Math 116 Practice for Exam 3

Transcription

1 Math 116 Practice for Exam 3 Name: Instructor: Generated April 13, 2018 Section Number: 1. This exam has 12 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 2. Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the exam. 3. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam. 4. Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 5. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). owever, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card. 6. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 7. You must use the methods learned in this course to solve all problems. Semester Exam Problem Name Points Score Winter anoi Tower 5 Winter Winter Okefenokee 12 Winter microphone 14 Winter peanut 14 Fall ice sculpture 12 Winter Winter ladybugs1 10 Winter owl and dove 9 Fall tank 12 Winter lagoon 11 Winter tomatoes 6 Total 121 Recommended time (based on points): 122 minutes

2 Math 116 / Final (April 21, 2016) DO NOT WRITE YOUR NAME ON TIS PAGE page [5 points] The anoi tower is made by rotating the region depicted below around the y-axis. The region is made up of infinitely many adjacent rectangles. The nth rectangle has width 1 1 and height a n = where n = 0,1,2,3,... The rectangle touching the y-axis corresponds to n = 0. Note that the y-axis is not to scale. n!(2n+1) y a 0 a 1 a 2 a 3 a 4 a x Compute the volume of the anoi Tower. Give an exact answer. Winter, 2016 Math 116 Exam 3 Problem 11 (anoi Tower)

3 Math116 / Final (April24, 2017) do not write your name on this exam page [8 points] The Taylor series centered at x = 0 for a function F(x) converges to F(x) for all x and is given below. F(x)= X ( 1) n x 4n+1 (2n)!(4n+1) n=0 a. [3 points] What is the value of F (101) (0)? Make sure your answer is exact. You do not need to simplify. Answer: F (101) (0) = b. [3 points] Find P 9 (x), the 9th degree Taylor polynomial that approximates F(x) near x = 0. c. [2 points] Use your Taylor polynomial from part b. to compute F(x) x lim x 0 2x 5 Winter, 2017 Math 116 Exam 3 Problem 10

4 o 8 o o o o [ p 0 V 0 %*R # $ P$ K t t V%('*%%* c WR # P $ Uh ('R # ') f% I\ $K'RB%nK'*RB R *&S, $ P $%P ')*%(Y A # $ P$ K frb')\n $% ') KTR # # $') ` :S, % \ $K')RB%nRB_%(' % 0* K\ S, V%('t I ks RB') P n*x v F (_* (\%S Rn GY A P$ RB $ c O e < '* O\ P*= ' R V # '* ^\ P$* *R%f #R Rn c # $ P K % R*Ù,RB_%(', * '= ( #RB' ')t '. F (_* (k%s Rn I K $J* <U e OgJª + vg + p%q0r ('k F (_* ( %S R\ I '*PRB % \ QR% Rn I Q $ P$ K %]L p%uvr 6 ('f(j(',rk nfs, *T N #PR S,VU (Y \ # $')#v %')R*J*PP K / ') 3 R%L 6 % S, * L 6 D p`r 6 '=S,VU L Q $') %RVP$UWn MJ< K ' K f" ') ')(&#R*Ù G * $'% ') (U L U pvr :S arb', <* W # ') %')R*J*PD!"#')% '( #R*Ù * $'!% ') (U L Winter, 2004 Math 116 Exam 2 Problem 6 (Okefenokee)

5 Math 116 / Exam 2 (March 2011) page 3 2. [14 points] A microphone at the point r = 0 detects sounds in a region enclosed by the cardioid r = cosθ. The microphone is placed in front of the stage at an auditorium to record a musical band. Let d denote the smallest distance you must leave between the audience and the microphone to avoid recording any noise from the public in attendance. a. [5 points] Write an integral that computes the area of the shaded region A in terms of θ 1, θ 2 and d. b. [4 points] Write a formula in terms of θ that computes the value of the slope of the tangent line to the cardioid. c. [3 points] Find an exact expression for the values of 0 θ < 2π at which the cardioid has a vertical tangent line. Full credit will not be given for decimal approximations. d. [2 points] Find the value of d. Show all your work. Winter, 2011 Math 116 Exam 2 Problem 2 (microphone)

