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Midterm

appm134spring2014exam3Exam


Department
Applied Math
Course Code
APPM 1345
Professor
All
Study Guide
Midterm

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APPM 1345 Exam 3 Spring 2014
On the front of your bluebook, please write: a grading key, your name, student ID, section, and in-
structor’s name. This exam is worth 100 points and has 7 questions. Show all work and simplify answers.
Answers with no justification will receive no points. Please begin each problem on a new page. No notes,
calculators, or electronic devices are permitted.
1. (10 points) Find the 100th derivative of the following functions.
(a) f(t) = 7t(b) h(x) = ln(13x) + ln 1
13x
2. (16 points) For each of the following curves, find an equation of the tangent line at the given value.
(a) y=xln x, x =e(b) y= (tan x)ln x, x =π
4
3. (22 points) Find the values of xwhere the following curves have horizontal tangents.
(a) y=x2ln (2/x)
(b) y=log6x
x6
(c) y= (sin x)esin x,0x2π
4. (10 points) Let y=ex
ex+ 1.
(a) Show that yis an increasing function and therefore one-to-one.
(b) Find the inverse function of y.
5. (10 points) Evaluate the following limits.
(a) lim
u0tan1(eu)(b) lim
t1+cos1(ln t)(c) lim
x sin11x2
1 + 2x2
6. (20 points) Evaluate the following integrals.
(a) Ztan(3x)dx (b) Ze3x
1 + e6xdx (c) Zeπ
1
sin(ln x)
xdx
7. (12 points) Alvin mistakenly left his smartphone outside overnight in the Alaskan cold. The phone had a
temperature of 30C when he brought it inside in the morning. Alvin wants to check his messages but
the phone won’t work properly until it warms up to 0C. If the phone reaches a temperature of 25C
after 2 minutes in Alvin’s 20C house, how long will it take before Alvin can use his phone?
(Use the approximations ln 2 0.7,ln 3 1.1, and ln 5 1.6to calculate your answer.)
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