Math 132, Spring 2011 - Exam 2NAME:STUDENT ID NUMBER:This exam contains sixteen questions. The first fourteenare multiple choice questions and count for five pointseach. There is no partial credit on these questions, soread each question carefully, check your arithmeticand make sure that you have marked the answer you in-tended to mark. The last two questions, which are eachworth fifteen p oints, require written answers, and somepartial credit might be given. However, no credit will begiven for information that is not germane to the problemat hand. Please make sure to write your name and stu-dent ID number on the pages that include your answersto the last two questions. In fact, you will get onepoint on each of these two questions for writingyour name and ID number legibly.11. Use the Trapezoidal Rule with n = 4 to approximatethe integralZ511xdx(a) ln(5)(b) 5/3(c) 11/6(d) 19/7(e) 101/60(f) 103/30(g) 1.6094(h) 2.366722. Suppose that on the interval [0, 10], the function fis continuous, f(x) ≥ 0, f is decreasing, and f hasgraph which is concave up. For any value of n, listthe numbers Ln, Rn, Tn, E in increasing order, whereLnis the left endpoint approximation, Rnis the rightendpoint approximation, Tnis the trapezoidal ap-proximation, and E is the exact value of the integralZ51f(x) dx(a) Rn< E < Ln< Tn(b) Rn< E < Tn< Ln(c) Rn< Tn< E < Ln(d) Rn< Tn< Ln< E(e) Ln< E < Rn< Tn(f) Ln< E < Tn< Rn(g) Ln< Tn< E < Rn(h) Ln< Tn< Rn< E33. How large do we have to choose n so that the errorin using Simpson’s Rule to approximateZ1711x4dxis less than 10−4.(a) 1(b) 2(c) 16(d) 24(e) 134(f) 472(g) 26,368(h) 46,66844. ComputeZ∞e1x(lnx)3dx(a) diverges to ∞(b) diverges to −∞(c) diverges but not to ∞ or −∞(d) converges to e(e) converges to −e(f) converges to 1(g) converges to e − 1(h) converges to 1/255. Find the area of the region bounded by the curvesy = ex, y = x2− 1, x = −1 and x = 1.(a) e − 1/e + 4/3(b) e + 1/e + 1/3(c) e(d) 1/e(e) e − 1/e(f) e + 1/e(g) 1 + e(h) 2 + e66. Find the area of the region bounded by the curvesx = 1 − y2and x = y2− 1.(a) 2(b) 0(c) 2 − 2y2(d) 1 −√2(e) 1(f) 4/3(g) 8/3(h) 15/477. Compute the volume of the solid obtained by ro-tating the region in the first quadrant bounded byy = x2, y = 4 and x = 0 about the y-axis.(a)403√2π(b) 8π(c) 32π/5(d) 16/3(e) 16π/3(f) 16π2/3(g) 64π/3(h) 12π88. Compute the volume of the solid obtained by ro-tating the region in the first quadrant bounded byy = x2, y = 2 − x2and x = 0 about the x-axis.(a) 11π/5(b) 8π/3(c) 32π/15(d) 16π/5(e) 16π/3(f) 16π2/3(g) 64π/15(h) 12π/599. Compute the volume of the solid obtained by rotat-ing about the line x = −1 the region bounded byy = sin x, y = 0, 0 ≤ x ≤ π/2.(a) 1(b) π(c) 2π(d) 4π(e) (1 +√2)π(f)√3π/2(g)√3π(h) 2√3π1010. Find the exact length of the curve given parametri-cally byx = 1 + 3t2; y = 4 + t3where 0 ≤ t ≤√5.(a) 1(b) 19(c) 25(d) 45(e) 3√5(f) 5√5(g) 6√5(h) 9√51111. Find the exact length of the curve given by x =23y32,0 ≤ y ≤ 1.(a) tan−1(3/2)(b) tan−1(π/7)(c)4√23(d)4√2−13(e)9√24(f)4√227(g)4√2−23(h)3√221212. Find the average value of f(x) =1xln xon the interval[e, e2].(a)1e(b)1e−1(c)1e+1(d)1e2−e(e)2e2−e(f)ln 2e2−e(g)ln(ln2)e2−e(h)(ln2)2e2−e1313. Let f(x) = c/(1 + x2). For what values of c is f aprobability density function?(a) no value of c makes f a probability density func-tion(b) 0(c) 1(d) π/2(e) −π/2(f) 2/π(g) −2/π(h) π−11414. Suppose you are given the probability density func-tionf(x) =xe−x, x ≥ 00, x < 0Find P (1 ≤ X ≤ 2).(a) e2− e−2(b)2e−3e2(c)2e2−1e(d)2e2+1e(e)2e2−3e(f)3e−2e2(g)1e−1e−1(h)1e2−1e15Name: Student ID:15. Find the volume of the solid generated by revolvingthe region bounded by y = 1 − x2and the x-axisabout the line x = −4.(a) Use Method I (disc/washer method)(b) Use Method II (cylindrical shells method)16Name: Student ID:16. EvaluateZ20ln x dxShow all work.17Name: Student ID:18Name: Student
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