MATH 2070 Test 2 (Sections & )

Transcription

1 Multiple Choice: Use a #2 pencil and completely fill in each bubble on your scantron to indicate the answer to each question. Each question has one correct answer. If you indicate more than one answer, or leave a blank, the question will be marked as incorrect. In this section there are 14 multiple choice questions. Each question is worth 3 points for a total of 42 points. For future reference, circle your answers on this test paper as you will not receive your Scantron back with your test. The consumer demand for a 2.5-oz bag of cat treats is given by D( p) ln p thousand bags when p dollars per bag is the price of a 2.5-oz bag. Check: D(2) p 1.5(4.4) p.4 For the 2.5-oz bags of cat treats, the producer will supply S( p) p.4 thousand bags when p dollars per bag is the price of a 2.5-oz bag. Check: S(2) At what price will consumers no longer demand the 2.5-oz bag of cat treats? a. $.4 per bag b. $2.99 per bag c. $3.15 per bag d. There is no price at which consumers will no longer demand the cat treats. 2. What is the consumer expenditure when 2 thousand bags are purchased? a. 3.5 thousand dollars b. 6.1 thousand dollars c thousand dollars d thousand dollars 3. What is the producer surplus when the 2.5-oz bags of cat treats are sold for $ 2.25? a thousand dollars b thousand dollars c thousand dollars d thousand dollars P a g e 5 13

2 t The population of a state from 199 to 215 can be modeled as P( t) 7.21(1.315 ) hundred people, where t is the number of years since 199. Check: P(6) Use this information to answer the next three questions. 4. Find the average population of this state from 1998 to 22. a hundred people b hundred people c hundred people d hundred people 5. Find the average rate of change in the population of this state from 1998 to 22. a hundred people per year b hundred people per year c hundred people per year d hundred people per year 6. What is the correct interpretation of 5 P( t) dt 15.44? 5 a. From 199 to 1995, the population increased by hundred people. b. From 199 to 1995, the population increased an average of hundred people per year. c. From 199 to 1995, the average population was hundred people. d. From 199 to 1995, the average population increased by hundred people. P a g e 6 13

3 .1.1(2) 7. Evaluate the expression: N lim e lim e N N. a. b c d. Jo is planning to buy a house in five years. She plans to save for a down payment, and figures that she can invest R( t) 54 dollars per year into a savings account which offers 2.25% compounded continuously. Assume a continuous income stream. Use this information to answer the next three questions. 8. How much money will be in Jo s account at the end of the 5 year investment period? a. $ 38, b. $ 3, c. $ 28, d. $ 25, How much interest did Jo s account earn over the 5-year period? a. $ 67.5 b. $ 1, c. $ 3,4.7 d. $ 2, Assume Jo changes her mind and decides to make a one-time deposit today instead of investing continuously throughout the next 5 years. If she s still investing in the account that offers 2.25% interest compounded continuously, how much does she have to invest today to end up with the same amount of money at the end of the 5 year period? a. $ 24, b. $ 25, c. $ 27,. d. $ 38, P a g e 7 13

4 The rate at which people contract a virus during an epidemic is given by c( t ) people per week. The rate at which people recover from the virus is given by r( t ) people per week. For both functions, t is the number of weeks after the epidemic began. Use this information to answer the next two questions. 11. Which of the following notations represents the total area of the shaded regions in the graph above? 25 5 a. ( ) ( ) ( ) ( ) c t r t dt r t c t dt b. ( ) ( ) ( ) ( ) c t r t dt r t c t dt 25 5 c. ( ) ( ) 5 c t r t dt d. ( ) ( ) c t r t dt 12. What is the correct interpretation of 25 c( t) r( t) dt 1594? During the first 25 weeks after the epidemic began, a. the number of people with the virus increased by 1594 people. b. the number of people with the virus decreased by 1594 people. c. the rate at which people recover from the virus increased by 1594 people. d. the rate at which people contract the virus decreased by 1594 people. P a g e 8 13

