Department

Applied MathCourse Code

APPM 1345Professor

AllStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**1 pages of the document.**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

x−6

(c) y= (sin x)esin x,0≤x≤2π

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

u→0tan−1(eu)(b) lim

t→1+cos−1(ln t)(c) lim

x→∞ sin−11−x2

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 −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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