MAC 2311
August 7, 2003
Final Exam
Prof. S. Hudson
Name
Show all your work and reasoning for maximum credit. Do not use a calculator, book,
or any personal paper. You may ask about any ambiguous questions or for extra paper.
Hand in any extra paper you use along with your exam.
1) (10 pts) Find and classify the critical points (as relative max or min or neither). Follow
the directions (do not rely on a graph) and show your work.
a)
f
(
x
) =
xe

x
, using the first deriv test.
b)
f
(
x
) = cos(
x
) on (

π/
2
,
3
π/
2), using the second deriv test.
2) (10pts) Find all points on the curve where the tangent line is horizontal. Given:
x
= sin(
t
)
y
= sin(2
t
)
1
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3) (10pts) Choose ONE proof:
A) Prove that if
f
(
x
) = 0 on (
a, b
), then
f
is constant there. (Use the MV Thm).
B) Calculate the derivative
D
x
tan

1
(
x
) carefully, and simplify it, with a full explanation.
[Use the formula for
D
x
f

1
and the triangle, as done in class).
C) B) Calculate the derivative
D
x
cos(
x
) carefully, and simplify it, with a full explanation.
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 Spring '08
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 Calculus, Derivative, Continuous function, Limit of a function, derivative Dx cos

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