Math 116 Second Midterm November 17, 2010

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1 Math 6 Second Midterm November 7, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are problems. 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. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. Include units in your answer where that is appropriate. 6. You may use any calculator except a TI-9 (or other calculator with a full alphanumeric keypad). However, 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. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 8. Turn off all cell phones and pagers, and remove all headphones. Problem Points Score Total

2 Math 6 / Exam (November 7, ) page. [4 points] Indicate if each of the following is true or false by circling the correct answer. Justify your answer. a. [ points] If f(x)dx is divergent then f(x)dx is also divergent. x. f(x)dx can be convergent while f(x)dx is divergent. Example f(x) = b. [ points] If the median of a density function p(t) is, then p(t) is an even function. Example: p(x) = around the y axis..5 x.5 < x otherwise has median but it is not symmetric c. [4 points] A curve is parametrized by the functions x(t) = t and y(t) = t 4 + 3t for t. The concavity of the graph of the parametric curve is positive for < t <. d y dx = ( y x ) x = ( ) 4t 3 +6t t t = ( t 3) t = 4t t =. d. [ points] In polar coordinates, the coordinates (, π 3 ) and (, 7π 3 ) represent the same point. (, π 3 ) is in quadrant I and (, 7π 3 ) is in quadrant III e. [ points] If P (t) is a cumulative distribution function then P (t)dt converges. P (t) is an increasing functions and lim t P (t) = hence P (t)dt di- verges. f. [ points] The solutions to the differential equation dy dx = + y + 3x are increasing at every point. y = + y + 3x > hence y is an increasing function.

3 Math 6 / Exam (November 7, ) page 3. [4 points] The graph of the circle r = 4 and and the cardioid r = sin θ are shown below. 4 4 a. [3 points] Write a formula for the area inside the circle and outside the cardioid in the first quadrant. Area of the quarter of a circle= 4π Area of cardioid= 3π π ( sin θ ) dθ Area=4π 3π π ( sin θ ) dθ b. [7 points] At what angles θ < π is the minimum value of the y coordinate on the cardioid attained? No credit will be given for answers without proper mathematical justification. y(θ) = ( sin θ ) sin θ y (θ) = cos θ sin θ + ( sin θ ) cos θ = 4 cos θ sin θ cos θ Critical points (4 sin θ ) cos θ = cos θ = or sin θ = then θ = π 6, π, 5π 6, 3π. Minimum y coordinate at θ = π 6, 5π 6. c. [4 points] Write an integral that computes the value of the length of the piece of the cardioid lying below the x-axis. x(θ) = ( sin θ ) cos θ x (θ) = cos θ ( sin θ ) sin θ L = π ( cos θ ( sin θ ) sin θ) + (4 cos θ sin θ cos θ) dθ

4 Math 6 / Exam (November 7, ) page 4 3. [3 points] The phones offered by a cell phone company have some chance of failure after they are activated. Suppose that the density function p(t) describing the time t in years that one of their phones will fail is { λe λt for t. p(t) = otherwise a. [5 points] Find the cumulative distribution function P (t) of p(t). P (t) = t λe λt dt = e λt t = e λt { e λt for t. P (t) = otherwise b. [4 points] If the probability of a cell phone failing within a year and a half is 5, find the value of λ..5 λe λt dt = e.5λ e.5λ = 5 then λ = ln( 3 5 ).5 =.34 c. [4 points] The cell phone company offers its clients a replacement phone after two years if they sign a new contract. What is the probability that the client will not have to replace his or her phone before the company will give him or her a new one? λe λt dt = lim b e λt b = lim b e.68 e.34b = e.68 =.56 or P () = e.68 = 56.

5 Math 6 / Exam (November 7, ) page 5 4. [5 points] a. [5 points] Find the regions in the slope field of y = (y 3)y where the slopes are positive, negative or zero. Show all your computations. Positive : y > 3, y < Zero: y =, 3 Negative: < y < 3 b. [ points] Which of the following is the slope field of y = (y 3)y? Circle your answer (a) (b) B) c. [4 points] Find all the equilibrium solutions of y = (y 3)y. Use the slope field of the equation to determine the stability of each equilibrium solution. y = 3 unstable y = stable d. [4 points] Let y(x) be the solution to the differential equation y = (y x)y satisfying y() =. Use Euler s method with steps x = to approximate the value of y(). Show all your computations. (x, y ) = (, ) y = + ( )()( ) = 3 y = 3 + (3 3 )(3)( ) = 5.5 then y() 5.5.

