2. Find the marginal profit if a profit function is (2x 2 4x + 4)e 4x and simplify.

Transcription

1 Additional Review Exam 2 MATH 2053 The only formula that will be provided is for economic lot size (section 12.3) as announced in class, no WebWork questions were given on this. km q = 2a Please note not all questions will be taken off of this. Study homework and in class notes as well! Note there is another worksheet with Section 11.8 problems (e x and ln x derivatives) see this document for more problems! Finally, there are not optimization problems in this packet, these can be found in the extra practice from section Find g (x) when g(x) = 2e 4x+1 2. Find the marginal profit if a profit function is (2x 2 4x + 4)e 4x and simplify. 1

2 3. Find the rate of change of a population if the population is 8e 4x 5x 2, simplify. 4. Find the derivative of 4 6x+1 5. Find the equation of the tangent line to f(x) = e 2x + 3 and x = 0. 2

3 6. Th sales of a new high-tech item are given by S(t) = e 0.4t where t represents time in years. find the rate of change of sales after 5 years. 7. Using data in a car magazine, we constructed the mathematical model y = 100e t for the percent of cars of a certain type still on the road after t years. Find the percent of cars on the road after 5 years. Find the rate of change of the percent of cars on the road after 5 years. (Be clear with which number is which answer!) 8. Find the derivative of the following functions: (a) ln(5 4x) (b) 4 ln(4x) 4 + 5x 3

4 (c) log(6x) 9. The cost function (in dollars) for q units of a certain item is C(q) = 102q The revenue function (also in dollars) for the same item is (a) Find the marginal profit. R(q) = 102q + 54q ln(q). (b) Find the marginal profit when 8 units are sold. 4

5 10. A manufacturer sells video games with the following cost and revenue functions (in dollars), where x is the number of games sold C(x) = 0.17x x 3 R(x) = 0.554x x 3 Determine the interval(s) on which the profit function is increasing. 11. Assume that a demand equation is given by p = 148 q. Identify the open interval(s) for 0 q 148 where revenue is decreasing. 5

6 12. The projected year-end assets in a collection of trust funds, in trillions of dollars, where t represents the number of years since 2000, can be approximated by the following function where 0 t 50. A(t) = t 3 4.5t t Identify the open interval for 0 t 50 where A(t) is increasing (round to one decimal place as needed). 13. Determine the critical number(s) of the function graphed. 14. The function graphed is f (x), the derivative of f(x). List the intervals where the function f(x) is increasing. 6

7 15. Find the location(s) of all relative maxima of y = f(x) graphed. 16. Suppose the graph is f (x) which is the derivative of f(x). Find the location(s) of all relative extrema of f(x) and state whether each is a relative maximum or minimum. 17. Find the critical numbers for the function f(x) = 4x 3 3x 2 36x + 5 7

8 18. Suppose the total cost C(x) in dollars, to manufacture a quantity x of weed killer, in hundreds of liters, is given by C(x) = x 3 3x 2 + 6x Where is the total cost increasing? Where is it decreasing? 19. The cost (in dollars) of producing q headphones is given by C(q) = 3q 2 3q Identify the open interval where the average cost is increasing. 20. Find all relative extrema and locations of G(x) = x 3 + 3x x + 5 and tell whether each is a relative max or min. 8

9 21. Suppose f(x) is a function whose derivative is given by f (x) = 2(x + 3)(x 4) Find the locations of all relative extrema of f(x) adn tell whether each is a relative maximum or minimum. 22. Find the absolute minimum value of the function over the interval [0, 2] for f(x) = x 3 + 2x 2 4x The total profit P (x) (in thousands of dollars) from the sale of x hundred thousand pillows is approximated by P (x) = x x x 300 for x 5. Find the number of pillows that must be sold to maximize profit. 9

10 24. Find the minimum value of the average cost for the function over the interval (0, 10]. C(x) = x x Suppose that the cost function for a product is given by C(x) = 0.003x 3 + 7x + 10,629. Find the production level that will produce the minimum average cost per unit. 26. Consider the cost function C(q) = q and the price function p = 96 2q. Find the number, q, of units that produces maximum profit. 10

