Math 115 First Midterm October 13, 2015

Transcription

1 Math 5 First Midterm October 3, 05 EXAM SOLUTIONS. Do not open this eam until ou are told to do so.. This eam has pages including this cover. There are 0 problems. Note that the problems are not of equal difficult, so ou ma want to skip over and return to a problem on which ou are stuck. 3. Do not separate the pages of this eam. If the do become separated, write our initials (not name) on ever page and point this out to our instructor when ou hand in the eam. 4. Please read the instructions for each individual problem carefull. One of the skills being tested on this eam is our abilit to interpret mathematical questions, so instructors will not answer questions about eam problems during the eam. 5. Show an appropriate amount of work (including appropriate eplanation) for each problem, so that graders can see not onl our answer but how ou obtained it. 6. The use of an networked device while working on this eam is not permitted. 7. You ma use an calculator that does not have an internet or data connection ecept a TI-9 (or other calculator with a qwert kepad). However, ou must show work for an calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card. 8. For an graph or table that ou use to find an answer, be sure to sketch the graph or write out the entries of the table. In either case, include an eplanation of how ou used the graph or table to find the answer. 9. Include units in our answer where that is appropriate. 0. Turn off all cell phones, smartphones, and other electronic devices, and remove all headphones and smartwatches.. You must use the methods learned in this course to solve all problems. Problem Points Score Total 00

2 Math 5 / Eam (October 3, 05) page. [5 points] Below is the graph of a function f(). f() There are si graphs shown below. derivative f (). Circle the one graph that could be the graph of the i. iv ii. v iii. vi

3 Math 5 / Eam (October 3, 05) page 3. [ points] Angelica Neiring and Simona Koloji decide to enjo the fall weather b racing each other from the brass block M in the center of the Diag along a.5 kilometer (500 meter) route to the Huron River inside the Arb. Let A(t) (respectivel S(t)) be Angelica s (respectivel Simona s) distance along the route (in meters) t seconds after the start racing. Angelica and Simona are both wearing GPS watches that record data about their race. The table of values for the functions A and S below shows some of the resulting data, rounded to the nearest meter. Note that the data is not alwas recorded at regular intervals. t A(t) S(t) Use the data above to answer the questions below. Remember to show our work. a. [ points] Estimate Angelica s instantaneous velocit 3 minutes into the race. Include units. Solution: We estimate using average velocit based on nearb measurements: Avg velocit between 68 and 80 seconds: Avg velocit between 80 and 98 seconds: A(80) A(68) A(98) A(80) m/sec m/sec 8 So we estimate that Angelica s instantaneous velocit 3 minutes into the race was about.6 meters per second. (An estimate between the average velocities computed above would be reasonable. We rounded to two significant digits.) About.6 meters per second b. [ points] Estimate S (0). Solution: We estimate the derivative using nearb measurements: Average rate of change of S(t) for 4 t 0: Average rate of change of S(t) for 0 t 35: S(0) S(4) 0 4 S(35) S(0) So we estimate that S (0).6. (An estimate between the average rates of change computed above would be reasonable. We rounded to two significant digits here.) S (0).6 c. [ points] In the contet of this problem, what are the units on the quantit (A ) (50)? seconds per meter For questions d. and e. below, circle the one best answer or circle cannot be determined if there is not enough information to definitel determine the answer. You do not need to show our work or provide justification for our answers for these questions. d. [ point] Who was ahead 5 minutes into the race? Angelica Simona cannot be determined e. [ point] Who was running faster eactl one minute into the race? Angelica Simona cannot be determined

4 Math 5 / Eam (October 3, 05) page 4 f. [4 points] In describing the race later, Simona sas that her average velocit during the entire race was.8 meters per second while Angelica sas that after the first 5 minutes, her average velocit for the rest of the race was 3. meters per second. Assuming their statements and the table of values above are accurate, who won the race? Or is there not enough information to decide? Eplain our reasoning. (Circle one.) Angelica Simona Not enough information Eplanation: Solution: With an average velocit of.8 meters per second for the whole race of 500 meters, it took Simona seconds to complete the 500 meter race. On the.8 other hand, it took Angelica 300 seconds to run the first 737 meters and, with an average velocit of 3. meters per second for the rest of the race ( meters), it took her an additional seconds to finish the race. So Angelica s total time for the race was about seconds, which is less than Simona s time of about 89.9 seconds. Hence Angelica won the race.

