Berresford/Rockett, Brief Applied Calculus, 6e Chapter Derivatives And Their Uses Complete the table and use it to predict the limit, if it exists 6x  f ( x)  x lim f ( x)  ? x 0.5 x f ( x) 0.51 0.501 0.5001   0.5 ?   0.4999 0.499 0.49 A) –160.0 B) 80.0 C) –80.0 D) 0.5 E) does not exist Ans: C Use properties of limits and algebraic methods to find the limit, if it exists lim (8 x3  13x  3x  13) x 3 A) B) C) D) E) Ans: –121 121 141 –141 does not exist B x2  x without using a graphing calculator or making tables x 5 x  A) B) –5 C) D) E)  Ans: D Find lim ©2013 Cengage Learning All Rights Reserved Page 37 Berresford/Rockett, Brief Applied Calculus, 6e Use properties of limits and algebraic methods to find the limit, if it exists –7  x lim x 1 144 x  A) 14 B) 14 C)  14 D)  14 E) does not exist Ans: D Use properties of limits and algebraic methods to find the limit, if it exists x  x  14 lim x  –5 x2  2x A) B)  C)  D) E) does not exist Ans: B Use properties of limits and algebraic methods to find the limit, if it exists x  x  32 lim x 13 x  x  A) 17  12 B) 17 12 C) 12 17 D) 12  17 E) does not exist Ans: B ©2013 Cengage Learning All Rights Reserved Page 38 Berresford/Rockett, Brief Applied Calculus, 6e Use properties of limits and algebraic methods to find the limit, if it exists  x  h   x2 lim h 0 h A) B) 2x C) 9x D) 18x E) does not exist Ans: D A graph of y  f ( x ) is shown and a c-value is given For this problem, use the graph to find lim f ( x) x c c  2 A) B) C) D) E) Ans: –6 –4 does not exist A Use properties of limits and algebraic methods to find the limit, if it exists 16  x for x  lim f ( x), where f ( x)   x 3  x  x for x  A) B) C) –6 D) –5 E) does not exist Ans: E 10 Find lim+ f ( x) for x  –6 A) B) C) D) E) Ans: f ( x)  x+6 x+6 –1 –6 D ©2013 Cengage Learning All Rights Reserved Page 39 Berresford/Rockett, Brief Applied Calculus, 6e 11 Find lim f ( x) for the graph of f ( x ) given below + x 3 A) B) C) D) E) Ans: 12 -3 inf A Find lim– x  –1 A) B) C) D) E) Ans: 13 –1 C Find lim+ x A) B) C) D) E) Ans: x +1 –1  x – 6 –6 E ©2013 Cengage Learning All Rights Reserved Page 40 Berresford/Rockett, Brief Applied Calculus, 6e 14 For the given x-value, use the figure to determine whether the function is continuous or discontinuous at that x-value x5 A) discontinuous B) continuous Ans: A 15 Determine whether the function is continuous or discontinuous at the given x-value  x2  if x  –4  f ( x)   x  –4  9 x  123 if x  –4 A) discontinuous B) continuous Ans: B 16 Determine whether the given function is continuous If it is not, identify where it is discontinuous y  3x  x  A) discontinuous at x  B) discontinuous at x  C) discontinuous at x  5 D) discontinuous at x  10 E) continuous everywhere Ans: E 17 Determine whether the function is continuous or discontinuous at the given x-value x2  y , x  –7 x4 A) continuous B) discontinuous Ans: A ©2013 Cengage Learning All Rights Reserved Page 41 Berresford/Rockett, Brief Applied Calculus, 6e 18 Determine whether the given function is continuous If it is not, identify where it is discontinuous You can verify your conclusions by graphing the function with a graphing utility, if one is available x  3x  y x 1 A) discontinuous at x  B) discontinuous at x  1 C) discontinuous at x  D) discontinuous at x  1 E) continuous everywhere Ans: D ©2013 Cengage Learning All Rights Reserved Page 42 Berresford/Rockett, Brief Applied Calculus, 6e 19 By imagining tangent lines at points P1 , P2 , and P3 , state whether the slopes are positive, zero, or negative at these points A) At P1 : positive slope At P2 : negative slope B) At