Math 171
Exam III
Last Name(print):
Fall 2002
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You must show all appropriate work to receive full credit.
Work all problems on the paper provided. Any work on this sheet will not be graded.
Turn in your exam with your work.
Do not work on the back on any page.
SCHOLASTIC DISHONESTY WILL NOT BE TOLERATED.
The summation rules that I said I would give you on the exam.
n
X
1=n
n
X
i4 =
n
X
i=1
i=1
i=
i=1
n
X
n(n + 1)
2
i=1
i2 =
n(n + 1)(2n + 1)
6
n
X
i3 =
i=1
n(n + 1)
2
n(n + 1)(2x + 1)(3n2 + 3n − 1)
30
1. (8 points) Compute the exact value these
(a) cos(tan−1 5) =
(b) arccos(cos( 5π
4 ))
2. (10 points) A bacteria culture starts with 5000 critters and triples every three hours.
(a) Find a formula that gives the number of critters in the culture after t hours.
(b) How many critters are in the culture after 60 minutes?
3. (10 points) Find y ′ . YOU DO NOT HAVE TO SIMPLIFY.
(a) y = arctan(5x)
(b) y = (x2 + 2)x
4. (10 points)
The accelerationof a particle in the x-y plane is given by
t
a(t) = 2e + cos t, t + 3 sin t . If v(0) = 5i − 2j, find the velocity function.
5. (12 points) Compute the exact values of these limits.
(a) lim
n→∞
n
X
3 i
i=1
n n
+4
4
(b) lim (2 − x) x−1
x→1+
6. (8 points) On what interval is the function f (x) =
ln(x)
decreasing? Show all work.
x
More Questions On The Back
2
7. (10 points) For the function f (x) = 36 − x2 and the partition P = {0, 1, 4, 5}.
(a) Compute the Riemann sum, where x∗i is the left endpoint of the subintervals.
(b) Which of the following A, B, C, or D is true?
On the interval [0, 5] a left sum(sum using the left endpoint):
A) will always be an under estimate.
B) will always be an over estimate.
C) will be neither an over estimate nor an under estimate.
D) may be on or the other but more information is needed.
8. (10 points) If the domain of f (x) is all real numbers and
3
2
f ′′ (x) = (x − 9)2 (x + 3)(x − 2)e2x −3x +6 , find the intervals where f (x) is concave up/down and
label the inflection values.
9. (12 points) Below is the graph of the derivative f ′ (x) of a function f (x) on the interval
−5 ≤ x ≤ 5. Assume that the function f (x) is continuous on the interval. Use the graph to
answer the following questions.
5
4
3
2
1
−6 −5 −4 −3 −2 −1
−1
1
2
3
4
5
6
−2
−3
−4
(a) Classify the critical values of f (x) as local max, local min, or neither.
(b) Find the intervals where f (x) is decreasing.
(c) Find the intervals where f (x) is concave down.
10. (10 points) A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides
and a 2-inch margin at the top. What are the dimensions of the poster that will give the largest
printed area?
Printed Area