MATH 140 NAMEFinal Exam STUDENT NUMBERMay 04, 2004 INSTRUCTORSECTION NUMBERThis examination will be machine processed by the University Testing Service. Use only anumber 2 pencil on your scantron. On your scantron identify your name, this course (Math140) and the date. Code and blacken the corresponding circles on your scantron for yourstudent I.D. number and class section number. Code in your test form.There are 15 multiple choice questions worth a total of 90 points. For the problems 1 to 15,five possible answers are given, only one of which is correct. You should solve the problem,circle the letter of your answer in the exam form and blacken the corresponding space onthe scantron. Mark only one choice; darken the circle completely (you should not be ableto see the letter after you have darkened the circle). Check frequently to be sure the problemnumber on the test is the same as the problem number of the scantron. There are 6 partialcredit questions (60 points). In order to obtain full credit for the partial creditproblems, all work must be shown. Credit will not be given for an answer notsupported by work. The point value for each partial credit question is given in parenthesesto the right of the question number.THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINATION.M.C. ( 90 pts.)16. (10 pts.) 17. (10 pts.)18. (6 pts.) 19. (10 pts.)20. (12 pts.) 21. (12 pts.)TotalDo notwrite inthe box tothe left.MATH 140 Final Exam PAGE 21. Find the equation for the tangent line to the curve defined by y = x cos(πx)atx =12.a) y =12x −14b) y = −12x +14c) y =π2x −π4d) y = −π2x +π4e) y = −πx +π22. Find the limit oflimx→−1x2+3x +2x2− x − 2a) 0b) 1c)13d) −13e) The limit doesn’t exist.MATH 140 Final Exam PAGE 33. Let f(x) be a function defined byf(x)=cosx + c if x ≤ 0= x2+2x +2 if x>0Determine the constant c so that the function f(x) is continuous.a) c = −1b) c =0c) c =1d) c =2e) c =34. y is a differentiable function of x satisfying x3− xy + y3=1. Finddydxat (0, 1).a)13b) −13c) 3d) −3e) 1MATH 140 Final Exam PAGE 45. Which of these are inflection points of function f (x) whose second derivative is given byf00(x)=x2(x − 1)(x +2)3.I. x =0, II. x =1, III. x = −2.a) I and II only.b) II and III only.c) I and III only.d) I, II and III.e) No inflection point.6. Suppose f0(x)=x(x +1)2(x − 3). Which of the following statement is ture?a) f(x) is an increasing function over (0, 3).b) f (x) has a local minimum at x =3c) f (x) has a local maximum at x = −1andx =0.d) f (x) is an decreasing function over (−∞, 0) and (3, ∞).e) f (x) has critical numbers at x =1, 0, −3.MATH 140 Final Exam PAGE 57. A water tank has the shape of a circular cylinder of radius 2 ft and height 6 ft. If water isbeing pumped into the tank at a rate of 2 ft3/min, how fast is the water level rising?a)12b)112πc)π4d)12πe)148. Find the vertical and horizontal asymptotes for y =2x2− x − 1x2− 1, if it has any.a) vertical asymptote x = 1, horizontal asymptote y =2b) vertical asymptote x = 1, horizontal asymptote y =0c) vertical asymptote x = −1, horizontal asymptote y =2d) vertical asymptote x = −1, horizontal asymptote y =12e) vertical asymptote x =1,x = −1, No horizontal asymptoteMATH 140 Final Exam PAGE 69. Use Newton’s method with the initial approximation x0= 2 to find x1, the next approxima-tion to a root of x5− 34 = 0.a)7940b)8140c)7740d)16180e)834010. If F (x)=Zx20√1+8t3dt, find the value of F0(1).a) 6b) 2c) 3d) 4e) 8MATH 140 Final Exam PAGE 711. IfZ30f(x) dx =12andZ60f(x) dx = 42, find the value ofZ63[2f(x) − 3] dx.a) 50b) 51c) 53d) 56e) It cannot be determined.12. Find the value of the integral.Zπ30sec x tan x (1 + sec x) dxa) 4b) 3c)112d)52e)72MATH 140 Final Exam PAGE 813. Find the average value of f(x)=cosx on the intervalh0,π2i.a)π2b)2πc) 0d) 1e) π14. Find the area of the region between the curve f (x)=x2− 1andthex-axisfor0≤ x ≤ 2.a) 4b) 2c)113d)23e)73MATH 140 Final Exam PAGE 915. Find the volume of revolution obtained by taking the region bounded by the curvesy =√x, x =2,x=3,y=0rotated around the x-axis.a)32b)5π2c) 5d)3π2e)9π2MATH 140 Final Exam PAGE 1016. (10 points) Evaluate the following indefinite integrals.a)Z1√x(1 +√x)2dxb)Zcos√θ√θ sin2(√θ)dθMATH 140 Final Exam PAGE 1117. (10 points) What is the largest possible area for a right triangle whose hypoteneuse is 5 cmlong?MATH 140 Final Exam PAGE 1218. (6 points) Estimate the Riemann sum for f(x)=sinx, 0 ≤ x ≤ π,withn = 4 subintervals,takingthesamplepointstoberightendpoints.MATH 140 Final Exam PAGE 1319. (10 points) Let R be the region of the plane enclosed by the curve x − y2= 1 and the linex + y =1.a) Sketch the region and label all points of intersections.b) Find the area between the curves x − y2=1andx + y =1.MATH 140 Final Exam PAGE 1420. (12 points) Let R be the region in the first quadrant bounded by the curves y = x3andy =2x2.a) Write down an integral to express the volume of the solid generated by rotating Rabout x-axis. DO NOT EVALUATE THE INTEGRAL.b) Write down an integral to express the volume of the solid generated by rotating Rabout y-axis. DO NOT EVALUATE THE INTEGRAL.MATH 140 Final Exam PAGE 1521. (12 points) Let f (x) be a function satisfyingf(x + h) − f(x)=x2h + h2+3h, for all x, ha) Use the formal limit definition of the derivative to find f’(x).(No credit will be given to solutions that do not use the limit definition of the deriv-ative).b) Suppose that f(0) = 2. Find a formula for f(x).Hint: f(x) is an antiderivative of f’(x). Use your answer to part