Math 1020 ANSWER KEY TEST 3 VERSION A Fall 2016

Transcription

1 Printed Name: Section #: Instructor: Please do not ask questions during this eam. If you consider a question to be ambiguous, state your assumptions in the margin and do the best you can to provide the correct answer Any communication with any person (other than the instructor or the designated proctor) during this eam in any form, including written, signed, verbal or digital, is understood to be a violation of academic integrity. All devices, such as computers, cell phones, cameras and PDAs must be turned off while the student is in the testing room. The only calculators to be used are TI-83, TI-83+, TI-84 or TI-84+. You may NOT borrow or share a calculator with another person taking this test. Statement of Academic Integrity: I have not and will not give or receive improper aid on this test. In signing below, I acknowledge that I have read, understand, and agree to these testing conditions. Student s Signature: (This test will not be accepted for grading unless it bears the signature of the student.) General Directions and Calculator Hints are located on the last page. FR#1 FR # FR #3 FR #4 FR #5 scantron Free Response Total Multiple Choice Total Total Possible points Points Earned Formula: f ( ) = f ( c) + f ( c)( c ) L Page 1 of 1

2 MULTIPLE CHOICE: 69 points Use a # pencil and completely fill each bubble on your scantron to answer each multiple choice question. (For future reference, circle your answers on this test paper.) Each question is worth 3 points. There is no penalty for guessing on multiple choice questions. If you indicate more than one answer, or you leave a blank, the question will be marked as incorrect. C B D C B B D A B D C D B C D C A A D C A B A Use the following to answer the net two questions. The graph of a function, defined on a closed interval, is given below. 1. Point D is defined as. a. a relative minimum, but not an absolute minimum. b. an absolute minimum, but not a relative minimum. c. both a relative minimum and an absolute minimum. d. neither a relative minimum nor an absolute minimum.. Point A is defined as. a. a relative maimum, but not an absolute maimum. b. an absolute maimum, but not a relative maimum. c. both a relative maimum and an absolute maimum. d. neither a relative maimum nor an absolute maimum. 3. For a continuous smooth function f ( ), in which f () = 0 and f () > 0, which one of the following statements about f ( ) must be true? a. f ( ) has an inflection point at =. b. f ( ) has a zero at =. c. f ( ) has a relative maimum at =. d. f ( ) has a relative minimum at =. Page of 1

3 4. Identify the graph of the second derivative, f ( ), of the given function f ( ). a. b. c. d. 5. Graphs A, B, and C are shown below. Identify the graphs of f ( ), f ( ), and f ( ) and complete the following sentence. The graph of f ( ) is and the graph of its second derivative, f ( ), is. a. graph A; graph B b. graph A; graph C c. graph B; graph A d. graph B; graph C Page 3 of 1

4 Use the function 3 f ( ) = on the interval 3 3 to answer the net two questions. Checkpoint: f () = 3 6. The graph of the function f ( ) on the given interval has a relative maimum at input value. a. = b. = c. = d. = The graph of the function f ( ) on the given interval is decreasing most rapidly at input value. a. =.30 b. = c. = d. = If f ( t) = g( t) h( t) and g(6) = 4, h(6) = 10, g (6) = 10, and h (6) =, find f (6) =. a. 9 b. -0 c. 40 d Given g( ) 3 e 9 = +, h( ) ln(4 1) df = +, and f ( ) = g( ) h( ), find. d = a. b. c. d. (6 + 9 e ) (6 + 9 e ) ln(4 + 1) + (3 + e ) (6 + e ) ln(4 + 1) + (3 + e ) (6 + e ) Page 4 of 1

5 10. For the composite function, g( h( )) = , identify the inside function h( ) =. a. 5 b. 18 c. 3+ d Find f ( ) = for f ( ) = +. 5 a ( 9 + 4) 1 b. 5( 9+ 4) c. ( + ) ( ) 5 18 d. ( ) 1 1. The derivative of f ( ) = (3 5 ) is f ( ) =. a. b. c. d. 1 (ln ) (3 5 ) + (6 5 ) 1 (ln ) (6.5 ) 1 (ln ) (6.5 ) + (3 5 ) 1 (ln ) (3 5 ) + (6.5 ) Page 5 of 1

