Calculus late transcendental 4th edition smith test bank

Chapter Find the equation of the tangent line to y  x – x at x  A) y  –9 B) y  C) y  –9 x D) y  x Ans: A Difficulty: Moderate Section: 2.1 Find an equation of the tangent line to y = f(x) at x = –3 f  x   x3  x  x A) y = –6x + 18 B) y = 22x – 45 C) y = 6x + 18 Ans: D Difficulty: Moderate Section: 2.1 D) y = 22x + 45 Find an equation of the tangent line to y = f(x) at x = f ( x)  x  A) y = 21x – 64 B) y = –96x – 251 C) y = 96x – 251 Ans: C Difficulty: Moderate Section: 2.1 Find the equation of the tangent line to y  14 x 25 25 14 B) y  – x 25 25 Ans: C Difficulty: Moderate A) D) y = 96x + 251 at x  x3 y C) D) 14 x 25 25 14 y x 25 25 y– Section: 2.1 Find the equation of the tangent line to y  x + at x  –3 A) y  x + 10 B) y  x + 10 C) y  x + 20 Ans: B Difficulty: Moderate Section: 2.1 D) y  x + 20 Compute the slopes of the secant lines between the point at x = and points close to it (such as x = 0, x = 2, x = 0.9, x = 1.1) and use these results to estimate the slope of the tangent line at x = Round to two decimal places x 1 x 1 A) –1.30 B) –0.70 C) –0.20 D) 0.50 Ans: D Difficulty: Moderate Section: 2.1 Page 90 Chapter Compute the slope of the secant line between the points x = 2.9 and x = Round your answer to the thousandths place f ( x)  sin(2 x) A) –0.981 B) 1.852 C) –4.372 D) –1.963 Ans: B Difficulty: Easy Section: 2.1 List the points A, B, C, D, and E in order of increasing slope of the tangent line A) B, C, E, D, A B) A, E, D, C, B C) E, A, D, B, C Ans: B Difficulty: Easy Section: 2.1 D) A, B, C, D, E Use the position function s(t )  4.9t  meters to find the velocity at time t  seconds A) 3.1 m/sec B) –9.8 m/sec C) –1.8 m/sec D) –4.9 m/sec Ans: B Difficulty: Moderate Section: 2.1 10 Use the position function s(t )  t – meters to find the velocity at time t  seconds 1 m/sec D) m/sec Difficulty: Moderate Section: 2.1 A) m/sec B) m/sec C) Ans: D 11 Find the average velocity for an object between t = sec and t = 2.1 sec if f(t) = –16t2 + 100t + 10 represents its position in feet A) 34.4 ft/s B) 36 ft/s C) 32.8 ft/s D) 146 ft/s Ans: A Difficulty: Moderate Section: 2.1 Page 91 Chapter 12 Find the average velocity for an object between t = –1 sec and t = –0.9 sec if f(t) = 5sin(t) + represents its position in feet (Round to the nearest thousandth.) A) 2.702 B) 3.108 C) 2.907 D) –2.907 Ans: C Difficulty: Moderate Section: 2.1 13 Estimate the slope of the tangent line to the curve at x = –2 A) –1 B) –2 C) D) Ans: B Difficulty: Easy Section: 2.1 14 Estimate the slope of the tangent line to the curve at x = A) 1 D) Difficulty: Easy Section: 2.1 B) –2 Ans: D C) Page 92 Chapter 15 The table shows the temperature in degrees Celsius at various distances, d in feet, from a specified point Estimate the slope of the tangent line at d  and interpret the result d 12 18 15 C m  5; The temperature is increasing C per foot at the point feet from the specified point m  –0.67; The temperature is decreasing 0.67 C per foot at the point feet from B) the specified point m  –1.5; The temperature is decreasing 1.5 C per foot at the point feet from the C) specified point m  18; The temperature