Course CodeAPPM 1345
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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
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
3. (22 points) Find the values of xwhere the following curves have horizontal tangents.
(a) y=x2ln (2/x)
(c) y= (sin x)esin x,0≤x≤2π
4. (10 points) Let y=ex
(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.
t→1+cos−1(ln t)(c) lim
1 + 2x2
6. (20 points) Evaluate the following integrals.
(a) Ztan(3x)dx (b) Ze3x
1 + e6xdx (c) Zeπ
7. (12 points) Alvin mistakenly left his smartphone outside overnight in the Alaskan cold. The phone had a
temperature of −30◦C 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 0◦C. If the phone reaches a temperature of −25◦C
after 2 minutes in Alvin’s 20◦C 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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