Math 116 Midterm Exam November 14, 2016

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1 Math 6 Midterm Eam November 4, 6 UMID: Instructor: Initials: Section:. Do not open this eam until ou are told to do so.. Do not write our name anwhere on this eam. 3. This eam has pages including this cover. There are 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. 4. Do not separate the pages of this eam. If the do become separated, write our UMID on ever page and point this out to our instructor when ou hand in the eam. 5. 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. 6. Show an appropriate amount of work (including appropriate eplanation) for each problem so that graders can see not onl our answer, but also how ou obtained it. Include units in our answer where that is appropriate. 7. You ma use a TI-84, TI-89, TI-Nspire or other approved calculator. 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. If ou use graphs or tables to find an answer, be sure to include an eplanation and sketch of the graph, and to write out the entries of the table that ou use. 9. Turn off all cell phones, pagers, and smartwatches, and remove all headphones. Problem Points Score Total

2 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page. [9 points] While assembling a large fan as a holida gift, ou discover the following lovel diagram in the instruction manual. The propeller and base of the fan are illustrated b the polar curve r = + sin(3θ) and the circle r =, respectivel. - - a. [5 points] Write down, but do not evaluate, an integral that gives the arc length of the part of the propeller outside the circle in the first quadrant. b. [4 points] Write down the Cartesian equation of the tangent line to the propeller at the point (, ) = (, ).

3 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 3. [5 points] Find constants A and B so that the function h(w), defined for w > b is a solution to the differential equation h(w) = Aw 3 + w, w dh dw 3wh + B = satisfing h() = 3. Show all our work, and write our final answers in the spaces provided. A = B = 3. [5 points] In a recent presidential election between candidate A and candidate B, Shamcorp s rival compan Hawk-I tried fiing the election b changing the votes on some of the ballots. For the last three hours of the election (between 5pm and 8pm), the compan gained access to the huge ballot bo containing million ballots. Hawk-I emploees removed ballots from the ballot bo continuousl at a rate of 4 million ballots per hour. Those ballots were removed in proportion to the current ratio in the bo. Hawk-I emploees then instantl changed the the ballots voting for candidate B to vote for candidate A (leaving an votes for candidate A unchanged) before immediatel returning the ballots to the bo. Assume that the ballot bo alwas contains million votes, and that the ballot bo onl contains votes for candidates A and B. Write a differential equation that models a(t), the number of ballots voting for candidate A, in millions, in the ballot bo t hours after Hawk-I began changing votes. da dt =

4 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 4 4. [ points] Each graph below is a slope field for one of the differential equations. Beneath each slope field, write the letter of the differential equation in the blank. There is onl one correct differential equation for each slope field. Assume A > and B < are constants. You do not need to show our work. An ambiguous answers will be marked incorrect. (N) d d = (A ) (B ) (P) d = (A + )(B + ) (Q) d d = (A )(B ) d (R) d = ( A)(B ) d (V) d d = ( + B) d (S) d = ( A) ( + B) (W) d d = ( B) (T) d d = (A ) (Z) d d = ( + A) Answer: Answer: Answer: Answer:

5 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 5 5. [7 points] The Intern has designed an eperiment to stabilize the highl radioactive compound Porcinate. In his eperimental setup, the amount P (t) of Porcinate in moles, t hours after the eperiment began, should satisf the differential equation dp dt tp ln(p ) =. Use separation of variables to find a solution P (t) satisfing P (3) = e. 6. [5 points] The Intern is also studing a compound called Bovinate. The amount B(t) of Bovinate in moles, t hours after an eperiment began, satisfies the differential equation db dt = B( B)(t + B). a. [3 points] List all equilibrium solutions of the differential equation. Indicate whether each is stable or unstable. b. [ points] If the initial amount of Bovinate were.5 moles, what would happen to the amount of Bovinate in the long run?

6 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 6 7. [9 points] For π 4 < θ < 3π 4, consider the polar curve r = sin(θ) cos(θ) + sin(θ). The curve has an asmptote, the dashed line in the picture, as θ approaches π 4 and 3π 4. a. [4 points] Write down, but do not evaluate, an integral that gives the area inside the loop in the first quadrant. b. [ points] Find a formula for the quantit + in terms of the variable θ. Write our answer in the space provided. + = c. [ points] Find the limit of + as θ ( ) 3π. No justification is needed. 4 d. [ point] Write down the Cartesian equation of the asmptote. No justification is needed.

