buckley (jdb4687) - Review - beshaj - (53521) This print-out should have 60 questions. Multiple-choice questions may continue on the next column or page - find all choices before answering. Find the x-intercept of the tangent line at the point P (1, 3) on the graph of f when lim h0 001 10.0 points Let F be the function defined by x2 1 F (x) = . |x 1| Determine if lim F (x) x1 exists, and if it does, find its value. 1. limit does not exist 1 f (1 + h) 3 = 4. h 1. x-intercept = 1 2. x-intercept = 1 3. x-intercept = 4. x-intercept = 7 4 5. x-intercept = 1 4 1 4 6. x-intercept = 7 2. limit = 1 004 3. limit = 2 Find dy/dx when 4. limit = 2 y x = 4 xy . 5. limit = 1 002 10.0 points Determine if n o lim ln(8 + x) ln(6 + 5x) x exists, and if it does find its value. 4 3 4. limit does not exist 003 y x 2 dy q = 1. dx 4 + xy q y x +2 dy q 2. = dx 4+ x y q y 3. limit = ln 5 5. limit = ln q 2 xy + 1 dy q 3. = dx 1 + 2 xy q y 2 x 1 dy q 4. = dx 12 x 3 1. limit = ln 4 2. limit = ln 10.0 points 1 5 10.0 points q 2 xy + 1 dy q 5. = dx 1 2 xy q y x 2 dy q = 6. dx 4 x y buckley (jdb4687) - Review - beshaj - (53521) 005 10.0 points 2. 151/4 31 16 3. 151/4 65 32 4. 151/4 61 32 2 Determine f (e) when f (x) = x2 (6 + (ln x)3 ). 1. f (e) = 14e 2. f (e) = 16e 5. 151/4 2 3. f (e) = 17e 008 4. f (e) = 15e 10.0 points A function y = f (x) is known to have the following properties 5. f (e) = 13e (i) f (0) = 0, 006 10.0 points The dimensions of a cylinder are changing, but the height is always equal to the diameter of the base of the cylinder. If the height is increasing at a speed of 5 inches per second, determine the speed at which the volume, V , is increasing (in cubic inches per second) when the height is 2 inches. 1. dV = 12 cub. ins./sec dt 2. dV = 14 cub. ins./sec dt 3. dV = 13 cub. ins./sec dt 4. dV = 15 cub. ins./sec dt 5. dV = 16 cub. ins./sec dt 007 10.0 points Use linear appproximation to estimate the value of 151/4 . (Hint: (16)1/4 = 2.) 1. 15 1/4 63 32 (ii) f changes concavity at x = 0, (iii) f < 0 on (, 3), f > 0 on (3, ). Which of the following could be the graph of f when dashed lines indicate asymptotes? 6 5 1. 4 3 2 1 0 -1 -2 -3 -4 -5 -6 4 2 4 2 2 2 4 2 4 4 5 2. 4 3 2 1 0 -1 -2 -3 -4 -5 -6 4 2 4 2 2 4 buckley (jdb4687) - Review - beshaj - (53521) 6 5 3. 4 3 2 1 0 -1 -2 -3 -4 -5 -6 6 5 4. 4 3 2 1 0 -1 -2 -3 -4 -5 -6 6 5 5. 4 3 2 1 0 -1 -2 -3 -4 -5 -6 3 5. limit does not exist 4 4 2 2 2 4 010 f (x) = tan(x) 2x \u0012 \u0013 on , . Find all interval(s) on which f 2 2 is increasing? \u0010 \u0011 \u0010 \u0011 1. , 0 , , 4 4 2 \u0010 \u0011 2. , 4 4 \u0010 \u0011 \u0010 \u0011 , , 3. , 2 6 6 2 \u0010 \u0011 \u0010 \u0011 4. , , , 2 4 4 2 \u0010 \u0011 \u0010 \u0011 , 0, , 5. 2 4 4 4 2 2 4 4 4 2 011 4 2 2 2 4 10.0 points is continuous for all x. Determine 2 3 4 cos x + ex . lim x0 2 sin2 x 1. limit = 1 1. c = 1 2. c = 1 3. c = 4 2. limit = 3 4 4. c = 4 3. limit = 1 4 5. none of these 4. limit = 0 10.0 points Find the largest value of c so that the function g defined by ( x2 + 2x c2 , x > 2, g(x) = cx 3, x 2, 4 009 10.0 points Let f be the function defined by 4 4 2 2 1 2 6. limit = 2 012 10.0 points buckley (jdb4687) - Review - beshaj - (53521) 4 y Determine if the limit 6 6 x + 1 lim x0 x x exists, and if it does, find its value. 