6 Math 116 / Exam 2 (March 20, 2013) page [14 points] a. [6 points] Find a definite integral that computes the shaded area outside the circle r = 5 2 and inside the curve given by r = 2+sin2θ in the graph below. b. [4 points] Find the polar coordinates (r,θ) of the points where the line y = 3 x intersects the graph of r = 2+bsin2θ. ere the constant 0 < b < 2. Your answers may include b. P 1 = P 2 = Winter, 2013 Math 116 Exam 2 Problem 8 (peanut)

7 Math 116 / Exam 2 (March 20, 2013) page 12 c. [4 points] i) Find the equation in polar coordinates of the line x = 0. ii) Find the equation in polar coordinates of the line y = 4. Winter, 2013 Math 116 Exam 2 Problem 8 (peanut)

8 Math 116 / Final (December 17, 2009) page 6 4. [12 points] During a party, the host discovers that he has been robbed of his favorite gold hallway clock, and immediately calls the police. The last time the host noticed the clock was still hanging, it was 6:00 p.m. When the police arrive, Officer Tom notices a decorative ice sculpture. When he first arrives at 9:00 p.m., the ice sculpture s temperature is 27 o F. After questioning others at the party, Officer Tom again takes the temperature of the ice sculpture at 10:00 p.m., and finds it to be 29 o F. The temperature of the room has remained a constant 68 o F all day. a. [2 points] Let S be the temperature of the sculpture, measured in degrees Fahrenheit. Assuming the sculpture obeys Newton s Law of eating and Cooling, write a differential for ds dt, where t is the number of hours since 9:00 p.m. Your answer may contain an unknown constant, k. b. [7 points] Using separation of variables, and the information provided about the sculpture, solve the differential equation to find S(t), where t is the number of hours since 9:00 p.m. Your answer should contain no unknown constants. c. [3 points] Company FunIce provided the sculpture, and it was delivered by their employee Bill. For each ice sculpture they produce, the company guarantees the temperature will be exactly 18 o F upon delivery. If the sculpture was 18 o F at delivery, and Bill left directly afterwards, should he be considered as a possible suspect of the robbery? Briefly justify your answer. Fall, 2009 Math 116 Exam 3 Problem 4 (ice sculpture)

9 Math 116 / Exam 2 (March 23, 2015) page 5 5. [8 points] The graph of a slope field corresponding to a differential equation is shown below. y t -1-2 For all parts of this problem, no work is required and very little partial credit will be given. a. [2 points] On the slope field, sketch a solution curve passing through the point (0,0). b. [2 points] If you approximated the value of y(1) using Euler s method starting at the point (0, 0), would your approximated value be an overestimate or underestimate? Circle your answer. OVERESTIMATE UNDERESTIMATE c. [4 points] Which of the following differential equations could correspond to the slope field above? Circle all that apply. dy dt dy dt dy dt dy dt dy dt = (t 1)(y 1.1)(y +1.8)2 = (t 1)(y +1.1)(y 1.9)2 = (t 1)(0.9 y)(y +1.8)2 = (t 1)(y 0.9)(y +1.9)4 = (1 t)(y 1.1)(y +1.9)4 Winter, 2015 Math 116 Exam 2 Problem 5

10 Math 116 / Final (April 23, 2015) page 9 9. [10 points] Vic is planning to put ladybugs in his garden to eat harmful pests. The ladybug expert at the gardening store claims that the number of ladybugs in his garden can be modeled by the differential equation dl dt = L 20 L2 100 where L is the number of ladybugs, in hundreds, in Vic s garden, t days after they are introduced. a. [4 points] Find the equilibrium solutions to this differential equation and indicate their stability. b. [2 points] If Vic starts his garden with 50 ladybugs, what will the long term population of ladybugs in his garden be according to the differential equation above? The long term population is c. [4 points] For what value of b is the function L(t) = 5e bt( 4+e bt) 1 a solution to this differential equation. b = Winter, 2015 Math 116 Exam 3 Problem 9 (ladybugs1)

