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t3(4)

t3(4) - (a Let p denote the price per job and x the number...

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Methods of Calculus MAC3233 Test 3 March 25, 1996 Instructions: This is a closed-book, closed-notes test. Do not write your answers on this paper, use separate answer sheets. No calculators are allowed to be used. To receive credit, you must show all work . For the optimization problems, you are required to show why your answer is a minimum or maximum. Some numerical answers may involve the number e . (1) (10 points each) Find f 0 ( x ) for the following (a) f ( x ) = (3 x 4 - 2 x + 1)( x + 4) 3 (b) f ( x ) = 3 x + 2 x 4 9 x 2 - 1 (c) f ( x ) = ( e x - 1 + x - 2 ) 4 (d) f ( x ) = e (3 x +5) (2) (15 points) Find ∂f ∂x and ∂f ∂y for f ( x ) = x + e xy (3) (15 points) f ( x ) = (4 x - 1) e 3 x . (a) f 0 ( x ) = ? (b) f 00 ( x ) = ? (c) Find the x coordinate of each possible extremum. (d) Use f 00 ( x ) to say whether each answer in (c) is a relative maximum or minimum. (4) (15 points) When the price for cleaning roofs was \$150, the Acme Roof Cleaning Service had 40 roof cleaning jobs per month. When the rate was raised to \$160, the number of jobs dropped to 38 per month.
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Unformatted text preview: (a) Let p denote the price per job, and x the number of jobs per month. Assuming a linear demand curve, write the formula for the demand curve. (b) Write the formula for the monthly revenue function. (c) Suppose the company has ﬁxed costs of \$500 per month and the vari-able cost is \$30 per job. Write the formula for the monthly proﬁt function. Find the price per job that will maximize the company’s monthly proﬁt. (5) (15 points) Suppose a ﬁrm’s proﬁt from producing and selling x units of a product is given by P ( x ) = 32 x 2-9 x 3 thousands of dollars. (a) Find the marginal proﬁt, dP dx . (b) The production level at t weeks from the present is expected to be x = 4 + 3 t . Find the time rate of change of proﬁt, dP dt . (c) How fast (with respect to time) are proﬁts changing when t = 4? 1...
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