P3 : positive slope At P1 : zero slope At P2 : negative slope C) At P3 : positive slope At P1 : zero slope At P2 : positive slope D) At P3 : negative slope At P1 : positive slope At P2 : positive slope E) At P3 : positive slope At P1 : positive slope At P2 : negative slope At P3 : negative slope Ans: C ©2013 Cengage Learning All Rights Reserved Page 43 Berresford/Rockett, Brief Applied Calculus, 6e 20 Which graph represents f ( x) if the graph of f ( x ) is displayed below? A) B) C) D) ©2013 Cengage Learning All Rights Reserved Page 44 Berresford/Rockett, Brief Applied Calculus, 6e E) Ans: C 21 For the given function, find the average rate of change over the specified interval f ( x)   x  x over  –2,  A) B) –19 C) 19 D) 13 E) –13 Ans: E 22 Find the average rate of change of f  x   x  between x  and x  A) B) C) D) 11 E) Ans: A 23 Find the instantaneous rate of change of the function f  x   x  x at x  A) 30 B) 26 C) 41 D) 42 E) 29 Ans: E ©2013 Cengage Learning All Rights Reserved Page 45 Berresford/Rockett, Brief Applied Calculus, 6e 24 For the function in this problem, find the instantaneous rate of change of the function at the given value f ( x)  x  5x  5; x  A) B) 41 C) 31 D) 67 E) 77 Ans: D 25 For the function in this problem, find the slope of the tangent line at the given value f ( x)  x  x  9; x  A) B) 14 C) –4 D) E) 19 Ans: A 26 Find the slope of the tangent at x  –1 f ( x)  x  x A) –14 B) –4 C) –10 D) E) Ans: C 27 For the function in this problem, find the derivative, by using the definition f ( x)  x  x  A) x  3x  B) x  3x C) 10x D) 5x  E) 10 x  Ans: E 28 Find the slope of the tangent to the graph of f (x) at any point f ( x)  x  x A) 18x  B) 18x  C) 9x  D) 9x2  x E) 3x Ans: A ©2013 Cengage Learning All Rights Reserved Page 46 Berresford/Rockett, Brief Applied Calculus, 6e 56 Find the indicated derivative and simplify  6x2 dy for y  x  4x2  dx A) x  3x  x    x  4x  2 x  3x  x    x  4x  2 x  3x  x    x  4x  2 x  3x  x    x  4x  2 x  3x  x    x  4x  2 B) C) 2 E) 2 D) 2 4 2 2 Ans: C 57 x2  x2 3x  x  Find the derivative of f  x    x  3 A) B) C) D) E) x2  f   x   6x   x  3 x2  x  2 x2    x  3 x2 x2  4x  x 2 f   x  6x   x  3 x2  x  2 f   x   x6 x2    x  3 x2 x2  4x  x 2 f  x  7x   x  3 x2  x  2 f   x   x5 Ans: C ©2013 Cengage Learning All Rights Reserved Page 55 Berresford/Rockett, Brief Applied Calculus, 6e 58 Find the indicated derivative and simplify  x   x   f ( x) for f ( x)  x2  A) 11x  62 x  18 x B) 6 6 2   11x  34 x  18 x E)  x  68 x  18 x D) 6 x  34 x  18 x C) 2 6  11x  68 x  18 x 6  Ans: C 59 Find the derivative of A) B) 1 4x –1 C)  x –1 D) E) x+1 x–1 x2 1 x x  –1  x–1 Ans: E ©2013 Cengage Learning All Rights Reserved Page 56 Berresford/Rockett, Brief Applied Calculus, 6e 60 If the cost C (in dollars) of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given by 2000 p C ( p)  , 130  p find the rate of change of C with respect to p A) 4000000 130  p  B) 260000 130  p  C) 2 16900 130  p  D) 2000 130  p  E) 130 130  p  Ans: B 61 The number of bottles of whiskey that a store will sell in a month at a price of p dollars 2250 per bottle is N ( p )  Find the rate of change of this quantity when the price is p2 $9 A) –18.60 B) 204.55 C) –18.75 D) 18.50 E) –9.30 Ans: A 62 After x months, monthly sales of a compact disc are predicted to be S ( x)  x (125  x3 ) thousand Find the rate of change of the sales after months in thousands per month A) –48 B) 452 C) 420 D) 476 E) 468 Ans: C ©2013 Cengage Learning All Rights Reserved Page 57 Berresford/Rockett, Brief Applied Calculus, 6e 63 Find f ( x) and f ( x) f ( x)   x  x A) f ( x)   15 