6 Use the following to answer the net two questions. A city s population can be modeled by f ( ) thousand people where is the number of years since 1970, 0 1. Use the rate of change graph, f ( ), shown below to answer the following questions. 13. Between 1970 and 198, the population of the city was at a relative maimum years after a. 0 b..703 c d Between 1970 and 198, the population of the city was decreasing most rapidly years after a. 0 b..703 c d The population of the United States can be modeled as P( = ) (1.01 ) million people, where is the number of years since 1970, Checkpoint: P() M ( ) = gives the decimal proportion of people in the U.S. who live in the Midwest, where is the number of years since 1970, Checkpoint: M () = (e.g., M () = means that 7.86 percent of the U.S. population lived in the Midwest in 197.) How quickly was the population in the Midwest population changing in 005? a..88 million people per year b (10 ) million people per year c million people per year d million people per year Page 6 of 1

7 Use the following information to answer the net three questions. c( t ) dollars gives the annual premium for a $100,000 Whole-Life insurance policy for a t year old male. n( c ) people gives the number of people who purchase a $100,000 Whole-Life insurance policy when the annual premium is c dollars. For a 40 year old male, the annual premium for a $100,000 Whole-Life insurance policy is 1400 dollars, and that premium is increasing by 75 dollars per year of age. When the annual premium for a $100,000 Whole-Life insurance policy is 1400 dollars, the number of people who purchase a policy is 800, and this number is decreasing by people per dollar in premium price. 16. How can these two functions be combined into a meaningful function? a. multiplication: c( t ) n( c) b. composition: c( n( t )) c. composition: n( c( t )) d. division: c( t) n( c) 17. For a 40 year old male, how quickly is the number of people who purchase a $100,000 Whole-Life insurance policy changing with respect to the age of the purchaser? a. decreasing by 150 b. increasing by 00 c. decreasing by d. increasing by What are the output units for the above problem? a. people per year of age b. dollars per person c. people per dollar d. dollars per person per year of age 19. When a company supplying screen printed licensed T-shirts for Cubs fans makes,500 shirts each hour, its hourly production costs are increasing by 1.5 dollars per T-shirt. Epress the company s marginal cost in a sentence of interpretation. a. The additional cost of producing the,500 th T-shirt each hour is 1.5 dollars. b. The average cost of producing,500 T-shirts each hour is 1.5 dollars per T-shirt. c. The average cost of producing the,501 T-shirts each hour is 1.5 dollars per T-shirt. d. The additional cost of producing the,501 st T-shirt each hour is 1.5 dollars. Page 7 of 1

8 Use the following to answer the net two questions. R( ) million dollars gives the revenue for Royal Caribbean Cruises, years since 000. In 01, Royal Caribbean Cruises had revenue of 7,650 million dollars and its revenues were increasing by 50 million dollars per year. 0. Estimate the change in revenue between 01 and 014. a million dollars b. 50 million dollars c. 500 million dollars d million dollars 1. Write a linearization for the revenue function at input 01. a. R ( ) = 7, ( 1) b. R ( ) = ,650( 1) L c. R ( ) = 7, ( 50) d. R ( ) = ( 7,650) L L L Use the following to answer the net two questions. f ( ) dollars gives the stock price for a company, days after its initial public offering, Use the graph of f ( ) above and linearization to estimate the stock price 9 days after the company s initial public offering. a dollars b dollars c dollars d dollars 3. According to the graph, the estimate of the stock price 9 days after the company s initial public offering is. a. an underestimate b. an overestimate c. equal to f (9) d. less than f (8) Page 8 of 1