is increasing 18 C per foot at the point feet from the D) specified point Ans: C Difficulty: Moderate Section: 2.1 A) 16 The graph below gives distance in miles from a starting point as a function of time in hours for a car on a trip Find the fastest speed (magnitude of velocity) during the trip Describe how the speed during the first hours compares to the speed during the last hours Describe what is happening between and hours Ans: The fastest speed occurred during the last hours of the trip when the car traveled at about 70 mph The speed during the first hours is 60 mph while the speed from to 10 hours is about 70 mph Between and hours the car was stopped Difficulty: Moderate Section: 2.1 17 Compute f(2) for the function f ( x)  x3  5x A) 58 B) 43 C) 38 D) –43 Ans: B Difficulty: Moderate Section: 2.2 Page 93 Chapter 18 Compute f(5) for the function f ( x)  x 4 2 40 40 40 B) C) – D) – 841 841 841 Ans: D Difficulty: Moderate Section: 2.2 A) 19 Compute the derivative function f(x) of f ( x)  15 (3 x  7) 3 f ( x )  B) (3 x  7) Ans: A Difficulty: Moderate A) f ( x )  C) D) 3x  5 (3 x  7) 15 f ( x )  (3 x  7) f ( x )  Section: 2.2 20 Compute the derivative function f(x) of f ( x)  x  A) f ( x)  B) f ( x)  Ans: B 4 x C) 2x  2x 2 x2  Difficulty: Moderate D) Section: 2.2 Page 94 f ( x)  2 x x2  2 x f ( x )  4x  Chapter 21 Below is a graph of f ( x ) Sketch a graph of f ( x) Ans: Difficulty: Moderate 9+ Section: 2.2 Page 95 Chapter 22 Below is a graph of f ( x ) Sketch a graph of f ( x) Ans: Difficulty: Difficult Section: 2.2 Page 96 Chapter 23 Below is a graph of f ( x) Sketch a plausible graph of a continuous function f ( x ) Ans: Answers may vary Below is one possible answer Difficulty: Moderate Section: 2.2 Page 97 Chapter 24 Below is a graph of f ( x) Sketch a plausible graph of a continuous function f ( x ) Ans: Answers may vary Below is one possible answer Difficulty: Difficult Section: 2.2 25 Compute the right-hand derivative D f (0)  lim h 0 f (h)  f (0) h 0 h  –8 x – if x  f ( x)   10 x – if x  f (h)  f (0) and the left-hand h derivative D f (0)  lim A) C) D f (0)  10 , D f (0)  –8 B) D) D f (0)  –8 , D f (0)  10 Ans: A Difficulty: Moderate Section: 2.2 Page 98 D f (0)  –9 , D f (0)  –9 D f (0)  1, D f (0)  Chapter 26 The table below gives the position s(t) for a car beginning at a point and returning hours later Estimate the velocity v(t) at two points around the third hour t (hours) s(t) (miles) 0 15 50 80 70 Ans: The velocity is the change in distance traveled divided by the elapsed time From hour to the average velocity is (70 − 80)/(4 − 3) = −10 mph Likewise, the velocity between hour and hour is about 30 mph Difficulty: Easy Section: 2.2 27 Use the distances f(t) to estimate the velocity at t = 2.2 (Round to decimal places.) t 1.6 f(t) 43 1.8 2.2 2.4 2.6 38 32.5 28 23.5 18.5 2.8 13 A) 2250.00 B) 12.73 C) –22.50 D) –25.00 Ans: C Difficulty: Easy Section: 2.2 4 x + x if x  28 For f ( x)   find all real numbers a and b such that f (0) exists  ax  b if x  a  8, b any real number A) a  11, b  B) Ans: D Difficulty: Moderate C) D) Section: 2.2 Page 99 a  3, b any real number a  3, b  Chapter 55 Find the second-degree