Math 132Midterm Examination 2 – March 5, 20126 multiple choice, 4 long answer. 100 points.General Instructions: Please answer the following, without use of calculators. Youmay refer to a 3x5 card, but no other notes. Part I of the exam is multiple choice,while Part II is long answer.Part I Instructions: If you do not have a pencil to fill out your answer card, please askto borrow one from your proctor. Write your Student ID number on the six blank lineson the top of your answer card, and shade in the corresponding bubbles to the rightof each digit.Fill in the bubble corresponding to each of the following 6 questions. Each is worth 4points. On Part I, no partial credit will be given.1. Find the Trapezoid Rule approximation using 4 subintervals ofZ1−1x2dx.(a) 0(b) 1/4(c) 1/3(d) 1/2(e) 2/3(f) 3/4(g) 1(h) 5/4(i) 4/3(j) 3/22. Find the Simpson’s Rule approximation using 4 subintervals ofZ1−1x2dx.(a) 0(b) 1/4(c) 1/3(d) 1/2(e) 2/3(f) 3/4(g) 1(h) 5/4(i) 4/3(j) 3/23. Consider the system consisting of 3 point masses:10 kg at (3, −1)20 kg at (2, 10)100 kg at (1, 0)The center of mass is:(a) (0, 0)(b) (215,315)(c) (213,313)(d) (615,915)(e) (613,913)(f) (1715,1915)(g) (1713,1913)(h) (2, 3)(i) None of the above/does not exist.4. Simpson’s Rule applied to the integralZe11xdx with n = 20 will be closest to:(a) 0(b) 1/10(c) 2/10(d) 3/10(e) 4/10(f) 5/10(g) 6/10(h) 7/10(i) 8/10(j) 9/10(k) 15. Find the average value of sin x over the interval [0, π].(a) 0(b) 1/2(c) π/5(d) 2/π(e) 2/3(f) π/4(g) 3/π(h) 1(i) None of the above/does not exist.6. The decay of a certain radioactive isotope of the element rabbitonium is governed by thedifferential equation y0= −ky. At t = 0 you have 300 mg of radioactive rabbitonium.At t = 45 minutes, you are left with only 100 mg of radioactive rabbitonium.Then k is per minute.(a) ln 2/15.(b) ln 3/15.(c) ln 2/30.(d) ln 3/30.(e) ln 2/45.(f) ln 3/45.(g) ln 2/60.(h) ln 3/60.(i) None of the above.Name: Id #: Math 132Part II Instructions: Answer the following on the exam sheet, showing all your work.Correct answers without correct supporting work may not receive full credit. You mayuse the back of each page for additional answer space (please clearly indicate if youhave done so), or scratch work.Please put your name and student id number on each page of Part II now.1. Differential equations(a) (8 points) Solve the differential equation y0= x+xy subject to the initial conditiony(0) = 5.(b) (8 points) At time t = 0, there is 1000 liters of water in a tank, with 80 kg of saltdissolved in it. Distilled water flows into the tank at 10 L/min, and water flowsout of the tank at the same rate. The tank is continually stirred, and the salt iskept mixed evenly through the tank.Set up a differential equation (you needn’t solve it) for the mass of salt in thetank at time t. (Your answer should be of the form y0= .)Name: Id #: Math 1322. Arc lengths and approximate integration(a) (6 points) Set up a definite integral representing the length of the curve y = x3between x = 0 and x = 4.(b) (10 points) The first several derivatives of f(x) =√1 + x2are as follows:f0(x) =x√1 + x2, f00(x) =1(1 + x2)3/2, f(3)(x) =−3x(1 + x2)5/2,f(4)(x) =12x2− 3(1 + x2)7/2, f(5)(x) =45x − 60x3(x2+ 1)9/2.Find (with justification) an n such that the Simpson’s Rule approximation SnforZ4−1√1 + x2dx has error at most 0.001.Name: Id #: Math 1323. Calculations(a) (6 points) Find an upper bound for2e−(x+1)2+ 12 sin(x + 1)2 on the interval[−3, 3].(b) (7 points) EvaluateZx2cos x dx.(c) (6 points) EvaluateZ10x1 + x2dx.(d) (6 points) EvaluateZ1−1x tan−1x dx.Name: Id #: Math 1324. Volumes and centroidsIn both problems on this page, we consider the region between the x-axis and the graphof y = exfor 0 ≤ x ≤ 2.(a) (11 points) Find the volume of the solid formed by rotating the given regionaround the y-axis.(b) (8 points) Find the center of mass x with respect to x of the solid formed byrotating the given region around the x-axis.Half credit will be received for instead finding the center of mass x of the given(unrotated)
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