Math 1020 ANSWER KEY TEST 3 VERSION B Spring 2018

Transcription

1 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018 Printed Name: Section #: Instructor: Please do not ask questions during this exam. 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 exam 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, watches, 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-8 or TI-8+. 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 # FR#5 scantron Free Response Total Multiple Choice Total Total Possible points Points Earned Formula: f () x = f() c + f ()( c x c ) L C A D B B D A C C A B A D B C D B C D D C B A C D Page 1 of 1

2 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018 MULTIPLE CHOICE: 6 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, unless noted otherwise. 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. x 1. Consider the function f( x) = 5e + ln( x) 7x. Find the second derivative f ( x). a. c. x 1 f ( x) = 5e + x 7 b. = d. f ( x) 5e x x f ( x) = 5xe x x 5 f ( x) = + x x 1 1. [ pts] For f( x) x = e, identify gh ( ) and hx ( ) = = such that f( x) = ghx ( ( )). a. c. gh ( ) e; hx ( ) x h = = b. x gh ( ) h; hx ( ) e = = d. gh ( ) = e; hx ( ) = x gh ( ) = e; hx ( ) = x 3. [ pts] Find the derivative of the function: f( x) = e x a. x = b. f ( x) e 8x = c. f ( x) e x x f ( x) = e ( x ) + e ( 8x) d. f ( x) = 8xe x. For f x ( ) ln(x 3) =, find f ( x) =. a. 1 16x b. 8x x 3 c. x 3 d. ( x 3) ln(8 x ) x + Page of 1

3 Math 100 ANSWER KEY TEST 3 VERSION B Spring Find f ( x) = if 3 5 = = =. f( x) gx ( ) hx ( ), and gx ( ) x, and hx ( ) 1x x 3 x x x 3 x x x x x x 1 3 a. ( 70 ) b ( ) ( 70 ) 3 x x x 3 x x x x x x 1 3 c. + ( 70 ) d. 1 3 ( ) ( 1 ) 6. gh ( ) = h and hx ( ) = ln( x). These functions can be combined as the composition ghx ( ( )). What is dg the derivative of the composition, i.e.? dx = a. 1 ln x 1 b. (ln( x )) 1 c ln( ) x x x x 1 ln( x) 1 x + d. ( ) 1 7. N( x) thousand pen drives gives the number of USB 16GB pen drives sold by a department store every month, where x is the price in dollars per pen, 8.99 x Using this information, the store s monthly revenue from the sale of pen drives is Rx ( ) = xnx ( ) thousand dollars, where x is the price in dollars per pen, 8.99 x Determine a formula for R ( x) = and find R (9.00) = thousand dollars per dollar if N(9.00) = 3.09, N (9.00) = a. N( x) + x N ( x) ; b. x N ( x) ; -.1 c. N ( x) ; d. x N( x) + x N ( x) ; Page 3 of 1

4 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018 Use the following information to answer the next three questions. Ct () million students were attending Florida public schools, t years after fall of 010. In fall 01, there were.69 million students attending Florida public schools and that number was increasing by 0.08 million students per year. St () thousand dollars per student gives the state spending per student, t years after fall of 010. In fall 01, state spending was 8.63 thousand dollars per student and that number was decreasing by 0.19 thousand dollars per student per year. 8. In fall 01, the total amount of state spending on Florida public school students was billion dollars. a b c d In fall 01, the total amount of state spending on Florida public school students was decreasing by. (Give units in next question.) a b c d [ pts] What are the units for the previous question? a. billion dollars per year b. dollars per student c. thousand dollars per student per year d. million students per year 11. nt () pairs is the number of pairs of boots that a factory produces in t hours, 0 t 16. Cn ( ) thousand dollars gives the factory s costs of producing n pairs of boots, 8 n 100. In 8 hours of production, the factory can make 500 pairs of boots and because a new and larger shift of workers begins at this time, the production of boot pairs is increasing by 15 pairs per hour. When making 500 pairs of boots, the factory costs are 8 thousand dollars and those costs are decreasing by 0.00 thousand dollars per pair of boots. At 8 hours of production time, the factory s costs are decreasing by thousand dollars per hour. a b c d Page of 1