polynomial (of the form ax2 + bx + c) such that f(0) = 0, f '(0) = 5, and f ''(0) = x2 x2 x2 x2  x B)   x C)  x  D)   x  2 2 Ans: A Difficulty: Moderate Section: 2.3 A) 56 Find a formula for the nth derivative f ( n ) ( x) of f ( x)  20n ! C) ( x + 10) n 1 2n ! f ( n ) ( x)  ( 1) n 1 B) D) ( x + 10) n Ans: D Difficulty: Difficult Section: 2.3 A) f ( n ) ( x)  (1) n 1 x + 10 20n ! ( x + 10) n 2n ! f ( n ) ( x)  ( 1) n ( x + 10) n 1 f ( n ) ( x)  ( 1) n 57 Find a function with the given derivative f ( x)  36 x8 A) f ( x)  36 x9 B) f ( x)  x9 C) f ( x)  36 x7 Ans: B Difficulty: Moderate Section: 2.3 D) f ( x)  288 x7 58 Let f (t ) equal the average monthly salary of families in a certain city in year t Several values are given in the table below Estimate and interpret f (2010) t 1995 2000 2005 2010 $1700 $2000 $2200 $2450 f (t ) f (2010)  ; The rate at which the average monthly salary is increasing each year in 2010 is increasing by $2 per year f (2010)  ; The average monthly salary is increasing by $2 per year in 2010 B) f (2010)  50 ; The rate at which the average monthly salary is increasing each C) year in 2010 is increasing by $50 per year f (2010)  50 ; The average monthly salary is increasing by $50 per year in 2010 D) Ans: A Difficulty: Moderate Section: 2.3 A) Page 108 Chapter 59 Find the derivative of f ( x)    1  x – 5x  x2 –  x  45 3/ x + 3/ 2 2x 45 B) f ( x)  –135 x + x3/ + 3/ 2 2x 45 C) f ( x)  135 x + x3/ – 3/ 2 2x 45 10 D) f ( x)  –135 x + x3/ + + 3/ 2 x 2x Ans: B Difficulty: Moderate Section: 2.4 A) f ( x)  –135 x – 60 Find the derivative of f ( x)  6x + 9x – 48 2 B) C) – (9 x – 5) 3 Ans: D Difficulty: Moderate A) 61 Find the derivative of f ( x)  –48 (9 x – 5) Section: 2.4 D) 9x x2 – –54 x – 36 54 x + 36 3 B) C) D) – 2 2 2x (6 x – 4) (6 x – 4) 2x Ans: A Difficulty: Moderate Section: 2.4 A) Page 109 Chapter 62 Find the derivative of the function 4x – x 4x2 –  – x  x 1/ A) C) D)   –   x – x 8x 4x2 –  – x  x –    x – x  8x  4x – 7  x – x  8x   – x  x –   4x – 7  x – x  8x   – x  x –  1/ B) 2 2 Ans: B 1/ 2 1/ 2 4x2 – Difficulty: Moderate  Section: 2.4  63 Find the derivative of f ( x)  –8 x + x 32 x +5 B) f ( x)  – x – Ans: C Difficulty: Moderate A) f ( x)  C) D) 32 x +5 16 f ( x)  – x + 10 f ( x)  – Section: 2.4 64 Find the equation of the tangent line to the graph of y = f (x) at x = –2 x+5 f  x  x +3 19 59 x+ 49 49 19 22 B) y  x+ 7 Ans: A Difficulty: Moderate A) y C) D) 19 17 x– 49 49 19 16 y  – x+ 7 y– Section: 2.4 65 Find an equation of the line tangent to h( x)  f ( x) g ( x) at x  –2 if f (–2)  , f (–2)  , g (–2)  –2 , and g (–2)  –1 A) y  3x – 12 B) y  x – 24 C) y  –9 x – 24 Ans: C Difficulty: Moderate Section: 2.4 Page 110 D) y  –9 x + 12 Chapter f ( x) at x  –3 if g ( x) f (–3)  –2 , f (–3)  , g (–3)  , and g (–3)  66 Find an equation of the line tangent to h( x)  A) y  –3 x + 13 B) y  x + 13 C) y  –3 x – 11 Ans: B Difficulty: Moderate Section: 2.4 D) y  x – 17 67 A small company sold 1000 widgets this year at a price of $10 each If the price increases at rate of $1.25 per year and the quantity sold increases at a rate of 250 widgets per year, at what rate will revenue increase? A) $312.5/year