5 Math 5 / Eam (October 3, 05) page 5 3. [9 points] A portion of the graph of a function f is shown below, along with three graphs obtained from f b one or more transformations. Below each of the three graphs is a list of possible formulas for that graph. Find the one correct formula for each graph, and write the corresponding letter in the answer blank provided. Note that the zeros of each graph are labeled and that the scales on both the vertical and horizontal aes are the same for all the graphs shown. f() H E E A. f( ) B. f( ) A. f( ) B. f( + ) A. f( + ) B. f( ) C. f( 3) C. f( 3 ) C. f( ( )) D. f( ( )) D. f( ) D. f( + ) E. f( ( 3)) E. f() E. f( ) F. f( + ) F. f(3) F. f( + ) G. f( + ) G. f(( + )) G. f() H. f( + 3) H. f( ( )) H. f( ( + )) I. f( + ) I. f(( )) I. f( ) J. f( + 3) J. f( ( + )) J. f( ( 4))

6 Math 5 / Eam (October 3, 05) page 6 4. [3 points] Algernon Braik is making scones. He knows that the height of a scone is a function of how much baking soda it contains. Let h(b) be the height in millimeters of a scone that contains B grams of baking soda. Assume that the function h is increasing and invertible, and that h and h are both differentiable. a. [ points] Algie looks in his baking soda container and finds that there are eactl 46 grams of baking soda remaining. Suppose he uses all of this baking soda to make 8 scones, and that the baking soda is equall distributed among all 8 of the scones. Write a mathematical epression involving h or h for the height (in millimeters) of each resulting scone. ( ) 46 h 8 b. [5 points] Below is the first part of a sentence that will give a practical interpretation of the equation h (6) 5 in the contet of this problem. Complete the sentence so that the practical interpretation can be understood b someone who knows no calculus. Be sure to include units in our answer. If Algie decreases the amount of baking soda per scone from 6 grams to 5.8 grams, then... Solution: If Algie decreases the amount of baking soda per scone from 6 grams to 5.8 grams, then the height of each scone will decrease b approimatel 3 millimeters. c. [3 points] Algie makes a batch of scones, with each scone containing k grams of baking soda (for some constant k). When the scones come out of the oven, he decides the are each 0 millimeters shorter than he would like. Write a mathematical epression involving k, h, and h for the number of grams of baking soda per scone he should use to get scones of the desired height. h (h(k) + 0) 60 d. [3 points] Algie does some calculations and determines that h (30) 40. Based on this information, which of the following statements must be true? Circle all of the statements that must be true or circle none of these. A. If Algie makes 40 scones, each with 30 grams of baking soda, then the scones will rise to a height of 60 millimeters. B. If Algie wants to make 40 scones, then he must use 60 grams of baking soda. C. If Algie wants to make scones of height 30 millimeters and he has 60 grams of baking soda, then the maimum number of scones he can make is 40. D. A scone containing.5 grams of baking soda rises to a height of 30 millimeters. E. A scone containing 30 grams of baking soda rises to a height of.5 millimeters. F. none of these