5 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018 Use the following information to answer the next two questions. [ pts each] f is a differentiable function, and f (3) = 5 and f (3) = The amount of change between f (3) and f (3.5) is approximately and an estimate of the value of f (3.5) is. a. -0.8;. b. -1.6; 3. c. 0.5; 5.5 d. 3; Write the linearization for f at input c = 3. a. f ( x) = 5 3( x+ 1.6) b. f ( x) = 3 1.6( x 5) L c. f ( x) = 3+ 5( x 1.6) d. f ( x) = 5 1.6( x 3) L L L 1. Give a possible sketch of a function f that satisfies ALL of the following restrictions: f (0) = 0 f ( x) > 0 for all x < 0 and f ( x) > 0 for all x > 0 f ( x) < 0 for all x < 0 and f ( x) > 0 for all x > 0 a. b. c. d. Page 5 of 1

6 Math 100 ANSWER KEY TEST 3 VERSION B Spring Consider the function ( ) 3 f( x) = 3x 5x+ 1. To identify possible locations of either relative maxima or relative minima, which one of the following equations should be solved? 3x 5x+ 1 = 0 b. 6x 5= 0 a. ( ) 3 c. 33 ( x 5x+ 1) ( 6x 5) = 0 d. ( x x ) = 0 Use the graph provided below to answer the following TWO questions. 16. An absolute minimum on the interval [-6.3,6.5] occurs at. a. x = 6.3 b. x = 3 c. x = 3 d. x = 6.5 e. none exists 17. A relative minimum on the interval [-6.3,6.5] occurs at. a. x = 6.3 b. x = 3 c. x = 3 d. x = 6.5 e. none exists 18. [ pts] An everywhere continuous function f has the following properties: f (7) = 15 f (7) = 3 f ( x) > 0, for x 7. Which one of the following describes the value for f (10)?. a. f (10) < b. f (10) = c. f (10) > d. f (10) does not exist Page 6 of 1

7 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018 Use the information below to answer the following three questions. [ pts each] 3 ht ( ) = t t t t days gives the average length of stay in U.S. community hospitals,t years after 1990, 0 t 1. Checkpoint: h () = The average length of stay in U.S community hospitals was decreasing the most rapidly years after a b c d At the time in which the average length of stay in U.S community hospitals was decreasing the most rapidly, how fast is it changing? a b c d What are the units to the previous question? a. years b. years per day c. days per year d. days Use the information below to answer the following two questions. [ pts each] The slope graph below gives the RATE OF CHANGE at month t (1 = January, etc.) of sales for a store specializing in swimming pools in the summer and ski gear in the winter.. According to the graph, on what intervals were the store s sales increasing? a. (1, 3.5), (6.7, 9.8) b. (3.5, 6.7), (9.8, 1) c. (1, 1.9), (5.1, 8.), (11., 1) d. (1.9, 5.1), (8., 11.) 3. According to the graph, for what value(s) of t did the store s sales reach a relative maximum? a. 6.7 b. 5.1, 11. c. 3.5, 9.8 d. 1.9, 8. Page 7 of 1

8 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018. [ pts] Consider the SECOND DERIVATIVE graph f ( x) shown below, for a differentiable function f. Find any and all infection point(s) of f. a. x = 1 only b. x = 3 c. x = 1, x = 5 d. x = 5 only 5. [ pts] Consider the graph of f( x) below. Which one of the following is the graph of f ( x)? a. b. c. d. Page 8 of 1