B) $3750/year C) $1250/year Ans: B Difficulty: Moderate Section: 2.4 D) $4062.5/year ( x + 2) 68 Find the derivative of f ( x)  x( x + 2)3 B) f ( x)  x( x + 2)3 Ans: C Difficulty: Moderate A) f ( x)  C) D) x( x + 2)3 f ( x)  x( x + 2)3 f ( x)  Section: 2.5 69 Find the derivative of f ( x)  x – A) f ( x)  B) f ( x)  Ans: D 2x x –2 4x x2 – Difficulty: Moderate C) f ( x)  D) f ( x)  –x x2 – x x2 – Section: 2.5 70 Differentiate the function f (t )  t t – A) f (t )  B) f (t )  Ans: C 7t – 12t 2 t7 – 3t 2 t7 – Difficulty: Difficult C) f (t )  D) f (t )  Section: 2.5 Page 111 13t – 12t 2 t7 – 21t t7 – Chapter x x +1 71 Find the derivative of f ( x)  A)  1    x x2   B)  x      x2   x x2    Ans: B     3 x2    x   f ( x)  B) f ( x)  (5 x + 9) Difficulty: Moderate f ( x)  B) C) D)  f ( x)   f ( x)   f ( x)   f ( x)   Ans: A x3 +  x       f ( x)  D) f ( x)  2 x3 +  x  16 x3 +  3x 2 x3 +  x3 +  x  2 x3 +  x3 +  x  x3 +  x3 +  x  Difficulty: Difficult x2  C) Section: 2.5 x3 +  3x x3 +    x2  x2  (5 x + 9)3 40 x 73 Differentiate the function A) 5x2 + –20 x   Section: 2.5 72 Find the derivative of f ( x)  Ans: A D) Difficulty: Moderate A) C)  1  x x2    x x2   Section: 2.5 Page 112 20 x (5 x + 9)3 8x (5 x + 9)3       Chapter 74 Find an equation of the line tangent to f ( x)  x2 – at x = A) y = –2x + B) y = –2x C) y = 2x + D) y = –2x + Ans: D Difficulty: Moderate Section: 2.5 75 Use the position function s(t )  t  65 meters to find the velocity at t = seconds m/s C) m/s D) m/s 9 Difficulty: Moderate Section: 2.5 A) m/s B) Ans: B 76 Compute the derivative of h( x)  f  g ( x)  at x = –8 where f (–8)  , g (–8)  , f (–8)  , f (2)  –7 , g (–8)  , and g (2)  A) h(–8)  36 B) h(–8)  72 C) h(–8)  –63 Ans: C Difficulty: Moderate Section: 2.5 D) h(–8)  16 77 Find the derivative where f is an unspecified differentiable function f (9 x ) A) 36 x3 f (9 x ) B) (36 x3  x ) f (9 x ) C) f (36 x3 ) Ans: A Difficulty: Moderate Section: 2.5 D) f (36 x3  x ) 78 Find the derivative where f is an unspecified differentiable function A) f  x  f  x  Ans: D B) f  x  f  x  Difficulty: Moderate C) 4  f  x  D) f  x 4  f  x  Section: 2.5 79 Find the second derivative of the function f ( x)  100  x A) f ( x)  B) f ( x)  Ans: C 100 x (100  x )3/ x  100 (100  x )3/ Difficulty: Moderate C) f ( x)   100 (100  x )3/ D) f ( x)   100 x (100  x )3/ Section: 2.5 Page 113 Chapter 80 Find a function g ( x) such that g ( x)  f ( x) f ( x)   x +  (2 x) A)  x3  x2 + x     B) g ( x)   x +  (16 x) Ans: D Difficulty: Moderate C) g ( x)   x +  D) x g ( x)  + 5 5 Section: 2.5 81 Use the table of values to estimate the derivative of h( x)  f  g ( x)  at x = x f(x) g(x) –1 –5 –4 –3 –4 –5 –6 –5 –3 –1 –1 A) h(6)  B) h(6)  –3 C) h(6)  –2 D) h(6)  Ans: A Difficulty: Moderate Section: 2.5 82 Find the derivative of f ( x)  5sin( x) – 3cos(3 x)  x f ( x)  5cos x + 9sin x  A) C) f ( x)  5cos x + 3sin 3x  B) D) Ans: A Difficulty: Easy Section: 2.6 f ( x)  –5cos x – 9sin x  f ( x)  cos x – 3sin x  83 Find the derivative of f ( x)  7sin x + x f ( x)  –14sin x cos x + 18 x A) C) f ( x)  14sin x cos x + x B) D) Ans: D Difficulty: Easy Section: 2.6 84 Find the derivative of f ( x)  –9 cos x x2 –18( x sin x  