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151 WebCalc Fall 2002-copyright Joe Kahlig
Quiz #20
MATH 151 Section
Name:
November 14, 2002
1. Find the intervals where the function f is concave up and where it is concave down.
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f (x) = e−x
The domain is all real numbers.
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f 0 (x) = −2xe−x now use the product rule to find f 00 (x)
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2
2
f 00 (x) = −2e−x + 4x2 e−x = (−2 + 4x2 )e−x
There is no place where f 00 (x) is undefined so we only have to check where f 00 (x) = 0
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Since e−x is always positive, we need 0 = −2 + 4x2 . Solving this gives x = ±
f 00
−5
− 0.5
q
concave down −
0.5
0
concave up: −∞, −
1
2
+
−
+
q
q q
1
2,
1
2
,
q 5
1
2, ∞
1
2
2. If the domain of f (x) is all real numbers and
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f 00 (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.
The possible inflection values are x = −3, 2, and 9.
f 00
+
−5
−
−3
0
+
+
2
5
9
10
Concave up: (−∞, −3), (2, 9), (9, ∞)
concave down: (−3, 2)
inflection values at x = −3 and x = 2
3. Give the asymptotes( vertical and horizontal) for these functions.
(x − 3)(40x + 3)
x2 − 9
Horizontal asym: y = 40
Vertical asym: x = −3
(a) y =
(b) y = ln(70 − 3x)
Horizontal asym: none
Vertical asym: x = 70
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