9 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018 FREE RESPONSE: 38 points Show work where possible. Read the directions on rounding, inclusion of units, and writing models and sentences at the back of the test Px ( ) =.x x 75.63x dollars gives the price for one share of Morgan Stanley (MS) stock, x years since the beginning of September of 000, 0 x Checkpoint: P () = a. On the given interval, find the MS stock price (for a single share), correctly rounded EXACTLY to three decimal places (stock prices are often rounded beyond two decimal places) at each of the following locations: Relative Maximum: $ Part a: points Part b: points Part c: points Relative Minimum: $ b. On the given interval, if the Absolute Maximum occurs at the same point as the Relative Maximum, circle yes ; if not, circle no. Fill in the blank, correctly rounded EXACTLY to three decimal places. yes / no On the given interval, what was the highest Morgan Stanley stock price per share? $ On the given interval, if the Absolute Minimum occurs at the same point as the Relative Minimum, circle yes ; if not, circle no. Fill in the blank, correctly rounded EXACTLY to three decimal places. yes / no On the given interval, what was the lowest Morgan Stanley stock price per share? $ c. Find the location of the inflection point. Then use that value to fill in the blanks in the following sentences, correctly rounded to three decimal places. Inflection Point x = _3.595 On the given interval, Morgan Stanley stock price per share was increasing most rapidly years after the beginning of September of 000. At that time, the price of one share of MS stock was 5.09 or dollars per share and was increasing by dollars per share per year. ( /10 pts ) Page 9 of 1

10 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018. Consider the graph of f( x ) below. Part a:.5 points Part b: 0.5 points a. Determine whether the given quantities are positive, negative or zero. Write POS, NEG, or ZERO in each cell of the table below. x f( x ) f ( x) f ( x) -0.5 NEG POS NEG NEG ZERO POS ZERO NEG NEG b. Find the location of an inflection point in which the function graph is increasing most rapidly. Identify the point by marking an X directly on the function curve at that point. ( / 5pts ) 3. After years of bee colony collapse, it has been found that honeybees have recently begun to increase in population. In April, 016, there were an estimated.800 million honeybee colonies in the US and that number was increasing by million honeybee colonies per month. a. Estimate the number of honeybee colonies in the US twelve months later, in April, 017. Include units with the answer. million honeybee colonies.800 million honeybee colonies + (1 months) (0.006 ) month =.87 million honeybee colonies b. If the number of honeybee colonies can be modeled by an exponential growth function, beginning in April, 016, is the estimate in part a) an underestimate or an overestimate? _ underestimate _ ( / pts ) Part a: 3 points Part b: 1 point Page 10 of 1

11 Math 100 ANSWER KEY TEST 3 VERSION B Spring 018. Jennifer is training for a marathon that takes place in 6 months. hx ( ) hours gives the cumulative amount of time Jennifer spends running, x weeks since the start of training for the marathon 0 x 6. Ch ( ) calories is the total number of calories burned while running for h hours, Part a: points Part b: 6 pts h () = 1 dh =.05 C (1) = 7350 dx = x dc = 350 dh = h 1 a. Four weeks into the marathon training, Jennifer has burned a total of 7,350 calories. Ch ( ()) = C(1) = 7,350 b. Four weeks into the marathon training, how quickly is the total number of calories burned changing with respect to the number of weeks? Show work and write the answer in a completely defined sentence of interpretation. dc dc dh calories hours = = (350 ) (.05 ) = calories per week dx dh dx hour week x= h= 1 x= Four weeks into the marathon training, the number of calories burned was increasing by calories per week. ( / 8 pts ) 5. Find the derivative of the following functions. Use exact numbers. It is not necessary to simplify. For full credit, clearly show work and use proper mathematical notation. a. f x (x+ 1) x ( ) 3 7 = + f = + (x+ 1) x ( x) ln(3) ln(7)7 Part a:.5 points Part b: 5.5 points b. f x x x x ( ) = (3 + 5) ln( ) f ( x) = (1x 8 x) ln( x) + (3x x + 5) x 3 1 ( /10 pts ) Page 11 of 1

12 Math 100 ANSWER KEY TEST 3 VERSION B Spring 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. General Directions: END OF TEST Show work where possible. Answers without supporting work (where work is appropriate) may receive little credit. Do not round intermediate calculations. Answers in context ALWAYS require units. Round your answers to 3 decimal places UNLESS the answer needs to be rounded differently to make sense in the context 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 extra 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. Page 1 of 1 If you need to CLEAR MEMORY, use nd +, 7:Reset, 1:All Ram, :Reset