MATH 140A FINAL EXAMINATION May 3, 2004Name ID # Section #There are 16 multiple-choice questions, 8 True/False questions, and 3 partial credit ques-tions. For the partial credit problems you must present your work clearly and under-standably; no credit will be given for unsupported answers. For True/False andmultiple-choice problems, please circle the correct answer in each question.THE USE OF CALCULATORS IS NOT PERMITTED IN THISEXAMINATION.There are 20 problems on 13 pages, including this one.Check your booklet now.The area below is for the instructor’s use.MC .................... (96)T/F ................... (16)18 ...................... (10)19 ...................... (14)20 ...................... (14)Total ................. (150)MATH 140A FINAL EXAMINATION PAGE 21. (6 pts.) limx→5x2(x − 5)3=a) 5b) 25c) −∞d) ∞e) The limit does not exist.2. (6 pts.) limx→2−|x − 2|x2− 4=a)−14b) 0c) 1d)14e) The limit does not exist.MATH 140A FINAL EXAMINATION PAGE 33. (6 pts.) limx→3x2− 3x√x + 1 − 2=a) 12b) 9c) 3d) 1e) The limit does not exist.4. (6 pts.) What is the instantaneous rate of change of the function f (x) = tan(πx2) at x =12?a) 0b) 1c)√2πd) 2πe) −πMATH 140A FINAL EXAMINATION PAGE 45. (6 pts.) What are the horizontal and the vertical asymptotes of the functionf(x) =2x(x + 2)2(x − 1)x2(x − 1)?a) H.A. none; V.A. noneb) H.A. y = 2; V.A. nonec) H.A. y = 2; V.A. x = 0 and x = 1d) H.A. y = 2; V.A. x = 0e) H.A. none; V.A. x = 06. (6 pts.) Let F (x) = f (g(x)). Suppose f(2) = 5, f0(2) = −1, f (3) = 7, f0(3) = 2, g(2) =3, g0(2) = 4, g(5) = 6, and g0(5) = −2. Find F0(2).a) −4b) 8c) 2d) 7e) −10MATH 140A FINAL EXAMINATION PAGE 57. (6 pts.) A right triangle has base b and height h. Suppose the base is decreasing at a rate of12m/sec, while the height is increasing at a rate of 2 m/sec. At what rate is the triangle’sarea increasing when b = 4 and h = 5?a) −12m2/secb)114m2/secc)214m2/secd)32m2/sece) 0 m2/sec8. (6 pts.) Suppose f(x) is a differentiable function and f0(x) = x3(x + 2)(x − 2)2, then f(x)has a local minimum ata) x = −2 onlyb) x = 0 onlyc) x = 0, x = 2d) x = −2, x = 2e) x = −2, x = 0, and x = 2MATH 140A FINAL EXAMINATION PAGE 69. (6 pts.) Suppose x and y are two positive numbers such that xy = 6, what is the minimumvalue of the sum 2x + 3y?a) 2b) 3c) 5d) 12e) 1510. (6 pts.) Suppose f(x) is a continuous function and thatZ30f(x) dx = 2,Z96f(x) dx = −1,Z90f(x) dx = 5then what isZ36f(x) dx?a) −4b) −2c) 4d) 6e) The value cannot be determined by the information given.MATH 140A FINAL EXAMINATION PAGE 711. (6 pts.)Z2−2(|x| + 1) dx =a) 0b) 2c) 4d) 8e) The definite integral does not exist because the integrand is not differentiable atx = 0.12. (6 pts.)Zπ40tan2x dx =a) 0b) 1c) 1 −π4d)√2e)13MATH 140A FINAL EXAMINATION PAGE 813. (6 pts.) A particle is moving along a straight line with a velocity given by v(t) = t2−2t + 4.What is the net change of its position between t = 1 and t = 3?a) −143b) −2c) 4d) 12e)26314. (6 pts.) EvaluateddxZ√xπt cos3(2t2) dt.a)√x cos3(2x)b)√x cos3(4x)c)14√x cos4(2x)d)12cos3(2x)e)18cos4(4x)MATH 140A FINAL EXAMINATION PAGE 915. (6 pts.) When using the substitution u =√x, the definite integralZ251sin3√x√xdx becomesa)12Z251sin3u dub) 2Z51sin3u duc)Z251sin3u dud)Z51u sin3u due)Z51sin3uudu16. (6 pts.) The average (mean) value of the function f(x) = 3x2− 2x + 2 on the interval [0, 3]isa) 7b) 8c)234d) 21e) 24MATH 140A FINAL EXAMINATION PAGE 1017. (16 pts., 2 pts. each) Suppose f is a function differentiable everywhere on (−∞, ∞), suchthat f(0) = 3, f0(0) = 0, f (2) = −1, f0(2) = 5, f (10) = 10, and f0(10) = −2. True or False:a) T F f must be continuous everywhere on (−∞, ∞).b) T F f must have at least 2 real roots.c) T F The average rate of change of f, between x = 0 and x = 10, is −15.d) T F There must exist a point x = c, 0 < c < 2, such that f0(c) = −2.e) T F limx→0−f(x) = 0.f) T F The graph y = f(x) might not have a tangent line at x = 0.g) T F The point x = 0 is a critical point of f .h) T F The graph y = f(x) might have a vertical asymptote somewhere.MATH 140A FINAL EXAMINATION PAGE 1118. (10 pts.) Use the substitution method of integration to evaluate the definite integralZ40(2 +√x)34√xdx.MATH 140A FINAL EXAMINATION PAGE 1219. (14 pts.) Consider the region(s) in the xy–plane enclosed by the curves y = x3and y = 3x.a) (3 points) Find the x and y–coordinates of the points of intersection of these curves.b) (3 points) Draw the region(s). Label the curves and their points of intersection.c) (8 points) Calculate the total area of the region(s).MATH 140A FINAL EXAMINATION PAGE 1320. (14 pts.) Consider the first quadrant region bounded by the curves y = x3and y = 3x. Setup integrals which measure the volumes of the solids obtained by rotating this region aboutthe indicated axes. DO NOT EVALUATE THE INTEGRALS.a) Rotate about the x–axis.b) Rotate about the