1. limit = 7 2. limit = 6 3. limit does not exist 4. limit = 7 5. limit = 6 013 1. area = 9 sq. units 2 2. area = 25 sq. units 6 3. area = 23 sq. units 6 4. area = 31 sq. units 6 5. area = 29 sq. units 6 10.0 points Determine g (x) when g(x) = is enclosed by the graphs of f (y) = y 2 , g(y) = 2 + y . Find the area of this shaded region. Z 4 3t2 tan t dt . x 015 10.0 points The shaded region shown in 1. g (x) = 6x sec x tan x 2. g (x) = 6x sec x tan x 3. g (x) = 3x2 tan x 4. g (x) = 3x2 tan x 5. g (x) = 3x2 sec x 6. g (x) = 3x2 sec x 7. g (x) = 6x sec2 x 8. g (x) = 6x sec2 x 014 10.0 points The shaded region in is enclosed by the graphs of y = x4 , x = y4 . Find the volume, V , of the solid obtained by rotating this shaded region about the x-axis. 1. V = 2 3 buckley (jdb4687) - Review - beshaj - (53521) 2. V = 8 9 018 5 9 5. V = 4 9 satisfies the hypotheses of Rolle's Theorem on the interval [0, 12], and if it does, find all numbers c satisfying the conclusion of that theorem. 016 10.0 points Evaluate the integral I = Z 0 1. I = 1 3x2 + 3 dx. 2x3 + 6x + 1 1 ln 9 2 1. hypotheses not satisfied 2. c = 5 3. c = 4 , 5 4. c = 8 5. c = 9 6. c = 8 , 9 2. I = 2 ln 10 3. I = 10.0 points Determine if the function f (x) = x 12 x 7 3. V = 9 4. V = 5 019 1 ln 10 2 10.0 points Determine if the function 4. I = ln 10 f (x) = 5. I = ln 9 017 10.0 points Find the value of the integral I = Z 0 1. I = 1 2. I = 1 2 1 3. I = 2 4. I = 1 5. I = 0 /4 4 tan x 3 dx . cos2 x x3 x1 satisfies the hypotheses of the Mean Value Theorem (MVT) on the interval [0, 3]. If it does, find all possible values of c satisfying the conclusion of the MVT. 1. c = 3 , 1 2. c = 1 3. c = 2 4. hypotheses are not satisfied 5. c = 1 , 2 6. c = 2 , 3 7. c = 2 , 4 020 10.0 points buckley (jdb4687) - Review - beshaj - (53521) Cyclist Joe accelerates as he rides away from a stop sign. His velocity graph over a 5 second period (in units of feet/sec) is shown in 20 6 3. avg. vel. = 2 2 + 2h + h3 ft/sec 3 3 4. avg. vel. = 2 2 + 2h + h2 ft/sec 3 3 2 2 5. avg. vel. = 2 + h + h2 ft/sec 3 3 16 2 6. avg. vel. = 2 + h + 2h3 ft/sec 3 12 022 8 10.0 points Find the value of f (e) when 4 f (x) = 1 2 3 4 5 Compute best possible lower estimate for the distance he travels over this period when dividing [0, 5] into 5 equal subintervals and using endpoint sample points. 4 , x2 x > 0, and f (1) = 2 , f (2) = 3 . 1. f (e) = 2e + 5 2. f (e) = 3e 1 3. f (e) = 5e + 7 1. 50 ft < distance 4. f (e) = 3e + 1 2. 42 ft < distance 3. 44 ft < distance 5. f (e) = 2e 5 4. 46 ft < distance 6. f (e) = 5e 7 023 5. 48 ft < distance 021 2 s(t) = t3 . 3 Find the average velocity of the particle over the time interval [1, 1 + h]. 2 1. avg. vel. = 2 + 2h + h2 ft/sec 3 2. avg. vel. = Which of the following is the graph of 10.0 points After t seconds the displacement, s(t), of a particle moving rightwards along the t-axis is given (in feet) by 2 2 + h + 2h3 ft/sec 3 3 10.0 points n \u0010 x 2 \u0011o f (x) = 3 x 2 ln 15. 