cos x ) C) x3 18( x sin x  cos x ) f ( x)  B) D) x3 Ans: C Difficulty: Moderate Section: 2.6 A) f ( x)  14sin x + 18 x f ( x)  14sin x cos x + 18 x f ( x)  Page 114 18( x sin x  cos x ) x3 18( x sin x  cos x ) f ( x)  x4 f ( x)  Chapter 85 Find the derivative of f ( x)  2sin x sec x A) f ( x)  B) f ( x)  Ans: B sec x tan x sec2 x C) tan x Difficulty: Moderate D) 2sec2 x tan x sec x tan x f ( x)  tan x f ( x)  Section: 2.6 86 Find the derivative of the function f (w)  w2 sec2 10w A) B) C) D) Ans: f (w)  20w sec2 (10w) tan(10w) f (w)  2w sec2 (10w)  20w2 sec2 (10w) tan(10w) f (w)  2w sec2 (10w)  20w2 sec(10w) f (w)  2w sec2 (10w)  20w2 sec2 (10w) tan (10w) B Difficulty: Moderate Section: 2.6 87 Find the derivative of the function   f ( x)  cos3 sin  x  x  Ans:   f ( x)  6 cos sin  x  x  Difficulty: Difficult 2    sin sin  x  8x    cos  x  8x     x  8x   7 x  48x  7 Section: 2.6 88 Find an equation of the line tangent to f ( x)  tan x at x  (Round coefficients to decimal places.) y  –1.53 x – 8.204 A) y  –2.341x + 10.52 B) Ans: C Difficulty: Moderate C) D) Section: 2.6 y  2.341x – 8.204 y  2.341x + 7.277 89 Find an equation of the line tangent to f ( x)  x sin x at x   A) y  5( x   ) B) y  –5( x   ) C) y  5 ( x   ) Ans: D Difficulty: Moderate Section: 2.6 Page 115 D) y  –5 ( x   ) Chapter 90 Find an equation of the line tangent to f ( x)  x cos x at x  –2 (Round coefficients to decimal places.) y  –1.402 x – 3.637 A) y  –2.235 x – 3.637 B) Ans: B Difficulty: Moderate C) D) Section: 2.6 y  –2.235 x + 3.637 y  –1.402 x + 3.637 91 Use the position function s(t )  cos 4t – 9t feet to find the velocity at t = seconds (Round answer to decimal places.) A) v(3) = –51.85 ft/s B) v(3) = –56.15 ft/s Ans: A Difficulty: Moderate C) D) Section: 2.6 v(3) = 56.15 ft/s v(3) = –57.38 ft/s 92 Use the position function s (t )  –5sin(4t ) – meters to find the velocity at t = seconds (Round answer to decimal places.) A) v(2) = –19.79 m/s B) v(2) = 8.32 m/s Ans: D Difficulty: Moderate C) D) Section: 2.6 v(2) = 0.73 m/s v(2) = 2.91 m/s 93 Use the position function to find the velocity at time t  t0 Assume units of feet and seconds sin 9t s(t )  , t  t A) v( )  ft/sec B) v( )  Ans: C  ft/sec Difficulty: Moderate C) v( )  – D) v( )  –  2 ft/sec ft/sec Section: 2.6 94 A weight hanging by a spring from the ceiling vibrates up and down Its vertical position is given by s (t )  4sin(5t ) Find the maximum speed of the weight and its position when it reaches maximum speed A) speed = 4, position = 20 B) speed = 20, position = Ans: B Difficulty: Moderate C) D) Section: 2.6 Page 116 speed = 5, position = speed = 20, position = Chapter sin x sin(–7t )  , find lim x 0 t 0 x 5t 95 Given that lim B) –35 C) – 5 Ans: C Difficulty: Easy A) Section: 2.6 D) – cos x  cos t   , find lim x 0 t 0 x –5t 96 Given that lim A) C) –5 D) 5 Difficulty: Easy Section: 2.6 B) – Ans: A –9t sin x  , find lim t  sin(–7t ) x 0 x 97 Given that lim A) 63 Ans: D C) D) 9 Difficulty: Easy Section: 2.6 B) – sin x tan(–7t )  , find lim x 0 t 0 x –4t 98 Given that lim Ans: B A) – C) D) – Difficulty: Moderate Section: 2.6 B) 99 For f ( x)  sin x , find f (74) ( x) A) cos x B) –cos x C) sin x