MATH 2070 Final Exam Mixed Practice (MISSING 6.5 & 6.6)

Transcription

1 The profit for Scott s Scooters can be estimated by P(x, y) = 0.4x + 0.1y + 0.3xy + 2y + 1.5x hundred dollars, where x is the number of mopeds sold and y is the number of dirt bikes sold. (Check: P(3,2) = 14.3) Use this to answer the next three questions. 1. Find and interpret P(14,10). a. When 10 mopeds and 14 dirt bikes are sold, the profit is $14,460. b. When 14 mopeds and 10 dirt bikes are sold, the profit is $14,460. c. When 14 mopeds and 10 dirt bikes are sold, the profit is $17,140. d. When 10 mopeds and 14 dirt bikes are sold, the profit is $17, How many dirt bikes must be sold if Scott wants to make a profit of $19,500 and has sold 10 mopeds? a. 15 b. 20 c. 215 d How quickly is profit increasing with respect to the number of dirt bikes sold when he sells 2 mopeds and 6 dirt bikes? a. 3.8 hundred dollars per dirt bike b. 4.9 hundred dollars per dirt bike c. 3 hundred dollars per dirt bike d. 8.1 hundred dollars per dirt bike 4. Consider the function f(x) = () (). Check: f(3) = Evaluate the definite integral f(x)dx. a b c d A scooter has been traveling at a rate of s(x) = 4x miles per hour, where x is the number of hours since noon. What distance did the scooter travel between 2 pm and 4 pm? a. 8 miles b. 16 miles c. 24 miles d. 32 miles A major entertainment complex has determined that they will invest R(t) = t million dollars per year t years from now, where 0 t 10. Assume a continuous income stream and an APR of 4%, compounded continuously. Use this information to answer the next two questions. 6. Find the 10-year future value of the investment. a million dollars b million dollars c million dollars d million dollars 7. Find the 10-year present value of their investment. a million dollars b million dollars c million dollars d million dollars

2 Use the function f(x) = x 6x + 5 as graphed to answer the next three questions. (Check: f(3) = 4) 8. Find the total signed area of the shaded regions of the graph of f(x). a. 13 b. 13 c. 25/3 d. 25/3 9. The function F(x) is an accumulation function of f(x). If F(0) = 4/3, what is the value of F(5)? a. 25/3 b. 7 c. 0 d. 43/3 10. Which of the following expressions gives the total area trapped between the graph of f(x) and the x-axis between x = 1 and x = 6. a. f(x)dx f(x)dx + f(x)dx b. f(x)dx + f(x)dx + f(x)dx c. f(x)dx d. f(x)dx 11. It is estimated that it will require a continuous income stream of R(t) = 850(1.03 ) thousand dollars per year to support a local homeless shelter t years from now, where 0 t 5. A very wealthy philanthropist is willing to provide the funds to support the shelter for the next 5 years, but he intends to make a single, lump-sum, donation instead of making a continuous stream of payments. Assuming his donation will be invested in an account earning 3% interest, compounded continuously, how large does his donation need to be? Round to the nearest dollar. a. $3,942,149 b. $4,245,316 c. $4,932,353 d. $5,321,344 The size of an insect population within a controlled experiment is growing at a rate of g(t) = 10ln(1.02) (1.02 ) hundred insects per day, t days after the experiment began, 0 t 15. Let A denote the average value of g(t) from 0 to 15, as illustrated by the dotted line on the graph. Use this information to answer the next two questions. 12. Calculate A. a b c d Which of the following correctly completes the interpretation of A? From day 0 to day 15,. a. the average size of the insect population was A hundred insects b. the size of the insect population increased by A hundred insects c. the average size of the insect population increased by A hundred insects per day d. the size of the insect population increased by an average of A hundred insects per day