x , f ( x)  30 x B) f ( x)  30 x, f ( x)  30 C) f ( x)  15x , f ( x)  30 x D) f ( x)   15x , f ( x)  30 E) f ( x)  –10, f ( x)  Ans: A 64 Find the third derivative y  x3  x  x A) 42 B) 42x C) 21 D) 21x E) Ans: A 65 Find the indicated derivative Find y (4) if y  x8  8x3 A) 336x B) 336x C) 336 x  48 x D) 1680 x5  48 x E) 1680x Ans: E 66 Find f ''( x) for the function A) 99 72 x B) 99 72 x C) 11 92 x D) 99 72 x 16 E) 11 92 x Ans: A x11 ©2013 Cengage Learning All Rights Reserved Page 58 Berresford/Rockett, Brief Applied Calculus, 6e 67 Find f '''( x) for the function A) 399 152 x B) 6783 152 x C) 399 172 x D) 6783 152 x 16 E) 399 172 x Ans: B 68 Find f (4) ( x) for the function A) 9009 92 x B) 9009 52 x C) 143 72 x D) 9009 52 x 16 E) 143 72 x 16 Ans: D x 21 x13 69 Find the second derivative h( x )  x  x A) 30 42x  x B) 42 42x  x C) 42 30x  x D) 30 42x  x E) 42 30x  x Ans: C ©2013 Cengage Learning All Rights Reserved Page 59 Berresford/Rockett, Brief Applied Calculus, 6e 70 Find f ''(5) for the function 4x A) 625 B) 500 C) 3125 D) 500 E) Ans: C 71 Find the third derivative y x A) –120 x5 B) 120 x6 C) D) 40 x5 E) –120 x6 Ans: E 72 Find the second derivative of the function ( x  3)( x  7) A) x3  x  21 B) x3  x C) 12 x  20 D) 12 x  E) x  20 x  21 Ans: D 73 Evaluate the expression A) B) C) D) E) Ans: d3 x dx x 1 42 –42 –210 210 E ©2013 Cengage Learning All Rights Reserved Page 60 Berresford/Rockett, Brief Applied Calculus, 6e 74 Find the second derivative of the function A) 2x  2x  56 (2 x  7)3 B) 112 (2 x  7)3 C) 112  (2 x  7)3 D) 28  (2 x  7) E) 28 (2 x  7) Ans: C  75 If the formula describing the distance s (in feet) an object travels as a function of time t (in seconds) is s  60  90t  17t What is the acceleration of the object when t  5? A) ft/sec2 B) –34 ft/sec2 C) –80 ft/sec2 D) 34 ft/sec2 E) 80 ft/sec2 Ans: B 76 After t hours, a car is a distance s(t )  60t  300 miles from its starting point Find the t4 velocity after hours A) 51 miles/hour B) 66 miles/hour C) 54 miles/hour D) 57 miles/hour E) 63 miles/hour Ans: D 77 If f ( g ( x))  x  3x  x A) B) x 3 C) x  3x  D) x  3x  E) x  3x  Ans: D and f ( x)  x , find g ( x) ©2013 Cengage Learning All Rights Reserved Page 61 Berresford/Rockett, Brief Applied Calculus, 6e 78 If f ( g ( x))  A) and g ( x)  x  x , find f ( x ) 8x  x x B) 8x  C) 8x  x D) 1  8x x E) 8x2  x Ans: A 79  x4 If f ( g ( x))    and f ( x)  x , find g ( x)  x4 A)  x  4 B) x  x  4 C) x2 D)  x  4 x4 x4 Ans: E E) 80 Find f '( x) for the given function f ( x)   ( x  1) A) 4 x  x  1 B) C) D) E) x  x  1  x  x  1 x  x  1 2 x  x  1 Ans: A ©2013 Cengage Learning All Rights Reserved Page 62 Berresford/Rockett, Brief Applied Calculus, 6e 81 Differentiate the given function (5 x) y 4 A)  5x  B)  4x  C)  5x  D)  5x   20x  E) 3 Ans: C 82 Find the derivative of the given function Simplify and express the answer using positive exponents only y  (4 x  x  2)6 A) 27(4 x  x  2)5 8 x   B) C) D) E) 108 x(4 x  x  2)5 16 x   54 x(4 x  x  2)5 8 x  5 27 x(4 x  x  2)5 16 x   108 x(4 x  x  2)5 8 x   Ans: C 83 Differentiate the given function k ( x)  72 (5x7  x  6)14 A) 28(5 x  x  6)13  35 x  x  B) C) D) E) 4(5 x  x  6)13  35 x  1 4(35 x6  1)13 4(5 x  x  6)15  x  1 2(5 x  x  12)13  35 x  1 Ans: B ©2013 Cengage Learning All Rights Reserved Page 63 Berresford/Rockett, Brief Applied Calculus, 6e 84 Differentiate the given function y  x5  3x 1/ A) 35 x  3  1/ B) x5  3x   1/ C) 35 x5  3x   x5  3  1/ D) x5  3x   35 x  3  3/ E)   x5  3x   35 x  3 Ans: D 85 Differentiate the given function p  q   (q  7) 4 A) 12q   q3   B) C) D) E)     3q q 7  12q q 7  3q q 7  4q q 