P t0 pt0 means the price has been declining p t0 pt0

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P (t)<0 P′(t)<0 means the price has been declining. P ′′ (t)>0 P″(t)>0 means that the curve is concave up-- P (t) P′(t) is increasing (becoming less negative) so prices are declining at a slower and slower rate. Question 11 1.5 / 1.5 pts A hot cup of coffee is left to sit on the kitchen counter. The temperature of the coffee is given by H=f(t) H=f(t), where H H is the temperature in degrees Fahrenheit and t t is the time in minutes since it was left on the counter. The coffee initially cools rapidly. As it approaches room temperature the rate of cooling slows down. Choose the mathematical expression that best matches this statement. f (t)<0 f′(t)<0 and f ′′ (t)=0 f″(t)=0 .
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f (t)<0 f′(t)<0 and f ′′ (t)<0 f″(t)<0 . f (t)>0 f′(t)>0 and f ′′ (t)>0 f″(t)>0 . Correct! f (t)<0 f′(t)<0 and f ′′ (t)>0 f″(t)>0 . f (t)>0 f′(t)>0 and f ′′ (t)<0 f″(t)<0 . Since the temperature is decreasing, f (t)<0 f′(t)<0. Since the rate of decrease is slowing, f (t) f′(t) is increasing (becoming less negative), and f(t) f(t) is concave up, so f ′′ (t)>0 f″(t)>0. Question 12 0 / 1 pts If the function f(t)<0 f(t)<0 on an interval then f (t) f′(t) is decreasing on that interval. You Answered True Correct Answer False
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False: The sign of f (t) f′(t) that tells us whether the function f(t) f(t) is increasing or decreasing. The sign of f(t) f(t) does not tell us about f (t) f′(t). To tell whether the first derivative, f (t) f′(t) is increasing or decreasing we need to know the sign of the second derivative, f ′′ (t) f″(t). Question 13 1 / 1 pts Marginal cost is the instantaneous rate of change of cost with respect to quantity. Correct! True False True. The instantaneous rate of change of a function at a point is the same as the derivative at a point; for the cost function, this is the marginal cost. Question 14 1.5 / 1.5 pts Let C(q) C(q) represent the cost and R(q) R(q) represent the revenue, in dollars, of producing q q items and let C (250)=150 C′(250)=150 and R (250)=145 R′(250)=145. Choose the statement that best matches this situation. Correct! The cost of producing the 251st item exceeds the revenue it will generate, so producing the 251st item will reduce the total profit.
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The marginal cost is greater than the marginal revenue when 250 items are produced, so total profit is negative when 250 units are produced. The revenue from the 251st item exceeds the cost to produce it, so producing the 251st item will increase the total profit. The marginal cost is greater than the marginal revenue, so the firm cannot make a profit. The marginal cost is greater than the marginal revenue when 250 items are produced, so profits are maximized when some quantity less than 250 is produced. Since C (250)>R (250) C′(250)>R′(250), the cost to produce the next unit exceeds the the revenue it generates, and producing that unit will decrease the total profit. The only data in the problem is about the increased cost and revenue for producing the 501st item. There is no information about whether total profits are maximized or whether total profits must be positive or negative.
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