7 Math 5 / Eam (October 3, 05) page 7 5. [8 points] Remember to show our work carefull throughout this problem. Algie and Cal go on a picnic, arriving at :00 noon. a. [5 points] Five minutes after the arrive, the notice that 5 ants have joined their picnic. More ants soon appear, and after careful stud, the determine that the number of ants appears to be increasing b 0% ever minute. Find a formula for a function A(t) modeling the number of ants present at the picnic t minutes past noon for t 5. Solution: Since this is an eponential function, there are constants c and b such that A(t) cb t. We can see immediatel that b.. We can then use the fact that we know that A(5) 5 to find c: c(.) 5 5, so c 5/(.) 5, which is approimatel.0. Alternativel, we can use a horizontal shift to sa that this is 5(.) t 5. A(t) 5(.) (t 5) 5. 5 (.) t b. [3 points] Algie and Cal notice that their food is, unfortunatel, also attracting flies. The number of flies at their picnic t minutes after noon can be modeled b the function g(t).8(.5) t. Algie and Cal decide the will end their picnic when there are at least 000 flies. How long will their picnic last? Include units. Solution: We wish to find t such that.8(.5) t 000. Then.8(.5) t 000 ln(.8(.5) t ) ln(000) ln(.8) + t ln(.5) ln(000) t ln(.5) ln(000) ln(.8) t ln(000/.8)/ ln(.5) 8.3. So the end their picnic about 8.3 minutes after noon (when it started). About 8.3 minutes 6. [6 points] Consider the function R(w) + (ln(w)) cos(w). Use the limit definition of the derivative to write an eplicit epression for R (π). Your answer should not involve the letter R. Do not attempt to evaluate or simplif the limit. Please write our final answer in the answer bo provided below. R (π) lim h 0 ( + (ln(π + h)) cos(π+h)) ( + (ln(π)) cos(π)) h

8 Math 5 / Eam (October 3, 05) page 8 7. [ points] Phillip Asaf and Genevieve Omicks both enjo hot chocolate when it s cool outside. The made a few measurements, and these appear in the table below. P (respectivel G) is Phil s (respectivel Gen s) consumption of hot chocolate (in quarts, measured to the nearest tenth of a quart) in a month when the average dail high temperature is H (in degrees Celsius, measured to the nearest degree). H( C) P (quarts) G (quarts) a. [8 points] Based on this data, could either student s monthl hot chocolate consumption be reasonabl modeled as a linear function of average dail high temperature? An eponential function? Neither? Carefull justif our answer in the space below. (Hint: At least one of these can be modeled b a linear or an eponential function!) Solution: First consider Phil s hot chocolate consumption. Suppose P p(h). To check whether p(h) could be modeled b a linear function, we compute the average rate of change of p over the intervals [3, 7] and [7, 5]. We have p(7) p(3) and p(5) p(7) Since these two average rates of change are quite different, Phil s hot chocolate consumption is not reasonabl modeled b a linear function. To check whether p(h) could be modeled b an eponential function, we compute the percent rate of change of p(h) over the intervals [3, 7] and [7, 5]. We have ( ) ( p(7) p(3) 6. ) and ( p(5) p(7) ) ( ) The difference between these percent rates of change is less than 0.%, so based on this data, p(h) can be reasonabl modeled b an eponential function. In particular, we can check that we obtain the data in the table for P using, for eample, 9.(0.9435) H Now consider Gen s hot chocolate consumption. Suppose G g(h). From the calculations g(7) g(3) and g(5) g(7) we conclude that Gen s hot chocolate consumption is not reasonabl modeled b a linear function. From the calculations ( ) ( g(7) g(3) 3.3 ) and ( g(5) g(7) ) ( ) we conclude that Gen s hot chocolate consumption can t be reasonabl modeled b an eponential function. ( (Note that for the eponential cases we could instead compare, for eample, p(7) p(3)) with p(5) p(7).) Answers: Circle one choice for each student. Phil s consumption P : linear eponential neither linear nor eponential Gen s consumption G: linear eponential neither linear nor eponential

9 Math 5 / Eam (October 3, 05) page 9 b. [4 points] For this investigation, their friend Madd measures temperature in degrees Fahrenheit, and she measures her hot chocolate consumption in cups. She finds a function M(f) which is the number of cups of hot chocolate she consumes in a month when the average dail high temperature is f degrees Fahrenheit. If Q(H) is the number of quarts of hot chocolate Madd consumes when the average monthl temperature is H degrees Celsius, write a formula for Q(H) in terms of M and H. Recall that there are 4 cups in a quart and that the conversion from Fahrenheit to Celsius is given b 5 9 ( 3) (where C and F describe the same temperature). Solution: H degrees Celsius is the same as 9 5 H+3 degrees Fahrenheit, and M ( 9 5 H + 3 ) gives Madd s hot chocolate consumption in cups. We divide this quantit b 4 to convert from cups to quarts. M ( 9 5 H + 3) Q(H) 4

