Chapter Derivatives And Their Uses 1. Complete the table and use it to predict the limit, if it eists. 6 f( ) 0. 1 lim f( )? 0.1 0.01 0.001 0.? 0.999 0.99 f ( ) 0.9 160.0 80.0 80.0 0. does not eist. Use properties of limits and algebraic methods to find the limit, if it eists. lim (8 1 1) 11 11 11 11 does not eist Ans: B. Find lim 0 without using a graphing calculator or making tables. 01 Cengage Learning. All Rights Reserved. Page
. Use properties of limits and algebraic methods to find the limit, if it eists. 8 lim 1 1 9 1 1 1 1 1 9 1 does not eist. Use properties of limits and algebraic methods to find the limit, if it eists. 91 lim does not eist Ans: B 6. Use properties of limits and algebraic methods to find the limit, if it eists. lim 1 98 1 1 1 1 1 1 1 1 does not eist Ans: B 01 Cengage Learning. All Rights Reserved. Page 8
. Use properties of limits and algebraic methods to find the limit, if it eists. h 9 9 lim h0 h 0 9 18 does not eist 8. A graph of y f ( ) is shown and a c-value is given. For this problem, use the graph to find lim f( ). c c 0 6 does not eist 9. Use properties of limits and algebraic methods to find the limit, if it eists. 16 for lim f ( ), where f ( ) for 6 6 does not eist 10. Find lim f( ) + 6 6 1 0 1 6 for +6 f( ). +6 01 Cengage Learning. All Rights Reserved. Page 9
11. Find lim f ( ) + Berresford/Rockett, Brief Applied Calculus, 6e for the graph of f( ) given below. 0 - inf 1. Find lim 1. 1 +1 1 0 1 1. 1 Find 6 0 6 lim 6 + 6. 01 Cengage Learning. All Rights Reserved. Page 0
1. For the given -value, use the figure to determine whether the function is continuous or discontinuous at that -value. discontinuous continuous 1. Determine whether the function is continuous or discontinuous at the given -value. if f ( ) 9 1 if discontinuous continuous Ans: B 16. Determine whether the given function is continuous. If it is not, identify where it is discontinuous. y discontinuous at discontinuous at 0 discontinuous at discontinuous at 10 continuous everywhere 1. Determine whether the function is continuous or discontinuous at the given -value. y, continuous discontinuous 01 Cengage Learning. All Rights Reserved. Page 1
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. 8 y 1 discontinuous at 1 discontinuous at 1 discontinuous at 1 discontinuous at 1 continuous everywhere 01 Cengage Learning. All Rights Reserved. Page
19. By imagining tangent lines at points P1, P, and P, state whether the slopes are positive, zero, or negative at these points. At P1 : positive slope At P : negative slope At P : positive slope At P1 : zero slope At P : negative slope At P : positive slope At P1 : zero slope At P : positive slope At P : negative slope At P1 : positive slope At P : positive slope At P : positive slope At P1 : positive slope At P : negative slope At P : negative slope 01 Cengage Learning. All Rights Reserved. Page
0. Which graph represents f ( ) if the graph of ( ) f is displayed below? 01 Cengage Learning. All Rights Reserved. Page
1. For the given function, find the average rate of change over the specified interval. f ( ), 0 19 19 1 1 over. Find the average rate of change of f 8 8 11 between and 8.. Find the instantaneous rate of change of the function 0 6 1 9 f 6 at. 01 Cengage Learning. All Rights Reserved. Page
. For the function in this problem, find the instantaneous rate of change of the function at the given value. f ( ) 9 ; 0 1 1 6. For the function in this problem, find the slope of the tangent line at the given value. f ( ) 9 9; 1 1 1 0 19 6. Find the slope of the tangent at 1. f ( ) 6 1 10 0. For the function in this problem, find the derivative, by using the definition. f ( ) 9 9 10 10 8. Find the slope of the tangent to the graph of f () at any point. f ( ) 9 6 18 6 18 6 9 6 9 6 01 Cengage Learning. All Rights Reserved. Page 6
9. Find f ' of f 8 f ' 8 f ' f ' f ' f ' Ans: B Berresford/Rockett, Brief Applied Calculus, 6e by using the definition of the derivative. 0. Write the equation of the line tangent to the graph of f () at 1. f ( ) 8 y y y y y 1. The population of a town is ). Find f ' 0 0 8 weeks. 8 6 1 9 f 1 00 people after weeks (for to find the instantaneous rate of change of the population after. An automobile dealership finds that the number of cars that it sells on day of an advertising campaign is S 18 (for 0 S' to find the instantaneous rate of change on day. 1 18 16 1 ). Find 01 Cengage Learning. All Rights Reserved. Page