2 6 5 1.4 3 2 1 0 -1 -2 -3 -4 -5 6 4 2 2 2 4 4 6 8 buckley (jdb4687) - Review - beshaj - (53521) 6 5 2.4 3 2 1 0 -1 -2 -3 -4 -5 7 6 5 3.4 3 2 1 0 -1 -2 -3 -4 -5 6 5 4.4 3 2 1 0 -1 -2 -3 -4 -5 6 5 5.4 3 2 1 0 -1 -2 -3 -4 -5 7 has a point of inflection. 6 1. x = 3 4 2. x = 0, 6 2 2 4 6 8 3. x = 0, 2 2 4. x = 3 4 5. x = 0, 6 6. x = 2 6 4 025 2 10.0 points If 2 4 6 f (x) = x2 + 6x , 8 2 which of the following determines the derivative of f (x)? 4 1. 6 4 (x + h)2 + 6(x + h) + x2 6x h (x + h)2 + 6(x + h) + x2 6x h h0 \u0013 \u0012 2 x + 6x + hx2 x 3. lim x0 h 2. lim 2 2 4 6 8 2 (x + h)2 + 6(x + h) + x2 6x x0 h 4. lim 4 (x + h)2 + 6(x + h) x2 + 6x h \u0013 \u0012 2 x + 6x + h + x2 x 6. lim h h0 6 5. 4 2 2 4 6 026 8 2 10.0 points Determine if 4 \u0011 \u0010p 2 9x + 4 3x lim x x 024 10.0 points Find all values of x at which the graph of 1 f (x) = x3 3x2 + 7 3 exists, and if it does, find its value. 1. limit = 5 6 buckley (jdb4687) - Review - beshaj - (53521) 2. limit = 1 6 1 4. I = cos8 x + C 8 3. limit does not exist 4. limit = 2 3 5. limit = 1 2 10.0 points Determine the integral Z \u00013 I = 2x x2 + 6 dx . 1. I = 5. I = 1 8 sin x + C 8 6. I = 1 6 sin x + C 6 029 1 6. limit = 3 027 1 2 (x + 6)2 + C 4 10.0 points Evaluate the integral Z 2 x+6 I = dx . x+2 1 030 10.0 points Evaluate the integral Z 1 2 I = 2 ex +ln x dx . 0 1 2. I = (x2 + 6)4 + C 2 \u0010 e 1\u0011 1. I = e 3. I = (x2 + 6)4 + C 2. I = ( e 1 ) 4. I = (x2 + 6)2 + C 3. I = 2 \u0010 e 1\u0011 e 5. I = 1 2 (x + 6)2 + C 2 4. I = 2(e 1) 6. I = 1 2 (x + 6)4 + C 4 5. I = 2( e 1 ) 028 10.0 points Determine the integral Z I = cos7 x sin x dx . 6. I = e 1 031 1 7 sin x + C 7 1 3. I = cos6 x + C 6 10.0 points Determine if Rolle's Theorem can be applied to the function 1 1. I = cos7 x + C 7 2. I = 8 2 f (x) = 5x 3 20 on the interval [8, 8], and if it can, find all numbers c satisfying the conclusion of that theorem. 1. c = 2 buckley (jdb4687) - Review - beshaj - (53521) 9 2. c = 2, 4 3. average vel. = 8 ft/sec 3. c = 0 4. average vel. = 10 ft/sec 4. Rolle's Theorem not applicable 5. average vel. = 11 ft/sec 034 5. c = 2, 4 10.0 points Find the value of f (1) when 032 f (t) = 4(3t + 2) 10.0 points and Determine if the function f (x) = 1 x x f (1) = 5, f (1) = 2 . 3 satisfies the hypotheses of the Mean Value Theorem (MVT) on the interval [2, 1]. If it does, find all possible values of c in (2, 1) satisfying the conclusion of the MVT. 1 1. c = 2 1. f (1) = 18 2. f (1) = 20 3. f (1) = 17 4. f (1) = 19 2. c = 1 5. f (1) = 16 3. c = 1 , 1 2 035 4. c = 1 10.0 points The derivative, f , of f has graph 1 5. c = 2 6. c = 0 7. hypotheses not satisfied 033 10.0 points a b c If a ball is thrown into the air with a velocity of 48 ft/sec, its height in feet after t seconds is given by y = 48t 16t2 . graph of f Find the average velocity of the ball for the time period beginning at t = 1 and lasting 1/2 seconds. Use it to locate the critical point(s) x0 at which f has a local minimum? 1. average vel. = 12 ft/sec 1. x0 = a , b 2. average vel. = 9 ft/sec 2. x0 = c , a buckley (jdb4687) - Review - beshaj - (53521) 3. x0 = a 4. x0 = a , b , c 5. x0 = c 6. none of a , b , c x0 = b , c 036 15 49 5. f (1) = 19 49 038 10.0 points Find the derivative of F when Z 2x F (x) = (t2 2t) dt . 