11 27. Consider the cost function C(q) = q and the price function p = 67 2q find the maximum profit. 28. A manufacturer of porcelain sinks has an annual demand of 12,300 sinks. It costs $2.25 to store a sink for a year, and it costs $350 to set up a factory to produce each batch. Find the number of batches of sinks that should be produced for the minimum total production cost. 29. A regional market has a steady annual demand for 15,900 cases of beer. It costs $3.25 to store a case for a year. The market pays $5.50 or each order that is placed. Find the number of orders for cases of beer that should be placed each year. 30. A large hospital has an annual demand for 50,000 booklets on healthy eating. It costs $0.75 to store one booklet for a year, and it costs $80 to place an order for a new batch of booklets. Find the number of orders that should be placed each year. How many booklets will be ordered per order? (No formula for the second part, think!) 11

12 31. (Implicit differentiation seems to be a difficult concept. These are straight forward questions. Make sure that you know the calculus behind this. However, make sure that you can do the applied business concepts with other questions found below.) Find dy for the following functions. dx (a) x 2 y 4 = 19 (b) 3 ln x = 4 ln y (c) x 3 e y = 13 (d) 5 x = 2 y

13 32. Find dy at the given points for the following functions. (note you need to use implicit differentiation) dx (a) 2x 2 3y 3 = 5x 3y at (0, 1) (c) 2x 1 = 3 2y 1 at (1, 1) (d) 5x 6 y 2 = 6 2x 6y 2 at (0, 1) 13

14 33. A food truck owner has approximated demand for burritos by 10p + 3q 2 = 75 where p is price in dollars and q is the quantity demanded, in hundreds. Use implicit differentiation to find and interpret dq when q = 3. dp 34. The demand to download a hit single by a certain band can be approximated by 100p 2 + 9q 2 = 900 where p is the price in dollars and q is the quantity demanded, in thousands. Find and interpret dq dp when the price is $1.80 and the demand is 8000 downloads. 14

15 35. For PepsiCo, Inc., the relationship between revenue R in millions of dollars, and n, the number of employees, in thousands can be approximated by 4R 2 = 961n 3. In 2007, PepsiCo, Inc., generated revenue of $39 billion, and had 185,000 employees, and was adding employees at the rate of 13,000 employees per year. Find the rate of change of revenue with respect to time. (Use units in your answer.) 36. During the 2012 NFL regular season, a team s winning percentage W (wins divided by total games, expressed as a decimal) and turnover differential T (turnovers taken away minus turnovers given away) can be roughly approximated by the equation 5T 211W = 105. Suppose that a team s turnover differential dropped by 2 turnovers one week. Find the rate of change of winning percentage with respect to time.(use units in your answer.) 15

16 37. A man who is 5 feet 10 inches tall is losing weight at the rate of 4 pounds per month. For a person of this height, the relationship between body mass index B and weight w (in pounds) is given by 4900B 703w. Find the rate of change of body mass index with respect to time. (Use units in your answer.) 38. The campus bookstore has estimated that its profit (in dollars) from selling x hundred basketball conference championship t-shirts is given by P = 48x x 528. The demand is currently 400 t-shirts, but euphoria over the championship is subsiding so the demand is dropping by 100 t-shirts per day. How is the profit changing with respect to time. (Use units in your answer.) 16

17 39. An oil production company is concerned about an oil slick that is forming on the surface of the water in the Gulf of Mexico. The slick is roughly in the shape of a circle with area A = πr 2 square kilometers where r is the radius. Find the rate at which the area of the oil slick is changing at the instant when the radius is 2 kilometers and increasing at 0.3 kilometers per day.(use units in your answer.) 40. During a local political race, the demand x (in thousands) for campaign buttons is related to the price p (in dollars) by the equation 25p x 2 = If the quantity demand is currently 3000 at a price of $5.60 and demand is increasing by 400 buttons per month, find the rate of change of price with respect to time. (Use units in your answer.) 17