3 A(t, w) thousand students denotes the number of applicants to a major land-grant university when the tuition is t thousand dollars and the university won w football games in the previous year. Some values associated with A(t, w) are given: A(12,8) = 16.2, = 1.4, = 2.2 Use this information to answer the next five questions. (,) (,) 14. Interpret = 1.4 When tuition is $12k and 8 football games were won in the previous year,. (,) a. the number of applicants is decreasing by 1400 students per year. b. the tuition is decreasing by 1400 dollar per football game won. c. the number of applicants is decreasing by 1400 students per football game won. d. the number of applicants is decreasing by 1400 students per thousand dollars of tuition. 15. Estimate the number of applicants to the university if the tuition is 14 thousand dollars and the number of football games won in the previous year is 8. a. 13,400 students b. 14,800 students c. 19,000 students d. 20,600 students 16. Suppose the university wishes to keep the number of applicants fixed at 16,200 and it won 10 football games in the previous year. How should the tuition be adjusted? a. It should be decreased by about $3143. b. It should be increased by about $3143. c. It should be decreased by about $1273. d. It should be increased by about $ Specify the units for. a. thousand dollars per football game won b. football games won per thousand dollars c. thousand students per football game won d. thousand students per thousand dollars 18. Based on the given information, which of the following is a true statement? a. The point (12,8) is a relative extreme point (maximum or minimum) of A(t, w). b. The point (12,8) is a saddle point of A(t, w). c. The point (12,8) is a critical point of A(t, w), but there is not enough information to determine type. d. The point (12,8) is not a critical point of A(t, w). w(x) = 0.05x 0.4x thousand gallons represents the amount of water available to a farmer for irrigation purposes x days from the beginning of growing season, 0 x 90. Use this model to answer the next three questions. 19. Interpret () = During the first 30 days of the growing season,. a. there was an average of 19.8 thousand gallons available for irrigation b. there was enough water available to irrigate the crops for an average of 19.8 days c. the average amount of water available for irrigation increased by 19.8 thousand gallons d. the amount of water available for irrigation increased by an average of 19.8 thousand gallons per day 20. Find the average rate of change of w(x) from x = 0 to x = 90. a. 4.1 b. 369 c d Find the average value of w(x) from x = 0 to x = 90. a. 4.1 b c d

4 Entomologists studied the female and male populations of ladybugs in a botanical garden over an 8-month period. They found that the female population was changing at a rate of f(x) = 0.8x 8x + 25 thousand ladybugs per month and the male population was changing at a rate of m(x) = 0.6x + 3.6x thousand ladybugs per month, where x is the number of months since they first began observing. The graph of their finings is shown below. (check: f(3) = 8.2 and m(3) = 12.8) Use this graph and functions to answer the next five questions. 22. Interpret f(x)dx = In the first 5 months of observation,. a. the female population decreased by thousand ladybugs b. the female population increased by thousand ladybugs c. the female population decreased by thousand ladybugs per month d. the female population increased by thousand ladybugs per month 23. Which of the following calculates the total combined area of the shaded regions marked B and C? a. f(x) m(x) dx b. f(x) + m(x) dx.. c. f(x) + m(x)dx f(x) + m(x)dx.. d. m(x) f(x)dx + f(x) m(x)dx 24. Find the area of the shaded region of the graph marked A. a b c d Which of the following is the correct interpretation of area A found in the previous question? a. The increase in the female population was A thousand ladybugs greater than the increase in the male population during the first two months. b. The increase in the male population was A thousand ladybugs greater than the increase in the female population during the first two months. c. The female population was A thousand ladybugs greater than the male population after two months. d. The male population was A thousand ladybugs greater than the female population after two months. 26. Given that there were a total of 10 thousand ladybugs at the beginning of the observation, how many ladybugs were there after 2 months? a thousand ladybugs c thousand ladybugs b thousand ladybugs d thousand ladybugs

