MATH 142 Business Math II, Week In Review
Spring, 2015, Problem Set 7 (Exam2 Review)
JoungDong Kim
1. Find f ′ (x) if f (x) = (5x − x2 + 4)10 (x3 + 5x2 + 10x)
2. Find f ′ (x) if f (x) = log7 [(3x + 7)4 (2x − 3)8 ]
3. The elasticity of product is 1.2 at a price level of $15. If the price is increase by 2%, the quantity
demanded will (increase/decrease) by
%.
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4. Use the given information to sketch the graph of f .
• f ′ (x) > 0 on (−∞, −6) ∪ (−2, 4)
• f ′ (x) < 0 on (−6, −2) ∪ (4, ∞)
• f ′′ (x) > 0 on (−4, 0)
• f ′′ (x) < 0 on (−∞, −4) ∪ (0, ∞)
f (x3 )
. If f (3) = 1, f ′ (3) = −2, g(3) = 4, g ′(3) = −5, f (27) = −1, and f ′ (27) = 6,
g(x)
what is h′ (3)?
5. Let h(x) =
2
6. The demand equation for a particular item that is currently selling for $4 is given by x = −0.05p2 +
20 for 0 ≤ p ≤ 20. What is the elasticity at this price level?
(x − 3)(x + 4)
. Find critical points,
(x − 8)4
and where the function is increasing, decreasing, and where the function attains a relative minimum or relative maximum.
7. Given f (x) is continuous over (−∞, 8) ∪ (8, ∞) and f ′ (x) =
3
8. Given that the domain of f (x) is all real numbers, use the graph of f ′ (x) below
a) On what interval(s) is f (x) increasing
b) On what interval(s) is f (x) decreasing
c) Where is the local extrema?
d) On what interval(s) is f (x) concave down?
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9. What is the absolute minimum value of f (x) = x3 ln x on (0.5, ∞).
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10. Find f ′ (x) if f (x) =
2x2 + 3x5 − 4x ln x
x
11. When a management training company prices its seminar on management techniques at $400 per
person, 1,000 people will attend the seminar. The company estimates that for each $5 reduction
in price, an additional 20 people will attend the seminar. How much should the company charge
for the seminar in order to maximize its revenue? What is the maximum revenue?
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12. The price-demand function for a particular product is p(x) = 508 − 5x where p(x) is the unit
price when x units are demanded. Use the marginal revenue function to approximate the revenue
realized from selling the 22nd item.
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13. Given the function f (x) = 1 − 9x + 6x2 − x3 , 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.
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14. Evaluate the following
2e−x + 3 − 5e4x
x→∞
3e4x
lim
90x2 − 3
x→−∞ 10x + π 4
15. Evaluate lim
16. If the domain of f (x) is all real numbers and f ′′ (x) = (x − 3)2 (x + 2). Find the value(s) of x
where any inflection point occur.
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