D) –sin x Ans: D Difficulty: Easy Section: 2.6 100 The total charge in an electrical circuit is given by Q(t )  2sin(2t )  t – The current is dQ the rate of change of the charge, i (t )  Determine the current at t  (Round dt answer to decimal places.) A) i (2)  –0.31 B) i (2)  –1.61 C) i (2)  –9.61 Ans: B Difficulty: Moderate Section: 2.6 Page 117 D) i (2)  –2.03 Chapter 101 Compute the slope of the line tangent to x + xy + y  57 at (–3, 2) 56 28 B) slope = – C) slope = – 5 28 Difficulty: Moderate Section: 2.7 A) slope = – Ans: B D) slope = 49 16 102 Find the derivative y ( x ) implicitly x2 y – y  x A) y ( x)  B) y( x)  Ans: B xy +  xy 2 x2 y – Difficulty: Moderate C) D) xy – 2 xy + 10 y( x)  2x2 y y( x)  – Section: 2.7 103 Find the derivative y ( x ) implicitly if y + xy  –5 7y y xy – x A) y  x   B) y  x   – Ans: C y xy y + 7x Difficulty: Moderate C) y  x   – 7y y xy + x D) y  x   – 7y y + x xy Section: 2.7 104 Find the derivative y ( x ) implicitly if –7 sin xy – x  y  x cos xy x y y( x)  –  B) x x cos xy Ans: D Difficulty: Moderate A) y( x)  C) D) Section: 2.7 Page 118 5cos xy y  7x x y y( x)  –  x cos xy x y( x)  – Chapter 105 Find y′(x) implicitly y 2 x + y + 3x  y A) y  x   y + xy + 3x 2 x + y + B) y  x   –4 y – xy – x 2 x + y – C) y  x   D) y  x   Ans: C 2x + y 2x + y –2 y – x x + y y + xy – x + y y2 + 6x 2x + y y + xy – x + y Difficulty: Difficult Section: 2.7 106 Find an equation of the tangent line at the given point x  25 y  at (–5, 1) 4 B) y  x  C) y  – x  x 15 15 15 Ans: C Difficulty: Moderate Section: 2.7 A) y  D) y  – 1 x 15 107 Find an equation of the tangent line at the given point x y  15 y  at (4, 1) 49 x 17 17 Difficulty: Moderate Ans: y  – Section: 2.7 108 Find the second derivative, y( x) , of –4 x3 – y  A) y y( x)  –  xy y  y  y( x)  –  B) 2y xy Ans: B Difficulty: Moderate C)  y  y( x)   2y xy D)  y  y( x)   2y xy Section: 2.7 Page 119 Chapter 109 Find the second derivative, y( x) , of y  x3 + x + cos y x  (cos y + 3)( y) A) C) y( x)  y – sin y x  (– cos y + 6) y  y( x)  B) D) y + cos y Ans: D Difficulty: Moderate Section: 2.7 24 x  (– cos y + 3) y y( x)  y + sin y 24 x  (– cos y – 6)( y) y( x)  y + sin y 110 Find the location of all horizontal and vertical tangents for x  xy  16 A) B) C) D) Ans: horizontal: none; vertical: (–4, 0), (4, 0) horizontal: (4, 0); vertical: (–4, 0), (4, 0) horizontal: (–4, 0), (4, 0); vertical: none horizontal: none; vertical: (4, 0) A Difficulty: Moderate Section: 2.7 111 Find the location of all horizontal and vertical tangents for x  xy  16  A) B) C) D)    horizontal:  –4, –2  ,  –4, 2  ; vertical: (0, 0) horizontal:  –4, –2  ,  –4, 2  ; vertical: none horizontal:  4, –2  ,  4, 2  ; vertical: (–16, 0) horizontal: –4, –2 , –4, 2 ; vertical: (–16, 0) Ans: C Difficulty: Moderate Section: 2.7 112 Determine if the function satisfies Rolle's Theorem on the given interval If so, find all values of c that make the conclusion of the theorem true f ( x)  100  x ,  –6, 6 A) x  B) x  100 C) x  –10, x  10 Ans: A Difficulty: Easy Section: 2.8 D) Rolle's Theorem not satisfied 113 Using the Mean Value Theorem, find a value of c that makes the conclusion true for f ( x)  x3  5x , in