7. x0 = b 8. 4. f (1) = 10 x 10.0 points After t seconds the displacement, s(t), of a particle moving rightwards along the x-axis is given (in feet) by 1. F (x) = 7x2 6x 2. F (x) = 7x2 + 6x s(t) = 7t2 4t + 2 . 3. F (x) = 5x2 + 6x Determine the average velocity of the particle over the time interval [1, 3]. 4. F (x) = 5x2 4x 1. average vel. = 24 ft/sec 5. F (x) = 5x2 + 4x 2. average vel. = 22 ft/sec 6. F (x) = 7x2 4x 3. average vel. = 25 ft/sec 4. average vel. = 21 ft/sec 5. average vel. = 23 ft/sec 037 find the value of f (1). 2. f (1) = 17 49 3. f (1) = 16 49 10.0 points What value of x maximizes F when Z x+2 1 F (x) = dt ? 2 t +3 x 040 10.0 points If f is a continuous function such that Z x 3x f (t) dt = 2 , x +6 0 18 1. f (1) = 49 039 10.0 points Find the linearization of f (x) = 1 7+x at x = 0. 1\u0010 1 \u0011 1 x 7 7 1 \u0011 1\u0010 1+ x 2. L(x) = 7 14 1 \u0011 1 \u0010 3. L(x) = 1 x 14 7 1. L(x) = buckley (jdb4687) - Review - beshaj - (53521) 1 \u0010 1 \u0011 4. L(x) = 1 + x 14 7 6. 1 1 5. L(x) = + x 7 7 dy = 3 cot x tan y dx 043 10.0 points If P (a, f (a)) is the point on the graph of 1 1 6. L(x) = x 7 7 041 11 f (x) = x2 + 5x + 2 10.0 points at which the tangent line is parallel to the line Find the differential, dy, of y = 2x + 6 , y = f (x) = tan(4x2 ) . determine a. 1. a = 0 2 1. dy = 8x sec (4x) + dx 3 2 2. dy = 4 sec (4x ) tan(4x ) 2. a = 3. dy = 8x sec2 (4x2 ) 3. a = 2 2 2 2 1 2 4. dy = 8x sec (4x ) dx 4. a = 5. dy = 4 sec2 (4x2 ) tan(4x2 ) dx 5. a = 1 2 2 6. dy = 4 sec2 (4x2 ) tan(4x2 ) + dx 042 Find 044 10.0 points The shaded region in the figure y 10.0 points dy when dx 3 sin x sin y = 5 . 1. dy = 3 tan x cot y dx 2. dy = 3 tan x cot y dx 3. dy = tan x cot y dx dy = cot x tan y 4. dx 5. dy = cot x tan y dx x is bounded by the graphs of f (x) = x3 2x2 4x + 8 , Find the area of this region. 1. area = 196 sq. units 3 g(x) = 4x + 8 . buckley (jdb4687) - Review - beshaj - (53521) 12 exists, and if it does, find its value. 2. area = 194 sq. units 148 sq. units 3 3. area = 4. area = 20 sq. units 100 sq. units 3 5. area = 045 1. limit = 7 2. none of the other answers 3. limit = 4. limit = 5. limit = 7 10.0 points Find the volume of the paraboloidgenerated by rotating the graph of y = 6 x between x = 0 and x = 1 about the x-axis. 048 10.0 points Determine if the limit lim (1 + 3x)csc(x) 1. volume = 17 cu.units x0 2. volume = 15 cu.units exists, and if it does, find its value. 3. volume = 16 cu.units 1. limit = e 4. volume = 18 cu.units 5. volume = 14 cu.units 2. limit = 3. none of the other answers 046 10.0 points 4. limit = 3 Determine lim+ x (4 5 ln(x)) . 5. limit = e3 x0 6. limit = 1. limit = 7. limit = e3x 2. none of the other answers 049 10.0 points If the region enclosed bythe y-axis, the line y = 2, and the curve y = x is revolved about the y-axis, the volume of the solid generated is 3. limit = 4. limit = 0 5. limit = 1 1. 6. limit = 5 047 8 3 16 3. 3 32 4. 5 2. 