7  Ans: C ©2013 Cengage Learning All Rights Reserved Page 64 Berresford/Rockett, Brief Applied Calculus, 6e 86 Differentiate the given function (7 x  1)  x y 17 A)   x  1  1   17 B)   x  1     17 C)  x  1     17  D)   x  1     17 E)  42  x  1  1   17 Ans: A 87 Differentiate the given function y (6 x  x  1) / A)    x8  x  1 2 B)  7   48 x  3  x8  x  1 C)    x8  x  1  48 x   D)    x8  x  1  x8  x  1 E)    x8  x  1  48 x   Ans: E 88 Find the derivative of the given function Simplify and express the answer using positive exponents only   y   x2 8x2  x A) B) C) D) E)     x 8x    40 x  21x  128x  56   x 8x    40 x  21x  128x  56  2 x 8x    40 x  21x  128 x  56  2 x 8x    40 x  21x  128 x  56   x3 8x   40 x3  21x  128x  56 3 3 3 3 3 3 Ans: E ©2013 Cengage Learning All Rights Reserved Page 65 Berresford/Rockett, Brief Applied Calculus, 6e 89 Use the Generalized Power Rule to find the derivative of the function x x  A) x3  3x  1 B) x4 1 x  3x   C) x4  x  3x   D) x4 1 x3  3x   E) x4 1 x3  3x   x4 1 Ans: E 90 Differentiate the given function y (6 x ) A) 252 (6 x ) B) 42  (6 x) C) 252  (6 x) D) 42 (6 x ) E) 42  (6 x)5 Ans: C ©2013 Cengage Learning All Rights Reserved Page 66 Berresford/Rockett, Brief Applied Calculus, 6e 91 Differentiate the given function y 4x A) 12  x B)  x C) 12  x D)  x E)  x Ans: D 92 A company's cost function is C ( x)  x  800 dollars, where x is the number of units Find the marginal cost function and evaluate it for x  30 Round your answer to two decimal places A) 1.18 dollars B) 2.35 dollars C) 50.99 dollars D) 17.65 dollars E) 66.33 dollars Ans: A 93 If $1800 is deposited in a bank paying r% interest compounded annually, years later its value will be V (r )  1800(1  0.01r )5 dollars Find V '(8) Round your answer to nearest cent A) 122.44 dollars B) 132.24 dollars C) 24.49 dollars D) 26.45 dollars E) 142.82 dollars Ans: A ©2013 Cengage Learning All Rights Reserved Page 67 Berresford/Rockett, Brief Applied Calculus, 6e 94 For the function displayed in the graph below, find all x-values at which the derivative does not exist A) B) C) D) E) Ans: –3, 0, –1, none B 95 For the function displayed in the graph below, find all x-values at which the derivative does not exist A) B) C) D) E) Ans: 96 0, none –1 –1, E For the function f ( x)  ( x + 2) , find the x-value at which the derivative does not exist A) - B) C) D)  E) none Ans: A ©2013 Cengage Learning All Rights Reserved Page 68 Berresford/Rockett, Brief Applied Calculus, 6e 97 Use the numerical derivative function on a graphing calculator to calculate the derivative of the function f ( x)  at x  Is the calculator correct? x A)  2; No, the calculator is not correct B) 0; Yes, the calculator is correct C) ; No, the calculator is not correct D) 0; No, the calculator is not correct E)  2; Yes, the calculator is correct Ans: D 98 If a function is continuous at a point, then it is also not defined at that same point? A) True B) False Ans: B ©2013 Cengage Learning All Rights Reserved Page 69 ... 43 Berresford/ Rockett, Brief Applied Calculus, 6e 20 Which graph represents f ( x) if the graph of f ( x ) is displayed below? A) B) C) D) ©2013 Cengage Learning All Rights Reserved Page 44 Berresford/ Rockett,... 17 E) does not exist Ans: B ©2013 Cengage Learning All Rights Reserved Page 38 Berresford/ Rockett, Brief Applied Calculus, 6e Use properties of limits and algebraic methods to find the limit,... Ans: f ( x)  x+6 x+6 –1 –6 D ©2013 Cengage Learning All Rights Reserved Page 39 Berresford/ Rockett, Brief Applied Calculus, 6e 11 Find lim f ( x) for the graph of f ( x ) given below + x 3 A)