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Unformatted text preview: SIMON FRASER UNIVERSITY MATH 155 Final Examination 12 April 2007, 08:30–11:30 Last Name Given Name(s) Student # Signature INSTRUCTIONS 1. Do not open this booklet until told to do so. 2. Write your last name, given name(s), and student number in the box above. Sign on the last line of the box. 3. This exam has 12 questions on 13 pages. Check to make sure that your exam is complete. 4. No book, paper or device other than usual writing instruments, this examination booklet, and a scientific calculator are allowed. Calculators with graphing and/or symbolic computation capabilities are not allowed. 5. During the examination, speaking to, communicating with, or exposing written papers to the view of other examinees is forbidden. 6. You may use the reverse side of the previous page for rough work or if you run out of space. 7. You may lose marks if your explanations are incomplete or poorly presented. 8. Stop writing when you are instructed to do so. Failure to follow instructions may result in penalties. Question Maximum Score 1 2 3 4 5 6 7 8 9 10 11 12 Total 8 8 6 9 8 6 8 11 8 9 11 8 100 MATH 155 Page 1 of 13 [2] 1. (a) Use the substitution u = sin x to find ￿ sin x cos x dx. [2] (b) Use the substitution u = cos x to find ￿ sin x cos x dx. [4] (c) The antiderivatives obtained in parts (a) and (b) are different. Write down the general relation satisfied by any two antiderivatives of the same function. Verify that this relation holds for the antiderivatives obtained in parts (a) and (b) above. MATH 155 Page 2 of 13 [8] 2. Find ￿ ex sin x dx. MATH 155 Page 3 of 13 [6] 3. Suppose that h hours after midnight, the outside air temperature (in degrees Celsius) can be modeled by the formula 1 T (h) = 6 − (h − 13)2 . 5 Find the average temperature between 5:00 A.M. and 2:00 P.M. MATH 155 Page 4 of 13 [9] 4. An orange has the shape of a perfect sphere (ball) of radius 5 cm. Suppose that we divide the orange into two pieces by a planar cut; the radius of the circle created by the cut is 4 cm. Use an integral to determine the volume of the smaller of the two pieces (a so-called spherical cap) that we obtained after the cut. MATH 155 Page 5 of 13 [8] 5. Evaluate ￿ 2 0 1 dx. (x − 1)2 MATH 155 Page 6 of 13 [6] 6. Suppose that you found an approximation of dx using the Trape1x zoidal Rule with n = 1, 000 intervals. Is this approximate value of the integral larger or smaller than the exact value of the integral? Explain your answer. You are not allowed to perform any actual numerical computations with the Trapezoidal Rule. Hint: Draw a graph! ￿ 2 MATH 155 Page 7 of 13 [8] 7. Compute an approximation of ln(0.9) with the accuracy of three places after the decimal point. You are only allowed to use the basic arithmetic operations +, −, ×, ÷. Hint: Use Taylor polynomials of increasing degree. Stop as soon as two successive approximations (computed to the full precision of your calculator) agree after being rounded to the target precision. Show also the values before rounding. MATH 155 Page 8 of 13 8. The rate at which the temperature of an object changes is proportional to (i.e., it is a constant multiple of) the difference between its own temperature and the temperature of the surrounding medium. [5] (a) Let T (t) denote the temperature of the object at time t. Let the constant of proportionality be denoted k . Assume that the temperature of the surrounding medium is constant and denoted by M . Describe the situation given in the previous paragraph by a differential equation. Indicate whether in your model the constant k is positive or negative. [6] (b) Assume that T (0) = 80, M = 30 and k = 0.2 or k = −0.2 depending on the choice that you made in part (a). Find T (10), i.e. the temperature of the object at time t = 10. MATH 155 Page 9 of 13 [3] 9. (a) The Logistic Growth model with harvesting is described by the differential equation ￿ ￿ dN N = rN 1 − −H dt K where N (t) is the population size at time t, and H is the constant harvest rate. Explain the meaning of the other constants present in the model. [5] (b) What is the maximum harvest rate Hmax that allows to maintain a positive population size? Hint: You need a locally stable equilibrium of positive value. Express Hmax as a function of the other parameters introduced in part (a). MATH 155 Page 10 of 13 10. The Levins model of metapopulations is given by the differential equation dp = cp(1 − p) − mp. dt [3] (a) Explain the meaning of p(t), c and m in the differential equation written above. [6] (b) Analyze the equilibria for the differential equation written above. Remember that there are two cases to distinguish. MATH 155 Page 11 of 13 11. Suppose that the dynamics of a certain population is given by the following Leslie matrix: 0 1.3 2.2 L = 0.6 0 0 0 0.6 0 At time t = 0 the number of females in the population is given by 947 N (0) = 473 . 236 [5] (a) Compute the sizes of the age groups at the end of the next breeding season. The population is approaching a stable age distribution. What may be a possible value of λ? (Hint: λ is a “nice” number.) MATH 155 Page 12 of 13 [6] (b) The question 11. continues here. Using the guess that you made in regards to λ in part (a), set up a system of linear equations whose solution is a stable age distribution. Find the exact pattern of the stable age distribution. MATH 155 Page 13 of 13 12. The function is defined for all pairs (x, y ) ∈ R2 . [4] (a) Find all critical points for f . f (x, y ) = x + 2y − xy 2 [4] (b) For each critical point found in part (a), determine its nature. ...
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