MATH 140 NAME Final Exam STUDENT NUMBER May 04 2004 INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service Use only a number 2 pencil on your scantron On your scantron identify your name this course Math 140 and the date Code and blacken the corresponding circles on your scantron for your student 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 on the scantron Mark only one choice darken the circle completely you should not be able to see the letter after you have darkened the circle Check frequently to be sure the problem number on the test is the same as the problem number of the scantron There are 6 partial credit questions 60 points In order to obtain full credit for the partial credit problems all work must be shown Credit will not be given for an answer not supported by work The point value for each partial credit question is given in parentheses to 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 Total Do not write in the box to the left MATH 140 Final Exam PAGE 2 1 1 Find the equation for the tangent line to the curve defined by y x cos x at x 2 1 1 a y x 2 4 1 1 b y x 2 4 c y x 2 4 d y x 2 4 e y x 2 2 Find the limit of x2 3x 2 x 1 x2 x 2 lim a 0 b 1 c 1 3 d 1 3 e The limit doesn t exist MATH 140 Final Exam PAGE 3 3 Let f x be a function defined by f x cos x c if x 0 2 x 2x 2 if x 0 Determine the constant c so that the function f x is continuous a c 1 b c 0 c c 1 d c 2 e c 3 4 y is a differentiable function of x satisfying x3 xy y 3 1 Find a 1 3 b 1 3 c 3 d 3 e 1 dy at 0 1 dx MATH 140 Final Exam PAGE 4 5 Which of these are inflection points of function f x whose second derivative is given by f 00 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 f 0 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 3 c f x has a local maximum at x 1 and x 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 5 7 A water tank has the shape of a circular cylinder of radius 2 ft and height 6 ft If water is being pumped into the tank at a rate of 2 f t3 min how fast is the water level rising a 1 2 1 12 c 4 b d 1 2 e 1 4 8 Find the vertical and horizontal asymptotes for y 2x2 x 1 if it has any x2 1 a vertical asymptote x 1 horizontal asymptote y 2 b vertical asymptote x 1 horizontal asymptote y 0 c vertical asymptote x 1 horizontal asymptote y 2 d vertical asymptote x 1 horizontal asymptote y 1 2 e vertical asymptote x 1 x 1 No horizontal asymptote MATH 140 Final Exam PAGE 6 9 Use Newton s method with the initial approximation x0 2 to find x1 the next approximation to a root of x5 34 0 a 79 40 b 81 40 c 77 40 d 161 80 e 83 40 Z 10 If F x 0 a 6 b 2 c 3 d 4 e 8 x2 1 8t3 dt find the value of F 0 1 MATH 140 Z Final Exam Z 3 11 If f x dx 12 and 0 PAGE 7 Z 6 0 3 a 50 b 51 c 53 d 56 e It cannot be determined 12 Find the value of the integral Z 3 sec x tan x 1 sec x dx 0 a 4 b 3 c 11 2 d 5 2 e 7 2 6 2f x 3 dx f x dx 42 find the value of MATH 140 Final Exam PAGE 8 h i 13 Find the average value of f x cos x on the interval 0 2 a 2 b 2 c 0 d 1 e 14 Find the area of the region between the curve f x x2 1 and the x axis for 0 x 2 a 4 b 2 c 11 3 d 2 3 e 7 3 MATH 140 Final Exam 15 Find the volume of revolution obtained by taking the region bounded by the curves y x x 2 x 3 y 0 rotated around the x axis a 3 2 b 5 2 c 5 d 3 2 e 9 2 PAGE 9 MATH 140 Final Exam 16 10 points Evaluate the following indefinite integrals Z a Z b 1 dx x 1 x 2 cos d sin2 PAGE 10 MATH 140 Final Exam PAGE 11 17 10 points What is the largest possible area for a right triangle whose hypoteneuse is 5 cm long MATH 140 Final Exam PAGE 12 18 6 points Estimate the Riemann sum for f x sin x 0 x with n 4 subintervals taking the sample points to be right end points MATH 140 Final Exam PAGE 13 19 10 points Let R be the region of the plane enclosed by the curve x y 2 1 and the line x y 1 a Sketch the region and label all points of intersections b Find the area between the curves x y 2 1 and x y 1 MATH 140 Final Exam PAGE 14 20 12 points Let R be the region in the first quadrant bounded by the curves y x3 and y 2x2 a Write down an integral to express the volume of the solid generated by rotating R about x axis DO NOT EVALUATE THE INTEGRAL b Write down an integral to express the volume of the solid generated by rotating R about y axis DO NOT EVALUATE THE INTEGRAL MATH 140 Final Exam PAGE 15 21 12 points Let f x be a function satisfying f …