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Chapter 3 | Derivatives
160. The cost function, in dollars, of a company that manufactures food processors is given by C ( x ) =200+ 7 x + x 2 7 , where x is the number of food processors manufactured. a. Find the marginal cost function. b. Use the marginal cost function to estimate the cost of manufacturing the thirteenth food processor. c. Find the actual cost of manufacturing the thirteenth food processor. 161. The price p (in dollars) and the demand x for a certain digital clock radio is given by the price–demand function p =10−0.001 x . a. Find the revenue function R ( x ). b. Find the marginal revenue function. c. Find the marginal revenue at x =2000 and 5000. 162. [T] A profit is earned when revenue exceeds cost. Suppose the profit function for a skateboard manufacturer is given by P ( x ) =30 x −0.3 x 2 −250, where x is the number of skateboards sold. a. Find the exact profit from the sale of the thirtieth skateboard. b. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard. 163. [T] In general, the profit function is the difference between the revenue and cost functions: P ( x ) = R ( x )− C ( x ). Suppose the price-demand and cost functions for the production of cordless drills is given respectively by p =143−0.03 x and C ( x ) = 75,000 + 65 x , where x is the number of cordless drills that are sold at a price of p dollars per drill and C ( x ) is the cost of producing x cordless drills. a. Find the marginal cost function. b. Find the revenue and marginal revenue functions. c. Find R ′(1000) and R ′(4000). Interpret the results. d. Find the profit and marginal profit functions. e. Find P ′(1000) and P ′(4000). Interpret the results.
164. A small town in Ohio commissioned an actuarial firm to conduct a study that modeled the rate of change of the town’s population. The study found that the town’s population (measured in thousands of people) can be modeled by the function P ( t ) = − 1 3 t 3 +64 t +3000, where t is measured in years. a. Find the rate of change function P ′( t ) of the population function. b. Find P ′(1), P ′(2), P ′(3), and P ′(4). Interpret what the results mean for the town. c. Find P ″(1), P ″(2), P ″(3), and P ″(4). Interpret what the results mean for the town’s population. 165. [T] A culture of bacteria grows in number according to the function N ( t ) =3000 ⎛ ⎝ 1+ 4 t t 2 +100 ⎞ ⎠ , where t is measured in hours. a. Find the rate of change of the number of bacteria. b. Find N ′(0), N ′(10), N ′(20), and N ′(30). c. Interpret the results in (b). d. Find N ″(0), N ″(10), N ″(20), and N ″(30). Interpret what the answers imply about the bacteria population growth. 166. The centripetal force of an object of mass m is given by F ( r ) = mv 2 r , where v is the speed of rotation and r is the distance from the center of rotation. a. Find the rate of change of centripetal force with respect to the distance from the center of rotation. b. Find the rate of change of centripetal force of an object with mass 1000 kilograms, velocity of 13.89 m/s, and a distance from the center of rotation of 200 meters. The following questions concern the population (in millions) of London by decade in the 19th century, which is listed in the following table.
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