Question : The total cost to produce x units of perfume is C(x) = (6x + 3)(9x + 4) : 2151601
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) The total cost to produce x units of perfume is C(x) = (6x + 3)(9x + 4). Find the marginal average cost function.
A) 54 - (12/x2)
B) 108x + 51
C) 108 - (51/x)
D) 54x + 51 + (12/x)
2) The total profit from selling x units of cookbooks is P(x) = (9x - 6)(7x - 9). Find the marginal average profit function.
A) 63x - 123
B) 63x - 54
C) 63 - (123/x2)
D) 63 - (54/x2)
3) The demand function for a certain product is given by:
D(p) = (8p + 200/5p + 15).
Find the marginal demand D'(p).
A) D'(p) = (-880/5p + 15)
B) D'(p) = (-880/(5p + 15)2)
C) D'(p) = (1120 + 80p/(5p + 15)2)
D) D'(p) = (880/(5p + 15)2)
4) A rectangular steel plate expands as it is heated. Find the rate of change of area with respect to temperature T when the width is 1.9 cm and the length is 2.6 cm if dl/dt = 1.0 x 10-5 cm/°C and dw/dt = 8.4 x 10-6 cm/°C.
A) 1.9 x 10-5 cm2/°C
B) 4.9 x 10-5 cm2/°C
C) 0.8 x 10-5 cm2/°C
D) 4.1 x 10-5 cm2/°C
5) The total revenue for the sale of x items is given by:
R(x) = (170√(x)/9 + x3/2).
Find the marginal revenue R'(x).
A) R'(x) = (85(9x-1/2 - 2x)/9 + x3/2)
B) R'(x) = (85(9x1/2 - 2x)/(9 + x3/2)2)
C) R'(x) = (85(9x-1/2 + 4x)/(9 + x3/2)2)
D) R'(x) = (85(9x-1/2 - 2x)/(9 + x3/2)2)
6) Murrel's formula for calculating the total amount of rest, in minutes, required after performing a particular type of work activity for 30 minutes is given by the formula R(w) = (30(w - 4)/w - 1.5), where w is the work expended in kilocalories per min. A bicyclist expends 7 kcal/min as she cycles home from work. Find R'(w) for the cyclist; that is, find R'(7).
A) 2.98 min2/kcal
B) 16.36 min2/kcal
C) 2.48 min2/kcal
D) 1.98 min2/kcal
7) Prairie dogs form an important part of the coyote's diet. As coyotes are hunting for prairie dogs, they must be careful to expend just the right amount of time at each burrow. If a coyote spends too little time at each burrow, it catches very few prairie dogs per kilocalorie of energy expended. Likewise, if the coyote spends too much time digging at a single burrow, it can expend a large amount of energy per prairie dog caught. The relation between energy expended and time spent at each burrow is approximated by E = ((1/t) + (20/t - 0.75) )t2 for t > 0.75 minutes, where t is in minutes and E is in kcal expended per prairie dog caught. How much time should a coyote spend at each burrow to minimize the energy expended per prairie dog caught. (Hint: pay close attention to the domain of the above function.)
A) 1.5 minutes
B) 10 minutes
C) .8 minutes
D) 2.0 minutes
8) Assume that the temperature of a person during an illness is given by:
T(t) = (9t/t2 + 1) + 98.6,
where T = the temperature, in degrees Fahrenheit, at time t, in hours. Find the rate of change of the temperature with respect to time.
A) (dT/dt) = (9(1 - t2)/(t2 + 1)2)
B) (dT/dt) = (9/t2 + 1)
C) (dT/dt) = (9(1 - t2)/t2 + 1)
D) (dT/dt) = (9(t2 - 1)/(t2 + 1)2)
9) The population P, in thousands, of a small city is given by:
P(t) = (300t/2t2 + 5).
where t = the time, in months. Find the growth rate.
A) P'(t) = (300(5 - 2t2)/(2t2 + 5)2)
B) P'(t) = (300(2t2 - 5)/(2t2 + 5)2)
C) P'(t) = (300(5 - 2t2)/2t2 + 5)
D) P'(t) = (300(5 + 6t2)/(2t2 + 5)2)
Provide an appropriate response.
10) True or false? The derivative of the product of two functions is the product of their derivatives.
11) True or false? If average product is increasing then the marginal product must be increasing.
12) True or false? If marginal cost is decreasing then the average cost must be decreasing.
13) True or false? If the average cost is decreasing then the marginal cost must be less than the average cost.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) Revenues of a company are increasing. One analyst says it is due to a decrease in sales. Is this necessarily true? Explain.
15) What is true when marginal revenue and marginal cost are equal?
16) What must be true about a demand function so that, at a given price per item, revenue will decrease if the price per item is increased?
17) Prove that if average product is increasing, then marginal product is more than average product.
18) If g(2) = -1, g'(2) = 5, f(2) = -1, and f'(2) = 3, what is the value of h'(2) where h(x) = f(x)g(x)? Show your work.
19) Find the error that was committed below when taking the derivative of f(x) = (2x + 11/x2 + 22). Be specific.
Dx((2x + 11/x2 + 22)) = (2(x2 + 22) + (2x + 11)(2x)/(x2 + 22)2) = (6x2 + 22x + 44/(x2 + 22)2)