the interval [1,1] A) c  1.129 B) One or more hypotheses fail Ans: C Difficulty: Easy Section: 2.8 Page 120 C) c  0.295 D) c  Chapter 114 Using the Mean Value Theorem, find a value of c that makes the conclusion true for    f ( x)  cos x,   ,   2 A) One or more hypotheses fail Ans: B Difficulty: Easy B) c  C) c   D) c  881 Section: 2.8 115 Prove that x3 + x –  has exactly one solution Ans: Let f  x   x + x – The function f(x) is continuous and differentiable everywhere Since f(0) < and f(1) > 0, f(x) must have at least one zero The derivative of f ( x)  x3 + x – is f ( x)  24 x + , which is always greater than zero Therefore f(x) can only have one zero Difficulty: Moderate Section: 2.8 116 Prove that x + x –  has exactly two solutions Ans: Let f  x   x + x – The function f (x) is continuous and differentiable everywhere Since f (–1) > 0, f (0) < and f (1) > 0, f (x) must have at least two zeros The derivative of f  x   x + x – is f ( x)  x3 + 16 x  x  x +  , which has one zero Therefore f (x) can only have two zeros Difficulty: Difficult Section: 2.8 117 Find all functions g such that g ( x)  f ( x) f ( x)  x g ( x)  63x6 B) g ( x)  x8 C) g ( x)  63x6  C, for some constant C D) g ( x)  x8  C , for some constant C Ans: D Difficulty: Easy Section: 2.8 A) 118 Find all the functions g ( x) such that g ( x)  B) g ( x)  –  c C) g ( x)  5 x 20 x 4x Difficulty: Moderate Section: 2.8 A) g ( x)  – Ans: D x6 Page 121 D) g ( x)  – c x5 Chapter 119 Find all the functions g ( x) such that g ( x)  5sin x g ( x)  5cos x  c A) g ( x)  –5cos x  c B) Ans: B Difficulty: Moderate C) D) Section: 2.8 g ( x)  –5cos x g ( x)  –5sin x  c 120 Determine if the function f ( x)  x3 + x – is increasing, decreasing, or neither A) Increasing B) Decreasing C) Neither Ans: A Difficulty: Easy Section: 2.8 121 Determine if the function f ( x)  –4 x – 5x + is increasing, decreasing, or neither A) Increasing B) Decreasing C) Neither Ans: C Difficulty: Easy Section: 2.8 122 Explain why it is not valid to use the Mean Value Theorem for the given function on the specified interval Show that there is no value of c that makes the conclusion of the theorem true ,  4, 6 f ( x)  x–5 Ans: The function is not continuous on the specified interval, so the Mean Value Theorem does not apply Note that f (4)  1 and f (6)  , so that f (6)  f (4)  ( 1)   (6)  (4) Also, f ( x)   ( x – 5) Since f ( x)  for all x in the domain of f, there is no value of c such that f (6)  f (4) f (c)  That is, there is no value of c such that f (c)  (6)  (4) Difficulty: Moderate Section: 2.8 Page 122 ... distance in miles from a starting point as a function of time in hours for a car on a trip Find the fastest speed (magnitude of velocity) during the trip Describe how the speed during the first hours... compares to the speed during the last hours Describe what is happening between and hours Ans: The fastest speed occurred during the last hours of the trip when the car traveled at about 70 mph The... Chapter 26 The table below gives the position s(t) for a car beginning at a point and returning hours later Estimate the velocity v(t) at two points around the third hour t (hours) s(t) (miles) 0 15