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Math 142 WIR, copyright Angie Allen, Spring 2013
Math 142 Week-in-Review #6 (Sections 5.2, 5.3, and 5.4)
1. Determine where the function f (x) =
−1
x2 + 3
is concave upward/downward and has any inflection points.
2. If f is continuous on (−∞, ∞), and we know f ′ (3) = 0, f ′′ (3) = 7, and f ′′ (x) is continuous near x = 3,
what can we conclude about f ? What if f ′′ (3) = 0?
Math 142 WIR, copyright Angie Allen, Spring 2013
√
4
2x3 + 3 x3
3. Find lim √
.
7 9
x→−∞
x −x
e6x − e−3x
.
x→∞ e−5x − e4x + 2ex
4. Find lim
4ex − 5
x→−∞ e−x + 2e3x − ex
5. Find lim
2
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Math 142 WIR, copyright Angie Allen, Spring 2013
6. Create a chart showing the graphical relationships between f , f ′ , and f ′′ .
7. Consider the graph of f (x) below.
(Courtesy of Joe Kahlig)
a) Arrange the derivatives at the given points from smallest to largest.
b) At what points do f ′ (x) and f ′′ (x) have the same sign?
Math 142 WIR, copyright Angie Allen, Spring 2013
ax46 − bx2 − c
where a > 0, b < 0, and c < 0.
x→∞
bx46 − a
8. Find lim
9. a) Sketch a function whose slope is positive and decreasing.
b) Sketch a function whose slope is getting less negative.
c) Sketch a function that is increasing whose slope is also increasing.
10. Sketch the graph of a function that satisfies all of the following conditions:
• Domain of f : (−∞, ∞)
• f (0) = 1
• f ′ (1) = f ′ (−1) = 0
• f ′ (x) < 0 on (−1, 1)
• f ′ (x) > 0 on (−∞, −1) and (1, 2)
• f ′ (x) = −1 on (2, ∞)
• f ′′ (x) < 0 on (−∞, 0)
• f ′′ (x) > 0 on (0, 2)
• lim f (x) → −∞
x→−∞
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Math 142 WIR, copyright Angie Allen, Spring 2013
11. Consider a function f (x) that is continuous on its domain of (−∞, 4) ∪ (4, ∞). Also, f ′ (x) =
5
−x − 2
and
(x − 4)3
2x + 10
. Find where any local maximum and minimum values of f (x) occur using the Second
(x − 4)4
Derivative Test, if possible. If it is not possible, then use the First Derivative Test.
f ′′ (x) =
3x3 − 15x2 − 18x
algebraically, if
4x4 − 4
they exist. If there are vertical asymptotes, use limits to describe the behavior near each vertical asymptote.
12. Find any holes and horizontal and vertical asymptotes of the curve y =
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Math 142 WIR, copyright Angie Allen, Spring 2013
2
13. Use calculus to find any asymptotes of the function f (x) = xe−8x , where the function is increasing/decreasing
and concave upward/downward, the values of any local extrema, and any inflection points. Use your information to graph f (x).
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Math 142 WIR, copyright Angie Allen, Spring 2013
4 + e2x
.
x→∞ 1 + 6e−x + 2e−3x
14. Find lim
15. An appliance store sells 200 microwaves monthly. If the store invests $x thousand in an advertising campain,
the ad company estimates that sales will increase to
N(x) = 3x3 − 0.25x4 + 200
where 0 ≤ x ≤ 9.
When is the rate of change of sales with respect to advertising increasing and when is it decreasing? What
is the point of diminishing returns and the maximum rate of change of sales? Sketch a graph of N(x) and
N ′ (x) and locate the point of diminishing returns.