APPM 1350 Final Spring 2010
On the front of your bluebook, please write: a grading key, your name, student ID, and
section and instructor. This exam is worth 150 points and has 7 questions. Show all work! 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. (24 points) Find the requested information.
(a) lim
x→∞ 1 + 2
xx
(b) lim
t→1
t−1
ln t−sin πt
(c) Find the equation for the line tangent to the following curve at the point (0,π ).
x2cos2y−sin y= 0
(d) Z−1/2
−1
t−2sin21 + 1
tdt
2. (30 points) Let f(x) = x2(x2−9). Answer each of the following questions about the graph of f(x).
(a) What are the critical points of f?
(b) On what interval(s) is f(x) increasing?
(c) Give the xcoordinate of all local maxima and minima. Indicate which is which and justify your
answer.
(d) On what interval(s) is f(x) concave down?
(e) Give the xcoordinate of any inflection point(s), if they exist. Justify your answer.
(f) Sketch the shape of the graph y=f(x).
3. (16 points) Order the following functions from slowest growing to fastest growing as x→ ∞. Justify
your answer by evaluating the appropriate limits.
exxx(ln x)x
4. (20 points)
(a) Establish the following formula using implicit differentiation and an appropriate right triangle.
d
dx tan−1x=1
1 + x2
(b) Integrate Zdt
√e2t−1
(c) Find cot sin−1−1
2−sec−1(2).
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