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1. Determine where the following function is increasing and decreasing, and where
its graph is concave up and concave down. Find the relative extrema and inflection
points, and sketch the graph of the function
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(2) The first derivative of a certain function is
(a) On what ingerval is f increasing? Decreasing?
(b) On what intervals is the graph of f concave up? Concave down?
(c) Find the x coordinates of the relative extrema and inflection points of f
(d) sketch a possible graph of f(x)
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(3). Suppose the demand q and price p for a certain commodity are related by the
linear equation q=240-3p (for 0≤p≤80).
(a) Express the elastiticy of demand as a function of p
(b) Calculate the elasticity of demand when the price is p=60, interpret your answer
(c) Calculate the elasticity of demand when the price is p=20, interpret your answer
(d) At what price is the elasticity of demand equal to -1? What is the economic
significance of this price?
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(4). A manufacturer estimates that the marginal cost of producing q units of a certain
commodity is C’(q)=3q2-24q+48 dollars per unit. If the cost of producing 10 units is
$5,000, what is the cost of producing 30 units?
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(5). Evaluate the following integrals
(a)
(b)
(c)
(d)
(6)
(a) Sketch the given region bounded by the curves y=x2-2x and y=-x2+4
(b). Find the volume of the solid of revolution formed by rotating the specified region
R about x axis, where R is the region under the curve y=x2+1 from x=-1 to x=2.
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(7). Felip is trying to choose between two investment opportunities. The first will cost
$50,000 and is expected to produce income at the continuous rate of $15,000 per
year. The second will cost $30,000 and is expected to produce income at the rate of
$9,000 per year. If the prevailing rate of interest stays constant at 6% per year
compounded continuously, which investment is better for Felip over the next 5
years?
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8
A manufacturer estimates that when x units of a particular commodity are produced,
the market price p (dollars per unit) is given by the demand function p=7+50e-x/200,
(a) What market price corresponds to the production of x=0 units?
(b) How much revenue is obtained when 200 units of the commodity are produced?
(c) How much more (or less) revenue is obtained when x=100 units are produced
than when x=50 are produced?
9
Money is deposited in a certain bank doubles every 13 years. The bank compounds
interest continuously. What annual interest rate does the bank offer?
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10
Use logarithmic differentiation to find the derivative f’(x)
(a)
(b)
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