10.0 points Determine if the limit \u0010 8 1 \u0011 lim x 8 ln(x 7) x8 buckley (jdb4687) - Review - beshaj - (53521) Find the absolute maximum value of 16 5. 5 050 10.0 points Two sides of a triangle are 8 inches and 10 inches in length and the angle between them is increasing at a rate of 1/6 radians per second. Find the rate at which the area of the triangle is increasing when = /3. 1. rate = 11 sq.ins/sec 3 2. rate = 7 sq.ins/sec 2 3. abs maximum value = 2 10.0 points When Sue uses first and second derivatives to analyze a particular continuous function y = f (x) she obtains the chart Determine the absolute maximum value of p f (x) = x 1 x2 2 on [1, 1]. 5 2 2. absolute max. value = 1 3 3. absolute max. value = 2 1 2 5. absolute max. value = 2 6. absolute max. value = 3 052 2. abs maximum value = 0 053 10.0 points 4. absolute max. value = 1. abs maximum value = 1 6. abs maximum value = 5 23 sq.ins/sec 6 1. absolute max. value = on [, ]. 5. abs maximum value = 3 4. rate = 4 sq.ins/sec 051 f (x) = 3 2 sin2 (x) 4. abs maximum value = 6 10 sq.ins/sec 3. rate = 3 5. rate = 13 y x < 3 + x = 3 4 0 3 < x < 0 x=0 1 1 02 + y y + + Which of the following can she conclude from her chart? A. f has a point of inflection at x = 2. B. f is concave up on (, 0). C. f is concave up on (0, 2). 1. C only 2. all of them 3. A and B only 10.0 points 4. B and C only buckley (jdb4687) - Review - beshaj - (53521) 5. A only 6. f : 6. B only 14 f : 055 f : 10.0 points Find the interval(s) where 7. none of them 8. A and C only 054 10.0 points The figure below shows the graphs of three functions: 2 f (x) = 8 + x + 28x2 3x3 x4 3 is concave down. \u0010 7\u0011 1. 4, 2 \u0010 7\u0011 2. 4, 4 \u0010 \u0011 7\u0011 \u0010 3. , , 4, 4 \u00107 \u0011 ,4 4. 4 \u0011 \u0010 \u0011 \u00107 , 5. , 4 , 4 056 10.0 points Find all points x0 at which 3+x f (x) = (x + 4)2 has a local minimum. One is the graph of a function f , one is its derivative f , and one is its second derivative f . Identify which graph goes with which function. 1. x0 = 3 2. x0 = 2 3. x0 = 2 1. f : f : f : 4. no such x0 exist 2. f : f : f : 5. x0 = 2 , 2 3. f : f : f : 4. f : f : f : 5. f : f : f : 6. x0 = 4 7. x0 = 3 , 4 057 10.0 points Find the derivative of f when p x f (x) = tan1 ln 25 + x2 . 5 buckley (jdb4687) - Review - beshaj - (53521) 1. f (x) = 5+x 25 + x2 2. f (x) = x 25 + x2 3. f (x) = 5+x 25 + x2 5x 4. f (x) = 25 + x2 4. has jump discontinuity at x = 0 5. is continuous for all x 060 lim x0 058 10.0 points Find the derivative of f when n o f (x) = x 7 sin(ln x) + cos(ln x) . 1. f (x) = 6 sin(ln x) 8 cos(ln x) 2. f (x) = 8 sin(ln x) + 6 cos(ln x) 3. f (x) = x{8 sin(ln x) + 6 cos(ln x)} 4. f (x) = x{6 sin(ln x) + 8 cos(ln x)} 5. f (x) = 6 sin(ln x) + 8 cos(ln x) 059 10.0 points Locate and classify all the discontinuities of the function f defined by sin 6x , x 6= 0 |x| f (x) = 6, x = 0. 1. has removable discontinuity at x = 0 2. removable discontinuity at x = 0, 3. jump discontinuity at x = 0, 10.0 points Find the value of the limit 5 5. f (x) = 25 + x2 15 if the limit exists. 1. limit = 2 5 2. limit = 1 5 3. limit = 3 5 5 4 cos 2x \u0001 5 sin 2 + x 4. limit doesn't exist 5. limit = 4 5