5 f(x) customers per hour gives the rate of change of the number of customers in a shopping mall x hours after the mall opens. The graph of f(x) is shown. Use this information to answer the next two questions. 27. How did the number of customers in the mall change, on average, over the first 3 hours of operation? a. The number of customers decreased, on average, by 20 customers per hour. b. The number of customers increased, on average, by 11 customers per hour. c. The number of customers increased, on average, by 25 customers per hour. d. Not enough information provided. 28. What was the average number of customers in the shopping mall during the first 3 hours of operation? a. 20 customers c. 42 customers b. 25 customers d. Not enough information provided 29. The function f(x, y) has a critical point at ( 2,0) with a corresponding output of f( 2,0) The second partial derivatives matrix of f(x, y) is given below. 2x 8y Use the determinant test to classify the critical point ( 2,0), if possible. 8y 8x a. The point ( 2,0) is a relative maximum c. The point ( 2,0) is a saddle point b. The point ( 2,0) is a relative minimum d. The determinant test is inconclusive The number of bacteria in a lab sample is increasing at a rate of b(x) = + e. 1 million bacteria per day, x days after the beginning of an experiment, 1 x 3. Check: b(2) = Use this information to answer the next four questions. When asked to find the area between b(x) and the horizontal axis from x = 1 to x = 3, rounded to 3 decimal places, a student decided to draw rectangles and increase the number of rectangles each time. When he drew 10 midpoint rectangles, the sum of their areas was When he drew 20, the sum was If he drew only 4 midpoint rectangles, what would the sum of their areas be? a b c d What would the area be if he drew an infinite number of rectangles? a b c d What are the units for the value found in the previous question? a. million bacteria per day b. days c. million bacteria per rectangle d. million bacteria 33. What you found in question 32 is equivalent to which of the following? a. b(x)dx b. b(x)dx c. b(3) b(1) d. b(1) b(3)

6 At a price of p dollars per wafer, the demand for Soylent Green is D(p) = 3(0.95 ) million wafers. Use this model to answer the next five questions. Check: D(7) = If Soylent Green is selling at a price of $12 per wafer, what is the consumers surplus? a million wafers b million wafers c million wafers d million wafers 35. Find the market price when 2.5 million wafers are demanded. a. $2.64 b. $3.55 c. $7.04 d. $ Given that D (p) = 3 ln(0.95) (0.95 ), find the price at which demand is unit elastic. a. $0.05 b. $2.85 c. $19.50 d. $ When the price of Soylent Green is $23, calculate the elasticity of demand. a. η = b. η = c. η = d. η = Classify the elasticity of demand, if possible when the price of Soylent Green is $23. a. Demand is elastic c. Neither elastic nor inelastic b. Demand is inelastic d. Not enough information f(x, y) million dollars gives a delivery company s profit when the company employs x hundred drivers and y hundred customer service representatives. A contour graph of f(x, y) is shown below. Use this information to answer the next four questions. 39. If the company currently employs 5 hundred drivers and 1 hundred customer service representatives, which one of the following would result in the greatest increase in profit? a. decreasing the number of drivers by 50 b. decreasing the number of customer service representatives by 50 c. increasing the number of drivers by 50 d. increasing the number of customer service representatives by If the company employs 3 hundred drivers and would like to see 50 million dollars in profit, how many customer service representatives should they employ? a. 1.1 hundred b. 2.6 hundred c. 3.3 hundred d. not possible

7 41. Which of the following best completes the interpretation of critical point A? When the company employs 500 drivers and 225 customer service representatives, their profit is at a. a. relative minimum of -5 million dollars b. relative minimum of 5 million dollars c. relative maximum of -5 million dollars d. relative maximum of 5 million dollars 42. How many critical points does the graph of f(x, y) indicate? a. two b. three c. four d. five Suppose the delivery company referred to in the previous set of problems finds it necessary to limit the number of employees according to g(x, y) = 0.55x + y = 0.1. The constraint has been sketched on the graph below. 43. How many drivers and customer service representatives should the company employ under the constraint so as to maximize profit? a. 490 drivers and 290 customer service representatives b. 150 drivers and 40 customer service representatives c. 100 drivers and 65 customer service representatives d. 100 drivers and 70 customer service representatives An MP3 song is downloaded from the internet, and a graph of the download rate for the first 10 seconds is given below. Use this information to answer the next two questions. 44. Four seconds after the download starts, the amount of data downloaded is. a. increasing least rapidly b. increasing most rapidly c. decreasing most rapidly d. at a relative minimum

8 45. The total amount of data downloaded is best represented by which of the following graphs? The producers of football jerseys will supply S(p) = 0 for p < 10 p 3p + 20 for p 10 thousand jerseys when the jersey are sold for p dollars each. Check: S(18) = 290 Use this model to answer the next two questions. 46. How many jerseys are supplied when the market price is $45 per jersey? a thousand jerseys b thousand jerseys c thousand jerseys d thousand jerseys 47. Calculate the producer s surplus when the price of a jersey is $45. a thousand dollars c thousand dollars b thousand dollars d thousand dollars The graph of a function f(t) is given below. Let F(x) = f(t)dt define a particular accumulation function of f(t). Answer the next three questions about this accumulation function. 48. Which one of the following statements correctly describes the behavior of the accumulation function F(x) as you move left to right across the interval 0 < x < 2? a. F(x) decreases slower and slower b. F(x) decreases faster and faster c. F(x) increases slower and slower d. F(x) increases faster and faster

