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APPM 1345 Final Exam Spring 2011
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name, (2) section number, and (3) a grading table on the front of your bluebook. Start each problem on a new page. Simplify your answers. A correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may receive partial credit. Unless otherwise indicated, show all work.
- (15 points) Differentiate the following functions. Simplify your answers.
(a) y = ln(sec^2 2 θ) (b) y = (1 − t^2 ) coth−^1 t (c) y = x^1 /(ln^ x)
- (15 points) Evaluate the following limits.
(a) lim x→ 0
ex/^2 ex/^2 + 1
(b) lim t→ 0
10 t^ − 1 t (c) lim x→ 0 (1 − 3 x)^1 /x
- (25 points) Evaluate the following integrals.
(a)
θ^2
csc^2
θ
dθ
(b)
1
5 x
dx
(c)
∫ (^) e
1
ln x x dx
- (25 points) Consider the function f (x) = x ln
x
(a) What is the domain of f? (b) Find f ′. (c) Find f ′′. (d) Find any absolute extrema of f. (e) Evaluate lim x→ 0 +^
f (x).
(f) Show that (^) ∫ x ln
x
dx =
x^2 4
x^2 2 ln
x
+ C.
- (10 points)
x
y
y x 4 x
2
(a) Find the area of the shaded region. (b) Find the average value of the function y = x 4 −x
2 on [0, 1].
- (10 points) Let g(x) = sin−^1 (x). Find an equation of the line tangent to g at x =
- (10 points) Consider the function f (x) = ln
x 3
(a) Find the inverse of f.
(b) Let G(x) =
∫ (^) x
1
ln
t 3
dt.
i. Find G′(x). ii. At what value(s) of x does G have a local minimum value?
- (10 points) Use the identity
tanh−^1 x =
ln
1 + x 1 − x
to find the derivative of tanh−^1 x. Simplify your answer.
- (10 points) Is the following function continuous at x = 0? Justify your answer.
h(x) =
sinh 2x x
, x 6 = 0
1 , x = 0
- (10 points)
The Cave of Chauvet-Pont-d’Arc in France contains the earliest known cave paintings. Discovered in 1994, the Paleolithic artwork retains about 2 .6% of the original carbon-14 content. If the half-life of carbon-14 is about 5700 years, approximately how old are the paintings?
(You may leave your answer unsimplified.)