7 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 7 8. [ points] For each of the following parts, circle the correct answer. Ambiguous answers will receive no credit. You do not need to show our work. a a. [3 points] For nonzero constants a and b, the curve r = is a line. What sin(θ) + b cos(θ) is the Cartesian equation of the line? = b+a = a b = b a = a+b = b a b. [3 points] Ramond Green left a bowl of ice cream in a 5 C sauna. Over the first ln() hours, the ice cream goes from C to C. Which of the following describes the change in Q(t), the temperature of the ice cream in C after t hours? dq dt = Q 5 dq dt = (5 Q) dq ( dt = ln() 5 Q ) dq dt = 5 Q dq dt = (Q 5) ( ) u u α c. [3 points] Let α > be a constant. What is the value of lim? u u α e α DIVERGES e /α α d. [3 points] Consider the differential equation = + β, where β is a constant, and let () be a solution satisfing () =. For which value of β does Euler s method with steps give the estimate (4)?

8 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 8 9. [ points] Suppose that f is function with the following properties: f is differentiable f() > for all f() d converges. For each of the following parts, determine whether the statement is alwas, sometimes, or never true b circling the appropriate answer. No justification is needed. a. [ points] 5 f() d converges. ALWAYS SOMETIMES NEVER b. [ points] (f()) /3 d converges. ALWAYS SOMETIMES NEVER c. [ points] (f()) 3/ d converges. ALWAYS SOMETIMES NEVER d. [ points] f ( ) d converges. ALWAYS SOMETIMES NEVER e. [ points] f () ( f() d converges. Note: f () f() = d ) d ln(f()). ALWAYS SOMETIMES NEVER

9 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page 9. [4 points] Fearing that she is losing authorit over her robot ward, Dr. Durant has installed a tracking chip in Steph s mainframe. The chip gives Steph s location separatel in - and -coordinates, where the units of the aes are miles, Dr. Durant s office corresponds to the origin (, ) = (, ), the positive -ais points north, and the positive -ais points east. On night, Dr. Durant noticed unusual levels of activit; t hours after midnight, Steph began moving according to the parametric equations = f(t) = g(t), where f(t) and g(t) are plotted below for t 5. f(t) g(t) (,.5) (4,.7) (, ) t t - - (5,.7) a. [ points] When was Steph farthest north and south on night? Write our answers in the blanks provided. You do not need to show our work. North: a.m. South: a.m. b. [3 points] What was Steph s speed at t = 4.9 on night? You ma use the fact that f (4.9) =. Include units. c. [ points] What direction was Steph moving at t = on night? Circle onl one answer. NORTH AND EAST EAST ONLY SOUTH AND EAST NORTH AND WEST WEST ONLY SOUTH AND WEST

10 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page (continued). Recall that on night, Steph s position was given b the parametric equations = f(t) = g(t), where f(t) and g(t) are plotted below for t 5. As before, Dr. Durant s office is at the origin (, ) = (, ), the positive -ais points north, and the positive -ais points east. f(t) g(t) (,.5) (4,.7) (, ) t t - - (5,.7) d. [3 points] How far awa was Steph from Dr. Durant s office at t = on night? On night, Steph s movements were even stranger, following the parametric equations = t f(s) ds = t g(s) ds. e. [ points] What direction was Steph moving at t = on night? Circle onl one answer. NORTH AND EAST EAST ONLY SOUTH AND EAST NORTH AND WEST WEST ONLY SOUTH AND WEST f. [ points] Did Steph come to a stop between midnight and 5 a.m. on night? If so, at what time(s) did she come to a stop?

11 Math 6 / Midterm (November 4, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page. [ points] Determine whether the following integrals converge or diverge. If an integral converges, find its eact value (i.e., no decimal approimations) and write it in the blank provided. If it diverges, circle DIVERGES and eplain wh. In an case, show all our work, indicating an theorems ou use, and using proper snta and notation. a. [6 points] e c d, where c > is a constant DIVERGES CONVERGES TO b. [6 points] d DIVERGES CONVERGES TO

Math 116 Midterm Exam November 14, 2016

Math 116 Midterm Exam November 14, 2016 Math 6 Midterm Eam November 4, 6 UMID: EXAM SOLUTIONS Initials: Instructor: Section:. Do not open this eam until ou are told to do so.. Do not write our name anwhere on this eam. 3. This eam has pages