6 Math 6 / Exam (November 7, ) page 6 5. [4 points] A particle moves on the unit circle according to the parametric equations x(t) = sin(bt ), y(t) = cos(bt ) and b >. for t π. Make sure to show all your work. a. [ point] What is the starting point of the particle? (x(), y()) = (, ). b. [ points] In which direction (counterclockwise/clockwise) is the particle moving along the circle? Justify. The particle moves around the unit circle making a counterclockwise angle θ = bt measured from the positive y axis. Since bt is increasing, the particle never changes direction. c. [5 points] Find an expression for the speed of the particle. Simplify it as much as possible. x (t) = tb cos(bt ) y (t) = tb sin(bt ) v(t) = (x ) + (y ) = ( tb cos(bt )) + ( tb sin(bt )) = tb d. [ points] At what value of t in [, π] is the speed of the particle the largest? v(t) = tb is the largest at t = π. e. [4 points] Find the equation of the tangent line to the parametric equation at t = π 3b. x( π 3b ) = 3 =.866, x ( π 3b ) = b ( π 3b ) = πb 3 ) y( π 3b ) =, y ( π 3b ) = b π 3b ( 3 = πb Parametric equation for the tangent line: πb x tan (t) = t y tan(t) =.5 πbt

7 Math 6 / Exam (November 7, ) page 7 6. [5 points] Each of the integrals below are improper. Determine the convergence or divergence of each. Make sure you include all the appropriate steps to justify your answers. Approximations with your calculator will not receive credit. a. [4 points] 5 sin x dx x 3 Since sin x then 3 5 sin x 7. This yields 3 5 sin x 7 x 3 x 3 x 3 5 sin x 7 dx dx x 3 x 3 converges b. [5 points] x (x 3 ) dx x (x 3 dx = lim ) b + b x (x 3 dx = lim ) b + 3(x 3 ) b = lim b + 3(b 3 ) diverges c. [6 points] dx (x 3 + 7) 3 (x 3 + 7) 3 x for x since for x we have x 3. Multiplying by 7 and adding x 3 we get 8x 3 x Hence x 3 +7 and by taking the cube root on both sides we get 8x 3 (x 3 +7) x. Hence 3 dx (x 3 + 7) 3 x dx diverges

8 Math 6 / Exam (November 7, ) page 8 7. [5 points] An aquarium containing liters of fresh water will be filled with a variety of small fish and aquatic plants. A water filter is installed on the tank to help remove the ammonia produced by the decomposing organic matter generated by plants and fish in the aquarium. The filter takes water from the tank at a rate of liters every hour. The water then is filtered and returned to the aquarium at the same rate of liters every hour. Ninety percent of the ammonia contained in the water that goes through the filter is removed. It is estimated that the fish and plants produce 3 mg of ammonia every hour. Assume the ammonia mixes instantly with the water in the aquarium.

9 Math 6 / Exam (November 7, ) page 9 a. [6 points] Let Q(t) be the amount in mg of ammonia in the fish tank t hours after the fish were introduced into the aquarium. Find the differential equation satisfied by Q(t). Include its initial condition. ( ) ( ) dq Q Q = 3 + (.) dt dq = 3.8Q Q() = dt b. [7 points] Find the amount of ammonia in the fish tank 3 hours after the fish were introduced into the aquarium. Make sure to include units in your answer. dq = 3.8Q dt dq 3.8Q = dt ln 3.8Q = t + C.8 Q(t) = De.8t = De.8t using Q() = you get Q(t) = 66.6( e.8t ) Q(3) = mg c. [ points] What happens to the value of Q(t) in the long run? lim t Q(t) = lim t 66.6( e.8t ) = The value of Q(t) converges to 66.6 mg.

Math 116 Second Midterm November 14, 2012

Math 116 Second Midterm November 14, 2012 Math 6 Second Midterm November 4, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has pages including this cover. There are 8 problems. Note that