5 Suppose that a quantity of radioactive nuclei decays at a rate of r( t).2(.998 t ) milligrams per year where t is the number of years since the sample began to decay. Check r(2) If the numerical method is used to determine how much of the radioactive nuclei should eventually decay, what value goes in the shaded cell of the table to the right? a b c d N 1 1 1, 1, N r( t) dt 14. Given that H is an antiderivative of h, which of the following is equivalent to 5 h ( x ) dx? a. lim h( N ) lim h( 5) N N b. lim h( 5) lim h( N) N N c. lim H ( N) lim H ( 5) N N d. lim H ( 5) lim H ( N) N N Check your Scantron now to make sure it will successfully run. Refer to the last page of the test for specifics. If it does, you will earn one point. (1 pt) When you are not working on the multiple choice portion of the test, turn your Scantron over so that it cannot be read by others in the room. P a g e 9 13

6 RE-READ the directions on the second page of the test regarding rounding, units, etc. Then read each question carefully. Provide only one clearly indicated answer to each question. If your answer is illegible, it will be graded as incorrect. Show all work. This portion is 57%. For each question, set up the specific mathematical notation that is being evaluated to obtain your answer. No credit will be awarded for simply copying generic formulas from the formula sheet. Little or no credit will be awarded for answers without the corresponding notation. 1. The graphs below shows the rate of change of revenue, r( x) 126.6x million 2 dollars per year, and the rate of change of cost, c( x) 46.34x 126.5x 725 million dollars per year, for a company, where x is the number of years since 21, x 4. Check: r(2) and c(2) GRAPH 1 GRAPH 2 GRAPH 3 a. Find the average rate of change of this company s costs from 21 to 214. Show the specific notation, round your answer to three decimal places and include units. (5 pts) b. Graphically illustrate your answer from part a on GRAPH 2. (3 pts) c. Find the change in the company s profit from 21 to 214. Show the specific notation and complete the sentence of interpretation. (6 pts) Notation: Complete the sentence of interpretation: From 21 to 214, the company s profit by. increased/decreased round to three decimal places units P a g e 1 13

7 3 2. Use the algebraic method to evaluate 6.25 dx. (9 pts) x Show all of the steps of the algebraic method. Use proper notation throughout your work. State your conclusion in the blanks provided dx x 3 The improper integral 6.25 dx to. x converges or diverges number, +, or - 3. A company made a profit of $15 million last year, and it expects its profits to grow by 5.3% per year over the next 4 years. One quarter of the profits will be continuously invested in an account paying 1.9% interest, compounded continuously. a. What is the income flow rate, R( t ), of the income stream into the account? Include units.(5 pts) R( t) b. Set up the specific INTEGRAL notation that represents the 4-year present value of this account. You do not need to compute this value. (3 pts) P a g e 11 13

8 4. Suppose the demand and supply functions for a certain brand of windshield wipers can be modeled as: 2 D( p) p 2 p thousand pairs thousand pairs when p < 6 S( p) 2.25p 8 p 125 thousand pairs when p 6 where $p per pair is the price of a pair of wipers. Check: D(3) = and S(6) = 182 The areas of four regions (M, N, P, Q) are labeled. Assume that the areas do not overlap. a. What is the price at the point of market equilibrium? $ per pair (3 pts) b. Find the amount consumers are willing and able to spend at market equilibrium. Show the specific notation, round your answer to three decimal places, and include units. (6 pts) c. Using only the letters M, N, P, and Q and the operations addition (+) or subtraction ( ), indicate how would each of the following would be found at market equilibrium. [For example, your answers will look like M, M + Q, P N, etc.] (4 pts) Consumer Surplus Producer Revenue Total Social Gain Consumer Willingness and Ability to Spend d. Determine the total social gain at market equilibrium. Show the specific notation, round your answer to three decimal places, and include units. (6 pts) P a g e 12 13

9 The functions are repeated for your convenience: 2 D( p) p 2 p thousand pairs and thousand pairs when p < 6 S( p) 2.25p 8 p 125 thousand pairs when p 6 where $p per pair is the price of a pair of wipers e. Determine the price at the point of unit elasticity of demand. Show work. (4 pts) Work: Answer: $ per pair Round to two decimal places f. Is the demand elastic or inelastic at the point of market equilibrium? Show the specific value used to justify your conclusion. (3 pts) Conclusion: Demand is at the point of market equilibrium. END OF TEST P a g e 13 13