. Differentiate the given function. 6 9 y 6 6 6 9 9 9. Find the derivative of w g w 0 w g w w g w g w w g w 0 w Berresford/Rockett, Brief Applied Calculus, 6e g w 0 w.. Find the derivative of the function. 1 y 9 1 18 18 1 18 9 9 Ans: B 1 6. For the function given, find f '( ). f 1 8 1 1 1 8 ( ) 1 8 01 Cengage Learning. All Rights Reserved. Page 8
. Find the derivative of the function. 8/ 10/ f ( ) 9 9 11/ 1/ 0 / / 0 11/ 1/ 0 / / 0 11/ 1/ 90 8. Find the derivative of 8 f f f f f Ans: B Berresford/Rockett, Brief Applied Calculus, 6e f. 9. Find the derivative of the function. y 6 8 6 8 6 6 8 6 Ans: B 0. Find the derivative of the function. 1 11 8 h( ) 11 19 1 6 0 10 1 11 8 0 10 0 10 1 11 8 0 190 9 1 1 09 6 1 11 19 1 1 09 6 1 0 190 9 1 01 Cengage Learning. All Rights Reserved. Page 9
1. Find the derivative of 1 1 h h h 1 1 h h Berresford/Rockett, Brief Applied Calculus, 6e h 6.. At the indicated point, find the instantaneous rate of change of the function. R( ) 1, 9 19 1. If 9 f 60, find 81 f 81 1 f 81 1 f 81 1 f 81 16 f 81 6 f.. Find the derivative at the given -value with the appropriate rule. y 8 at 9 8 6 8 0 01 Cengage Learning. All Rights Reserved. Page 0
. If f, find df. d df d df 19 d df 0 d df 18 d df 80 d Berresford/Rockett, Brief Applied Calculus, 6e 6. If 0 0, f find df d df d df 10 d df 10 d df d df. d. Suppose the Marginal Cost Businesses can buy multiple licenses for PowerZip data compression software at a total cost of approimately C dollars for licenses. Find the derivative of this cost function at 6. C 6 8 C6 C6 C6 1 6 6 C Ans: B 01 Cengage Learning. All Rights Reserved. Page 1
8. Suppose the number of people newly inflected on day t of a flu epidemic is f t 1 t t (for 0 t 1). Find the instantaneous rate of change of this number on day 10. f 10 00 f 10 f 10 0 f 10 0 f 10 60 9. Find the derivative of f 6 8 1 answer. 1 f 6 f 6 f 6 f 6 f 0. ds Find if s t 8t 8. dt 8 6t 6t t 8 9t 8t t 8 6t 8t t 8 9t 6t t 8 9t 8t t by using the Product Rule. Simplify your 01 Cengage Learning. All Rights Reserved. Page
1. Find the derivative, but do not simplify your answer. 9 8 8 9 9 y 6 9 1 6 81 8 9 6 9 9 8 8 9 9 6 1 6 81 8 9 6 9 9 9 6 9 9 1 6 81 8 9 6 9 9 8 8 9 9 6 9 1 6 81 8 9 1 6 81 8 9 6 9 9 8 8 9 9 6. Find the derivative of f z z 8 z 1 1 z 1 z Simplify your answer. f z z z 0 f z z 9z z f z z z 9 f z z 0z 1 f z z 1. 1 Find the derivative of 6. 1 6 6 1 6 6 1 6 Ans: B by using the Product Rule. 01 Cengage Learning. All Rights Reserved. Page
. Find the indicated derivative and simplify. C ( ) for C ( ) 1 1 1 1 1 1. Find the derivative of f f f f f f 1 0 0 0 0 1 0 by using Quotient Rule. Simplify your answer. 01 Cengage Learning. All Rights Reserved. Page
6. Find the indicated derivative and simplify. dy d for 1 6 y. Find the derivative of 6 f. 6 f 6 6 6 f 6 f 6 6 f 6 6 6 f 01 Cengage Learning. All Rights Reserved. Page
8. Find the indicated derivative and simplify. f ( ) for f( ) 6 11 618 6 18 6 6818 6 11 18 6 11 6818 6 9. Find the derivative of +1. 1 1 1 1 1 1 1 1 01 Cengage Learning. All Rights Reserved. Page 6
60. If the cost C (in dollars) of removing p percent of the particulate pollution from the ehaust gases at an industrial site is given by 000 p C( p) 10 p, find the rate of change of C with respect to p. 000000 10 p 60000 10 p 16900 10 p 000 10 p 10 10 p Ans: B 61. The number of bottles of whiskey that a store will sell in a month at a price of p dollars 0 per bottle is N( p). Find the rate of change of this quantity when the price is p $9. 18.60 0. 18. 18.0 9.0 6. After months, monthly sales of a compact disc are predicted to be S( ) (1 ) thousand. Find the rate of change of the sales after months in thousands per month. 8 0 6 68 01 Cengage Learning. All Rights Reserved. Page
6. Find f ( ) and f ( ). f ( ) 6 f ( ) 1, f ( ) 0 f ( ) 0, f ( ) 0 f ( ) 1, f ( ) 0 f ( ) 1, f ( ) 0 f ( ) 10, f ( ) 0 6. Find the third derivative. y 1 1 0 6. Find the indicated derivative. () Find y 8 if y 8. 6 6 6 8 1680 8 1680 66. Find f ''( ) for the function 11. 99 99 8 11 99 16 11 9 9 01 Cengage Learning. All Rights Reserved. Page 8
