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Question #1 FAMAT State Calculus Bowl 2007 Let f be the function defined by f ( x) 3x 2 4 x3 . 2 Let A = the sum of all the x-coordinates of any point where f ( x) has a tangent line parallel to the line y 9 x 8. Let B = the sum of all the x-coordinates for all points of inflection of f ( x ). Find A + B. Question #1 FAMAT State Calculus Bowl 2007 Let f be the function defined by f ( x) 3x 2 4 x3 . 2 Let A = the sum of all the x-coordinates of any point where f ( x) has a tangent line parallel to the line y 9 x 8. Let B = the sum of all the x-coordinates for all points of inflection of f ( x ). Find A + B. Question #2 FAMAT State Calculus Bowl 2007 The function f ( x) is continuous on a domain of [-4, 4] and is symmetric with respect to the origin. The first and second derivatives of f ( x) have the properties shown in the following chart. x O<x<1 x=1 1<x<3 x=3 3<x<4 / Positive DNE Negative 0 Negative f ( x) Positive DNE Positive 0 Negative f // ( x) Let A = the sum of all the x-coordinates of all relative extrema on the domain [-4, 4]. Let B = the sum of all points of inflection on the domain [-4, 4]. Find A + B -1 Question #2 FAMAT State Calculus Bowl 2007 The function f ( x) is continuous on a domain of [-4, 4] and is symmetric with respect to the origin. The first and second derivatives of f ( x) have the properties shown in the following chart. x O<x<1 x=1 1<x<3 x=3 3<x<4 / Positive DNE Negative 0 Negative f ( x) Positive DNE Positive 0 Negative f // ( x) Let A = the sum of all the x-coordinates of all relative extrema on the domain [-4, 4]. Let B = the sum of all points of inflection on the domain [-4, 4]. Find A + B -1 Question #3 FAMAT State Calculus Bowl 2007 Suppose g ( x) 3 5( x 2) 7( x 2)2 6( x 2)3 is the third degree Taylor polynomial for the function g about 2. Assume that g is differentiable for all orders of real numbers. Find the sum of g (2) and g // (2). Question #3 FAMAT State Calculus Bowl 2007 Suppose g ( x) 3 5( x 2) 7( x 2)2 6( x 2)3 is the third degree Taylor polynomial for the function g about 2. Assume that g is differentiable for all orders of real numbers. Find the sum of g (2) and g // (2). Question #4 FAMAT State Calculus Bowl 2007 Let A = the volume of the solid whose base is the circle x2 y 2 9 and whose cross-sections perpendicular to the x-axis are squares. Let B = the volume of the solid of revolution whose base is bounded by the lines f ( x) 1 x, g ( x) x 1, and x 0 and whose cross-sections are semicircles perpendicular to the x-axis. Find A + B. Question #4 FAMAT State Calculus Bowl 2007 Let A = the volume of the solid whose base is the circle x2 y 2 9 and whose cross-sections perpendicular to the x-axis are squares. Let B = the volume of the solid of revolution whose base is bounded by the lines f ( x) 1 x, g ( x) x 1, and x 0 and whose cross-sections are semicircles perpendicular to the x-axis. Find A + B. Question #5 FAMAT State Calculus Bowl 2007 A particle moves along the x-axis so that the force required to move it when it is x feet from the origin is f ( x) 5 3x 2lbs. Let A = the amount of work done (in ft-lbs) in moving the particle from x = 3 to x = 7. A bucket weighing 10 lbs. is filled with water weighing 15 lbs. The bucket is at the bottom of a 30 ft. well, hanging from a chain that weighs 0.2 lbs/ft. Let B = the amount of work (in ft-lbs) that is done by lifting the bucket to the top of the well. Find A + B. Question #5 FAMAT State Calculus Bowl 2007 A particle moves along the x-axis so that the force required to move it when it is x feet from the origin is f ( x) 5 3x 2lbs. Let A = the amount of work done (in ft-lbs) in moving the particle from x = 3 to x = 7. A bucket weighing 10 lbs. is filled with water weighing 15 lbs. The bucket is at the bottom of a 30 ft. well, hanging from a chain that weighs 0.2 lbs/ft. Let B = the amount of work (in