10 Math 5 / Eam (October 3, 05) page 0 8. [ points] A portion of the graph of a function f is shown below. 3 f() a. [ points] Give all values c in the interval 0 < c < 0 for which lim c f() does not eist. If there are none, write NONE. c, 8 b. [ points] Give all values c in the interval 0 < c < 0 for which lim f() does not eist. c + If there are none, write NONE. c NONE c. [ points] Give all values c in the interval 0 < c < 0 for which f() is not continuous at c. If there are none, write NONE. c, 3, 8 d. [6 points] With f as shown in the graph above, define a function g b the formula B A 5 g() if 0 f() if 0 < < 0 where A and B are nonzero constants. Find values of A and B so that both of the following conditions hold. g() is continuous at 0. lim g(). If no such values eist, write none in the answer blanks. Be sure to show our work or eplain our reasoning. Solution: To satisf the first condition, we first compute g(0) b plugging in 0 to the rational function B A to find lim g() g(0) B. In order for 0 g() to be continuous at 0, we must also have lim g() B 0 +. Now lim g() lim f() (from the graph), so B , and B. To satisf the second condition, we compute that lim g() lim B A A 4. In order for this limit to equal, we must have A/4 /, so A. A and B

11 Math 5 / Eam (October 3, 05) page 9. [3 points] Cal is jumping a rope being swung b Gen and Algie while Madd runs a stopwatch. There is a piece of tape around the middle of the rope. When the rope is at its lowest, the piece of tape is inches above the ground, and when the rope is at its highest, the piece of tape is 68 inches above the ground. The rope makes two complete revolutions ever second. When Madd starts her stopwatch, the piece of tape is halfwa between its highest and lowest points and moving downward. The height H (in inches above the ground) of the piece of tape can be modeled b a sinusoidal function C(t), where t is the number of seconds displaed on Madd s stopwatch. a. [4 points] On the aes provided below, sketch a well-labeled graph of two periods of C(t) beginning at t 0. Pa attention to both the shape of our graph and the location of important points. H (inches) H C(t) 0 t (seconds) b. [4 points] Find a formula for C(t). C(t) sin(4πt) c. [5 points] Now Gen takes a turn at jumping while Cal and Algie swing the rope. Madd resets the stopwatch and starts it over again. Let G(w) be the height (in inches above the ground) of the piece of tape when Madd s stopwatch sas w seconds. A formula for G(w) is G(w) cos(πw). Madd is 60 inches tall. For how long (in seconds) during each revolution of the rope is the piece of tape higher than the top of Madd s head? (Assume Madd is standing straight while watching the stopwatch.) Remember to show our work. Solution: We are looking for when (cos(πw) > 60. First we find one time when (cos(πw) cos(πw) 9 cos(πw) / πw arccos(/) (is one solution) πw π/3 w /6 There are man was to find the answer from here. One wa is to note that since we are looking at a cosine function that has not been shifted horizontall, for 0 < w < /6, the rope is above her head, so it s there for the first sith of a second. B smmetr around the peak, the rope reached that height /6 second before the timer started. Thus the rope is above her head for /6 /3 second during each revolution. 3 second

12 Math 5 / Eam (October 3, 05) page 0. [0 points] Below is the graph of f (), the derivative of the function f(). Note that f () is zero for, linear for < <, and constant for < < 0. f () For each of the following, circle all of the listed intervals for which the given statement is true over the entire interval. If there are no such intervals, circle none. You do not need to eplain our reasoning. a. [ points] f () is increasing. < < 0 < < < < < < 3 none b. [ points] f () is concave up. 0 < < < < < < 3 none c. [ points] f() is increasing. < < < < 0 0 < < < < < < 3 none d. [ points] f() is linear but not constant. 3 < < < < < < 0 0 < < < < < < 3 none e. [ points] f() is constant. 3 < < < < < < 0 0 < < < < < < 3 none