6. Find f '''( ) for the function 1. 1 99 1 68 8 1 99 68 16 1 99 8 Ans: B 1 Berresford/Rockett, Brief Applied Calculus, 6e 68. Find f () ( ) for the function 9 9009 9009 8 1 8 9009 16 1 16 1. 69. Find the second derivative. 6 1 h( ) 6 0 8 8 0 8 0 0 01 Cengage Learning. All Rights Reserved. Page 9
0. 1 Find f ''() for the function. 1 6 1 00 1 9 00 1 1. Find the third derivative. y 10 10 6 0 0 10 6 Berresford/Rockett, Brief Applied Calculus, 6e. Find the second derivative of the function 8 1 8 1 0 1 8 0 1 ( )( ).. d Evaluate the epression d 10 10 1. 01 Cengage Learning. All Rights Reserved. Page 60
. Find the second derivative of the function. 6 ( ) 11 ( ) 11 ( ) 8 ( ) 8 ( ). If the formula describing the distance s (in feet) an object travels as a function of time t (in seconds) is s 60 90t 1t. What is the acceleration of the object when t? 0 ft/sec ft/sec 80 ft/sec ft/sec 80 ft/sec Ans: B 6. 00 After t hours, a car is a distance s( t) 60t t miles from its starting point. Find the velocity after 6 hours. 1 miles/hour 66 miles/hour miles/hour miles/hour 6 miles/hour. If f ( g( )) and f ( ), find g. ( ) 01 Cengage Learning. All Rights Reserved. Page 61
8. If f ( g( )) 8 1 6 1 1 8 6 1 8 6 1 1 8 6 8 6 Berresford/Rockett, Brief Applied Calculus, 6e and g( ) 8 6, find f( ). 9. If f ( g( )) 1 and f ( ) g., find ( ) 80. Find f '( ) for the given function. f ( ) ( 1) 1 1 1 1 1 01 Cengage Learning. All Rights Reserved. Page 6
81. Differentiate the given function. ( ) y 0 Berresford/Rockett, Brief Applied Calculus, 6e 8. Find the derivative of the given function. Simplify and epress the answer using positive eponents only. 9 ( ) 6 y ( ) 8 108 ( ) 16 ( ) 8 ( ) 16 108 ( ) 8 8. Differentiate the given function. 1 k( ) ( 6) 8( 6) 1 6 ( 6) 1 6 1 6 1 ( 1) ( 6) 1 6 1 ( 1) 1 1 Ans: B 01 Cengage Learning. All Rights Reserved. Page 6
8. Differentiate the given function. y 1 1/ 1 1/ 1 1/ 1 1/ / 1 Berresford/Rockett, Brief Applied Calculus, 6e 8. Differentiate the given function. p q ( q ) 1q q q q 1q q q q q q 6 01 Cengage Learning. All Rights Reserved. Page 6
86. Differentiate the given function. 6 (1) y 1 6 1 1 1 1 6 1 1 1 6 1 6 1 1 1 1 1 1 8. Differentiate the given function. 1 y 8 / (6 1) 9 6 8 1 8 8 6 1 8 6 1 8 9 6 8 1 6 8 1 9 8 6 1 8 Berresford/Rockett, Brief Applied Calculus, 6e 88. Find the derivative of the given function. Simplify and epress the answer using positive eponents only. y 8 0 1 18 6 8 0 1 18 6 8 0 1 18 6 8 0 1 18 6 8 0 1 18 6 01 Cengage Learning. All Rights Reserved. Page 6
89. Use the Generalized Power Rule to find the derivative of the function 1. 1 1 1 1 1 1 90. Differentiate the given function. y 6 (6 ) (6 ) (6 ) (6 ) (6 ) (6 ) 01 Cengage Learning. All Rights Reserved. Page 66
91. Differentiate the given function. y 1 1 Berresford/Rockett, Brief Applied Calculus, 6e 9. A company's cost function is C( ) 800 dollars, where is the number of units. Find the marginal cost function and evaluate it for 0. Round your answer to two decimal places. 1.18 dollars. dollars 0.99 dollars 1.6 dollars 66. dollars 9. If $1800 is deposited in a bank paying r% interest compounded annually, years later its value will be V ( r) 1800(1 0.01 r) dollars. Find V '(8). Round your answer to nearest cent. 1. dollars 1. dollars.9 dollars 6. dollars 1.8 dollars 01 Cengage Learning. All Rights Reserved. Page 6
9. For the function displayed in the graph below, find all -values at which the derivative does not eist., 0, 1, none Ans: B 9. For the function displayed in the graph below, find all -values at which the derivative does not eist. 0, none 1 1, 96. For the function f ( ) ( + ) 9, find the -value at which the derivative does not eist. - 0 9 none 01 Cengage Learning. All Rights Reserved. Page 68
9. Use the numerical derivative function on a graphing calculator to calculate the 1 derivative of the function f( ) at 0. Is the calculator correct? ; No, the calculator is not correct. 0; Yes, the calculator is correct. 1 ; No, the calculator is not correct. 0; No, the calculator is not correct. ; Yes, the calculator is correct. 98. If a function is continuous at a point, then it is also not defined at that same point? True False Ans: B 01 Cengage Learning. All Rights Reserved. Page 69