9 FREE RESPONSE: 31 points Show work where possible. Read the directions on rounding, inclusion of units, and writing models and sentences at the back of the test. 1. Find the derivative of the following functions. Use eact numbers. It is not necessary to simplify. For full credit, clearly show work and use proper mathematical notation. A. f ( ) = (3 )(ln ) ( ) (ln 3)(3 )(ln ) (3 )( 1 ) f = + Part A: 3 points Part B: 3 points B. 4 f ( ) = = ( 4 3 5e ) ( 3 5e ) ( 3 ) ( ) f ( ) = 8 3 5e 6 5e ( / 6pts ). 3 g( t) = t t t million gallons gives the amount of gasoline consumed by federal government fleets, t years after 000, 5 t 1. Checkpoint: g () = When filling in the following blanks, answers should be correct to THREE decimal places. Answers that are not rounded correctly to three decimal places will be marked incorrect. A) Abs. Ma: pts first blank; 1 pt second blank B) Abs. Min: pts first blank; 1 pt second blank C) Increasing 1 pt Inflection Pont: 3 pts first blank; 1 pt second blank; 1 pt third blank A. The consumption of gasoline by Federal government fleets was at its highest years after 000, when consumption was million gallons. B. The consumption of gasoline by Federal government fleets was at its lowest 6.69 years after 000, when consumption was million gallons. C. The consumption of gasoline by Federal government fleets was increasing/decreasing (circle one) most rapidly years after 000 by million gallons per year. And at that time, consumption was million gallons. ( /1pts) Page 9 of 1

10 3. Consider the graph of f ( ) given below. A. Mark an X directly on the graph itself to indicate the location of a relative maimum. The marked relative maimum occurs at = 8, y= 8. or =, y= 10. Part A: 1 point Part B: 3 points B. Answer yes or no as to whether there is a relative minimum at each of the following input values. = 0 yes = no = 4 yes = 6 no = 8 no = 1 no ( / 4 pts ) 4. Find the first and second derivatives of f = e + ln( ) + 5. ( ) 3 Part A: 1.5 points Part B: 1.5 points A e or e or e f ( ) = B. ( ) = 4e ( / 3pts ) f TEST CONTINUES ON NEXT PAGE Page 10 of 1

11 5. n( t ) restaurants gives the total number of restaurants owned by a corporation, t years after 1996, 0 t 0. a( n ) thousand dollars gives the corporation s advertising epenses for n restaurants, 1 n 70. a( n( t)) thousand dollars gives the corporation s advertising epenses for its restaurants, t years after 1996, 0 t 0. n (14) = 44, a (44) = 40.90, dn dt t= 14 = 3.391, and da = dn = n 44 5 A. What were the corporation s advertising epenses for its restaurants in 010? Include units with the answer. a( n(14)) = a(44) = thousand dollars Part A: points Part B: 3 points B. How quickly were the corporation s advertising epenses for its restaurants changing, with respect to time, in 010? Show work and Include units with the answer. da da dn thousand dollars restaurants = = = thousand dollars per year dt t dn dt restaurant year = t= 14 n= ( / 5pts ) 6. 1 point for correctly filling out and bubbling the scantron with a # pencil, a correct XID, a correct test version AND the front of the test is completed with your signature on the academic integrity statement. END OF TEST Page 11 of 1

12 General Directions: Show work where possible. Answers without supporting work (where work is appropriate) may receive little credit. Do not round intermediate calculations. Answers in contet ALWAYS require units. Round your answers to 3 decimal places UNLESS the answer needs to be rounded differently to make sense in the contet of the problem OR the directions specify another type rounding OR the complete answer has less than 3 decimal places. When you are asked to write a model, include all components of a model: an equation, a description of the input including units, a description of the output including units, and the interval when known When asked to write a sentence of practical interpretation, answer the questions: when?, what?, and how much? using ordinary, conversational language. DO NOT use math words, terms, or unnecessary phrases. Always use a ruler when estimating values off of a graph. HINTS FOR TROUBLESHOOTING YOUR CALCULATOR: If you lose your L1, L, etc., you may reinsert them using STAT 5 (set-up editor) enter. The SCATTER PLOT will not show unless Plot 1 has been turned on and there is data in L1 and L. ZOOM 0 may not work for graphing if Plot 1 is turned on. DIM MISMATCH error usually means that the lists in L1 and L are not of equal length. DATA TYPE error usually means that you already have something in Y1 and you need to clear it before you can paste a new equation. INVALID DIM error usually means that your plot(s) are on, but that you have no data in the lists. Refer to the second hint above. If your batteries die, raise your hand and hold up your calculator. If your instructor has an etra calculator available, he/she will loan it to you for a few minutes. SYNTAX ERROR: Try GO TO. This will happen if you use a subtraction minus sign when you should use a negative sign. If you need to CLEAR MEMORY, use nd +, 7:Reset, 1:All Ram, :Reset Page 1 of 1