MATH 142 Business Math II, Week In Review
Spring, 2015, Problem Set 6 (5.2, 5.3, 5.4)
JoungDong Kim
1. Find f ′′ (x)
(a) f (x) = 4x5 + x2 + x + 3
(b) f (x) =
1
x+1
(c) f (x) =
√
x2 + 1
2. If f ′′ (x) = x(x − 2)4 (x − 4) is given, find all inflection values, find the largest open intervals on
which the graph of f is concave up and down.
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3. Find where the function is increasing, decreasing, concave up, and concave down. Find critical
points, inflection points, and where the function attains a relative minimum or relative maximum.
Then use this information to sketch a graph.
(a) f (x) = 1 − 9x + 6x2 − x3
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(b) f (x) = x2 · ex
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4. Find the limits.
x3 − 1
x→∞ 2x4 + 1
(a) lim
(b) lim
√
x
x→∞
2x2 + 1
x→∞ 3x2 − 1
(c) lim
x4 − x2 + x − 1
x→−∞
x3 + 1
(d) lim
(e) lim (1 + 2e−x )
x→∞
(f) lim (3 − 2ex )
x→−∞
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x→∞ 2 + e−x
(g) lim
2
x→∞ 1 + ex
(h) lim
ex − e−x
x→∞ ex + e−x
(i) lim
ex − e−x
x→−∞ ex + e−x
(j) lim
(k) lim
x→−∞
r
1−
1
x2
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5. The profit equation for a firm after the introduction of a new product is given by
P (t) =
5t4 + 0.1
t4 + 1
where P is profit in millions of dollars and t is years. What happens for large time?
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6. Use the graphing strategy to sketch a graph of f (x) = x4 − 2x3 + x2 .
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