9 49. Which of the following is NOT an accurate statement about the accumulation function? a. F(x) decreases on the interval 2 < x < 5. b. F(x) has a relative minimum at x = 2. c. F(x) has a zero at x = 5. d. F(x) has no inflection points. 50. Given the graph of f(t), which one of the following is the graph of F(x) = f(t)dt? 51. Given that f(x) =, use the numerical method to evaluate f(x)dx. Check: f(2) ( ) a. Diverges b. 0 c d The 550 contour curve for the function f(m, p) = 2m p + mp is shown in the graph. Also shown is a line tangent to the contour curve at the point (5,10). Use this information to answer the next two questions. 52. Calculate the slope of the line tangent to the curve at the point (5,10). Round to 3 decimal places. a b c d From the point (5,10), which of the following estimates how much p would need to change to compensate for a unit decrease in m if the value of f is to remain at 550? a. Δp (,) b. Δp (,) c. Δp (,) d. Δp (,)

10 54. An oil well is producing oil at a rate of r(x) thousand barrels per year x years after production begins. Interpret r(x)dx = 200. a. The well will continue to produce oil at the current rate for 200 years. b. Eventually, the well will produce 200 thousand barrels of oil. c. The well will produce 200 thousand barrels of oil per year forever. d. The well will never produce 200 thousand barrels of oil. 55. A farmer recently purchased a tract of land. He subdivided this land into ¼ acre plots, and measured the height in feet above sea level at the midpoint of each plot. The data is given in the table below, where the rows and columns are indicated with letters. a. Draw and label the 5.0 and 10.0-foot contour curves on the data table. (Hint: There are ft contours.) b. The table indicates a relative in column and row. c. The table indicates a saddle point in column and row. 56. The number of rabbits living on the UVic campus was increasing at a rate of r(t) = 18.42(1.771 ) rabbits per year, t years after 1980, 0 t 4. Check: r(4) = a. Find r(t)dt = b. Use the fact that there were 60 rabbits in 1981 to find R(t), the specific anti-derivative of r(t). Complete the model and show your work. R(t) = (units) gives the t years after 1980, 0 t 4. c. Show how each function can be used to find the change in the number of rabbits between 1981 and (You only need to write the notation. It is not necessary to complete the calculation.) Using r(t): Using R(t): d. How many rabbits were on the campus in 1984?

11 57. C(m, t) gives the average temperature in Clemson, SC, where m is the month (Jan=1, Feb=2, etc.) and t is the number of years after a. Find a cross-sectional model for the average temperature in Clemson, SC as a function of m in the year Use appropriate notation and report your model with coefficients rounded to 3 decimal places. (Assume the data is best represented by a quadratic model.) b. Use the unrounded model found in part (a) to estimate the average temperature in July of Round your answer to 3 decimal places and include units. 58. Evaluate the following improper integral. If the integral diverges state or, as appropriate, for your final answer. dx 59. Given g(x, y) = ye + 12x ln(y) + 6, find the following. It is not necessary to simplify your answer. g = = 60. Write a simplified value or algebraic expression for each of the following. a. (5t + 6t)dt b. 4t dt

12 61. The College of Western Idaho posted net profits of 125 million dollars in During the 5-year period from January 1, 2011 to January 1, 2016, they will continuously invest some or all of their profits, as described below, in an account bearing 2% interest, compounded continuously. Write the function, R(t), that describes the continuous income stream for their investment t years after the beginning of 2011 for each scenario. Include units. a. Assume the profit will decrease by 0.05 billion dollars per year and they will invest 60% of their profit. R(t) = b. Assume the profit will increase by 7% each year and they will invest all of their profit. R(t) = 62. Find each general anti-derivative. It is not necessary to simplify your answers. a. x + 4e dx b. 3x 4x 8 + dx 63. The first partial derivatives of the function f(a, b) are given by f = 4be and f = 4e + Use this information to find the following. It is not necessary to simplify.. a. f = b. f = 64. At the end of 2006 a company began continuously investing 1.5 million dollars per year into an account returning 4% APR, compounded continuously. At the end of 2011 they will use the money from this investment account to expand their company. How much money will they have available for the expansion? Show your work and include units with your answer.

