Math 161 - Solutions To Sample Exam 2 Problems Worksheet Page 7
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8Solutions To Sample Exam 2 Problems – Math 161
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7.96
A (x)
+
FINAL ANSWER:
5π
To get the maximum area, we need x = 7.96 and y = 50
(7.96) ≈ 18.7, meaning that the fence lengths for
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the rectangular portion of the garden should be 18.7 feet on each side, and that π(7.96) ≈ 25 feet of fencing
should be used along the semicircular boundary of the garden. The maximum area is given by
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A(7.96) = 100(7.96)
2π(7.96)
≈ 398 square feet.
8. For each of the following, use the provided graph to estimate the requested information.
(a) To the right is a graph of f (x).
f (x)
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i. Where is f (x) increasing? decreasing?
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ii. Where is f (x) concave up? concave down?
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iii. Locate any local extreme values of f (x).
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1
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3
4
5
6
−1
−2
−3
(b) To the right is a graph of g (x), the DERIVATIVE
g (x)
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of a function g(x).
2
i. Where is g(x) increasing? decreasing?
1
ii. Where is g(x) concave up? concave down?
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1
2
3
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5
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iii. Locate any local extreme values of g(x).
−1
−2
−3
(c) To the right is a graph of h (x), the SECOND
h (x)
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DERIVATIVE of a function h(x).
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i. Where is h(x) concave up? concave down?
1
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6
−1
−2
−3
Solution:
(a) Note that, in this part of the problem, we are given the graph of f itself.
i. The function f is increasing where its graph is rising from left to right, and decreasing where its
graph is falling from left to right. Therefore:
f is increasing on
[0, 1.7] and [4.5, 8]
f is decreasing on
[1.7, 4.5]
ii. Note: Since it is difficult to tell exactly where a function changes concavity by looking at its graph,
the intervals below are very approximate:
f is concave up on
(3.1, 6.5)
f is concave down on
(0, 3.1) and (6.5, 8)
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