11 Math 116 / Exam 1 (February 5, 2018) do not write your name on this exam page 3 2. [9 points] When Alejandra and Brontel were children they spent summer mornings chasing birds in flight. One memorable day they encountered an owl. The following graph shows the velocities a(t) of Alejandra (solid) and b(t) of Brontel (dashed), measured in meters per second, t seconds after the owl took off. The area of each region is given. v b(t) a(t) a. [1 point] ow far (in meters) do Alejandra and Brontel chase the owl? Answer: t b. [5 points] Suppose the owl ascends to a height of h meters according to h(t) = t where t is seconds since it went airborne. Let L(h) be the number of meters Alejandra is ahead of Brontel when the owl is h meters above ground. Write an expression for L(h) involving integrals and compute L (2). Answer: L(h) = L (2) = c. [3 points] The next bird to pass is a dove. This time Alejandra runs twice as fast and Brontel runs three times as fast as they did when chasing the owl. ow much faster (in m/s) is Brontel than Alejandra on average in the first 5 seconds? Answer: Winter, 2018 Math 116 Exam 1 Problem 2 (owl and dove)

12 Math 116 / Exam 1 (October 10, 2012) page [12 points] A tank whose base is at ground level has lateral walls in the form of a trapezoid 3 meters wide at the bottom, 4 meters wide at the top, and 3 meters high, and has a length of 7 meters, as shown in the figures below. The tank contains water up to a level of 1 meter. The density of water is 1000 kg per m 3. a. [8 points] Write an expression that approximates the work done in lifting a horizontal slice of water with thickness h that is at a height of h meters above the ground to the top of the tank. Use g = 9.8m/s 2 for the acceleration due to gravity. b. [4 points] Write an expression for the work required to pump all the water in the tank to the top of the tank. You do not need to evaluate the expression. Include units. Fall, 2012 Math 116 Exam 1 Problem 6 (tank)

13 Math 116 / Exam 1 (February 6, 2012) page [11 points] In the following problems show all your work to receive full credit. a. [7 points] The population of an invasive aquatic plant in a circular lagoon has density given by δ(r) = 20(1 e r2 ) kg/m 2, where r is the distance in meters from its center. The lagoon has radius R meters. Find the exact amount of plants living at the lake. b. [4 points] Let F(x) = x 0 e 2t 1dt. Find the exact value of the length of the curve on 0 x 1. Winter, 2012 Math 116 Exam 1 Problem 9 (lagoon)

14 Math 116 / Final (April 24, 2017) do not write your name on this exam page 2 1. [6 points] Suppose that the yield of the tomato plants in a particular Michigan garden is a function of the amount of water that the plants receive (from rainfall and irrigation). Let T(w) be the seasonal yield (in pounds) of the tomato plants in a season when the plants receive w gallons of water every week. A portion of the graph of T 0 (w) (the derivative of T(w)) is shown below. Note that T 0 (w) is linear for 50 w 70. Let A be the area of the region between the w axis and the graph of T 0 (w) for 0 w 30, and let B be the area of the region between the w axis and the graph of T 0 (w) for 30 w 50, T 0 (w) B A w a. [2 points] If the tomato plants yield 150 pounds of tomatoes when the plants receive 70 gallons of water every week, how many pounds of tomatoes would the plants yield in a season when they receive 30 gallons of water each week? (Your answer may involve the constants A and B.) Answer: b. [2 points] In order to maximize the yield of the tomato plants, how many gallons of water should the plants receive each week? (Round to the nearest 5 gallons.) c. [2 points] Consider the integral Answer: Z T 0 (w)dw. Rank the following four estimates of the value of this integral in order from least to greatest by writing them in the correct order on the answer blanks below: LEFT(10) RIGT(10) TRAP(10) MID(10) < < < Winter, 2017 Math 116 Exam 3 Problem 1 (tomatoes)