13 65. The rate of change in the annual profit of a major restaurant chain from 1970 through 2000 can be modeled by the equation p(x) = 0.46x x thousand dollars per year, x years after a. On the graph provided, sketch and shade three right rectangles of equal width that can be used to estimate the signed area between p(x) and the x axis from x = 0 to x = 30. b. Let w refer to the width of each rectangle and h, h, and h refer to the signed height of each rectangle you were asked to sketch (in order from left to right). Calculate each value and specify the units. width units: height units: w = h = h = h = c. Use these three rectangles to estimate and interpret the signed area between p(x) and the x axis from x = 0 to x = The quantity of CPUs that producers will supply is modeled by S(p) = p 0.01p + 24 thousand CPUs when p 50, 0 otherwise, where p dollars per CPU is the price. The quantity of CPUs demanded by consumers is modeled by D(p) = p 0.07p + 54 thousand CPUs, where p dollars per CPU is the price. a. Find each price and quantity indicated below. The market equilibrium price is $. The market equilibrium quantity is thousand CPUs. The price at which consumers will no longer purchase CPUs is $. The minimum price which producers are willing to accept for CPUs is $. b. Fill in each of the prices and quantity found above on the graph shown below.

14 c. Find the amount producers are willing and able to receive when the market price is $80 per CPU. Show your work. Shade the corresponding area on the graph below. d. Find the amount consumers are willing and able to spend at market equilibrium. Show your work. Shade the corresponding area on the graph below. e. Find Total Social Gain. Show your work. Shade the corresponding area on the graph below. 67. Use the algebraic method to find the EXACT value of the definite integral below. Clearly show each step in the process. Use proper mathematical notation. Simplify your answer as much as possible without converting to decimal form. 4 dx

15 68. Tailgate Essentials sells two types of tailgate tents, one plain tent and one with a team logo. The profit from the sale of these tents is given by P(x, y) = 83.45x y 0.17xy 0.22x 0.34y dollars, when x dollars is the price of a plain tent and y dollars is the price of a tent with a team logo. Check: P(40,60) = 0 a. Find the first partial derivatives of P(x, y). P = P = b. Write out the system of equations that can be used to find the critical point of P(x, y). c. Solve the system to find the critical point of P(x, y). Show all work. Round to the nearest cent. Critical point: x = $ y = $ P = $ d. Find the second partials derivative matrix and the determinant value for the critical point of P(x, y). 2 nd Partial Derivatives Matrix: Determinant Value = e. Classify the critical point as a relative maximum, relative minimum, or saddle point. Show how your answers to part (c) support your conclusion.

16 69. Refer back to the profit model for Tailgate Essentials. Suppose Tailgate Essentials wants to restrict the price difference between tents to that logo tent price is $80 more than the price of a plain tent. The price constraint is given by g(x, y) = y x = 80. a. Set up the system of equations that can be used to find the optimal point of P(x, y) subject to the constraint. b. Solve the system of equations to find the new optimal point. Show all work. Round to the nearest cent. x = y = P(x, y) = λ = c. Use the close point test to classify the optimal point as either a maximum or a minim. The complete the table and the sentence that follows. Close point Optimal Point Close point x y P(x, y) If Tailgate Essentials requires that the price of a logo tent be $80 more than the price of a plain tent, they will have a (maximum or minimum) profit of $ when they sell plain tents for $ each and logo tents for $ each. d. Estimate the optimal profit earned by Tailgate Essentials if they increase the price difference they are willing to allow between the two tents by $10. (Do NOT re-solve the problem. Estimate using previous solution.) Note: 1. There may be questions on the final that are not included in this review. 2. Section 6.5 and 6.6 are not in the review, but will be on the final exam. 3. This review is longer than the length of your final exam. 4. This is the only document that has material from Chapters 5 8 mixed up together.