ft-lbs) that is done by lifting the bucket to the top of the well. Find A + B. Question #6 FAMAT State Calculus Bowl 2007 0 Let A = the value of e x dx . x e 1 If the integral diverges, then A = 1. Let B = the value of e x dx . x e 1 If the integral diverges, then B = 2. Find A + B. Question #6 FAMAT State Calculus Bowl 2007 0 Let A = the value of e x dx . x e 1 If the integral diverges, then A = 1. Let B = the value of Find A + B. e x dx . x e 1 If the integral diverges, then B = 2. Question #7 FAMAT State Calculus Bowl 2007 Let A = the radius of convergence for the series Let B = the radius of convergence for the series ( x 3)2n . n 4 n1 ( x 2)n n 0 3n ln(2n) converges. 2 n Let C = the value to which the sequence of Find A + B + C. Question #7 FAMAT State Calculus Bowl 2007 Let A = the radius of convergence for the series Let B = the radius of convergence for the series ( x 3)2n . 4n n1 ( x 2)n n 0 3n ln(2n) converges. 2 n Let C = the value to which the sequence of Find A + B + C. Question #8 FAMAT State Calculus Bowl 2007 Of the four statements or expressions below, which one(s) is/are true? Mark the letters for the true responses on your answer sheet. 3 A. Calculating 1 /4 B. 0 2 x dx using the Trapezoidal Rule with 6 subdivisions on [1, 3] gives a value of tan 4 x sec4 xdx 235 . 27 11 . 35 C. The volume generated when the region bounded by the circle y 2 z 2 10 y 21 0 in the yz-plane is revolved about the z-axis (to make a torus) is 40 . D. The length of the curve 5y3 x 2 that lies inside the circle x 2 y 2 6 is 134 . 27 Question #8 FAMAT State Calculus Bowl 2007 Of the four statements or expressions below, which one(s) is/are true? Mark the letters for the true responses on your answer sheet. 3 E. Calculating /4 F. 0 1 x dx using the Trapezoidal Rule with 6 subdivisions on [1, 3] gives a value of 2 tan 4 x sec4 xdx 235 . 27 11 . 35 G. The volume generated when the region bounded by the circle y 2 z 2 10 y 21 0 in the yz-plane is revolved about the z-axis (to make a torus) is 40 . H. The length of the curve 5y3 x 2 that lies inside the circle x 2 y 2 6 is 134 . 27 Question #9 FAMAT State Calculus Bowl 2007 A = lim tan 2u csc 4u u 0 C = lim y 1 3 y 3y 5 3 B = lim x 6 x 4 2 x3 23x 2 12 x 36 x 4 6 x3 5 x 2 106 x 120 D = lim z z 2 5z z 2 7 z Find AB + CD Question #9 FAMAT State Calculus Bowl 2007 A = lim tan 2u csc 4u u 0 C = lim y 1 3 y 3y 5 3 Find AB + CD B = lim x 6 x 4 2 x3 23x 2 12 x 36 x 4 6 x3 5 x 2 106 x 120 D = lim z z 2 5z z 2 7 z Question #10 FAMAT State Calculus Bowl 2007 This question is a relay type question. The answer to part (I), A, will be used in part (II)…and so on through part (III). On your answer sheet, only put down the exact value of C. 2 (I). 0 x x cos dx A 2 (II). lim y 3 y 2 2 y 15 B y 2 8 y A 23 x / . f ( ) C. B (III). Let f ( x) x 2 cos Question #10 FAMAT State Calculus Bowl 2007 This question is a relay type question. The answer to part (I), A, will be used in part (II)…and so on through part (III). On your answer sheet, only put down the exact value of C. 2 (I). 0 x x cos dx A 2 (II). lim y 3 y 2 2 y 15 B y 2 8 y A 23 x / . f ( ) C. B (III). Let f ( x) x 2 cos Question #11 FAMAT State Calculus Bowl 2007 4x If f ( x) , then find the value of f 2 8x 1 2 3 25 f / ( 3) lim f ( x) x 3 0 f ( x) 8x 2 1dx Question #11 FAMAT State Calculus Bowl 2007 4x If f ( x) , then find the value of f 2 8x 1 2 3 25 f / ( 3) lim f ( x) x 3 0 f ( x) 8x 2 1dx Question #12 FAMAT State Calculus Bowl 2007 e / 2 Find the value of 1 sin(ln x)dx e / 2 1 cos(ln x)dx Question #12 FAMAT State Calculus Bowl 2007 e / 2 Find the value of 1 sin(ln x)dx e / 2 1 cos(ln x)dx Question #13 FAMAT State Calculus Bowl 2007 7 3 Let f ( x) 2 x . What is the value of the constant term in the full expansion of f / ( x)? x Question #13 FAMAT State Calculus Bowl 2007 7 3 Let f ( x) 2 x . What is the value of the constant term in the full expansion of f / ( x)? x
