. However, you must state when you use it.
- If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph that you use.
- Turn off all cell phones and pagers, and remove all headphones and hats.
- Remember that this is a chance to show what you’ve learned, and that the questions are just prompts.
Problem Points Score
Total 100
Some trig identities you may need:
sin^2 (x) =
cos(2x) 2 cos^2 (x) =
cos(2x) 2 sin(2x) =2 sin(x) cos(x) cos(2x) = cos^2 (x) − sin^2 (x)
tan(x) =
sin(x) cos(x) sec(x) =
cos(x)
Some logarithm identities that may be useful:
ln(A) + ln(B) = ln(AB) AND ln(A) − ln(B) = ln(A/B).
The Gaussian curve with average (mean) m and standard deviation s is
s
2 π
e−(x−m) (^2) / 2 s 2 .
Z-score table: x 0 0.1 0. 11 0. 12 0. 13 0. 14 0.15 0. 16 0. 17 A(x) 0. 5 0. 5398 0. 5438 0. 5478 0. 5517 0. 5557 0. 5596 0. 5636 0. 5675
Some integrals you may find helpful: f(x) cos(x) sin(x) ex^ √^1 1 − x^2
1 1 + x^2
sec^2 (x) sec(x) tan(x) ∫ f (x)dx sin(x) + C − cos(x) + C ex^ +C sin−^1 (x) + C tan−^1 (x) + C tan(x) + C sec(x) + C
- [15 points] For this problem, Answer true, false, or not enough information by circling your choice.
Use the graph
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.
2
f(x)
g(x)
and assume that:
0 f^ (x)dx^ converges and
- for 1 ≤ x < ∞ it’s true that (^1) x ≥ g(x) ≥ (^) x^12 ≥ f (x) ≥ 0. No partial credit, no explanation required.
a. [3 points]
0 f^ (x)dx^ converges.^ True^ False^ Not Enough Information
b. [3 points]
3 g(x)dx^ diverges.^ True^ False^ Not Enough Information
c. [3 points]
1 f^ (x)g(x)dx^ converges.^ True^ False^ Not Enough Information
d. [3 points]
0 g(x)dx^ diverges.^ True^ False^ Not Enough Information
e. [3 points]
0 f^ (x)^ −^ g(x)dx^ converges.^ True^ False^ Not Enough Information
- [20 points] Determine if each of the following integrals diverges or converges. If the integral converges, find the exact answer. If the integral diverges, write “DIVERGES.” Show ALL work and use proper notation. a. [10 points] (^) ∫ 10
1
x^2 − 1
dx.
b. [10 points] (^) ∫ 5
3
dx (x^2 − 9)^3 /^2
- [20 points] Let t be the number of minutes a student waits for the bus. The probability density function giving the distribution of t is
p(t) =
0 if t < 0 2 e−bt^0 ≤ t < ∞.
a. [10 points] Find the number b so that p(t) satisfies the requirements of a probability density function. Show proper work.
b. [10 points] The average length of time we need to wait for the bus is given by ∫ (^) ∞
0
tp(t)dt.
Evaluate this integral using your number “b” from part (a) to figure out how long this is.
- [20 points]
a. [10 points] Write the degree 3 Taylor polynomial for ln(x) around 1.
b. [10 points] Use part (a) and the maximum error for your approximation to find a range of possible values for ln(1.5).
