Chapter 2: Differentiation 1. Find the slope of the tangent line to the graph of the function below at the given point.

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1 Chapter : Differentiation 1. Find the slope of the tangent line to the graph of the function below at the given point. f( ) 10, (, ) 10 1 E) none of the above. Find the slope of the tangent line to the graph of the function at the given point. ( ) 5 +10, (, 10) f E) none of the above. Find the slope of the tangent line to the graph of the function at the given point. ( ) + 6, (, ) f E) none of the above. Use the limit definition to find the slope of the tangent line to the graph of f( ) 9 at the point (5,). 1 1 E) 1 5 Larson, Calculus: An Applied Approach (+Brief), 9e Page 59

2 5. Find the derivative of the following function using the limiting process. f ( ) E) none of the above Ans: B 6. Find the derivative of the following function using the limiting process. f( ) f( ) f( ) 9 6 f( ) f( ) 9 6 E) either B or D 9 1/. Find the derivative of the following function using the limiting process. f( ) 9 f( ) f( ) f( ) f( ) E) none of the above Larson, Calculus: An Applied Approach (+Brief), 9e Page 60

3 8. Find an equation of the line that is tangent to the graph of f and parallel to the given line. f y y 0 0 y y y 0 0 E) none of the above ( ) 5, Find an equation of the a line that is tangent to the graph of f and parallel to the given line. f y y 15 0 y y y 15 0 E) both B and D Ans: E ( ) 5, Larson, Calculus: An Applied Approach (+Brief), 9e Page 61

4 10. Identify a function f ( ) that has the given characteristics and then sketch the function. f(0) ; f '( ), f ( ) f ( ) f ( ) Larson, Calculus: An Applied Approach (+Brief), 9e Page 6

5 f ( ) E) f( ) + Larson, Calculus: An Applied Approach (+Brief), 9e Page 6

6 11. Find the derivative of the function. f ( ) f ( ) 6 f ( ) 6 f ( ) 6 8 f ( ) 6 E) none of the above Ans: B 1. Find the derivative of the function. f( ) +1 f ( ) 6 6 f ( ) f ( ) f ( ) E) none of the above Larson, Calculus: An Applied Approach (+Brief), 9e Page 6

7 1. For the function given, find f '( ). f E) 15 6 Ans: C ( ) Find the derivative of the function h ( ) E) Find the derivative of the function 5 8/ h'( ) 5 / h'( ) 5 / h'( ) 5 8/ h'( ) E) 5 / h'( ) Ans: C 16. Find the derivative of the function s'( t) s'( t) s'( t) 8 s'( t) 8 E) s'( t) Ans: B h ( ) st 5/. () 8. Larson, Calculus: An Applied Approach (+Brief), 9e Page 65

8 1. Find the derivative of the function. 1 f( ) f( ) f( ) f( ) f( ) E) none of the above Ans: C 18. Differentiate the given function. y E) Differentiate the given function. 5 y ( ) 80 5 ( ) 0 5 ( ) 80 5 ( ) 0 5 ( ) E) 0 ( ) Ans: C Larson, Calculus: An Applied Approach (+Brief), 9e Page 66

9 0. Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. y( ) and 0 and E) There are no points at which the graph has a horizontal tangent. 1. The graph shows the number of visitors V to a national park in hundreds of thousands during a one-year period, where t = 1 represents January. Estimate the rate of change of V over the interval 5,8. Round your answer to the nearest hundred thousand visitors per year hundred thousand visitors per year 8.5 hundred thousand visitors per year hundred thousand visitors per year 8. hundred thousand visitors per year E) 66.6 hundred thousand visitors per year Ans: C. Find the marginal cost for producing units. (The cost is measured in dollars.) C 05, $9800 $9850 $8800 $8850 E) $950 Larson, Calculus: An Applied Approach (+Brief), 9e Page 6

10 . Find the marginal revenue for producing units. (The revenue is measured in dollars.) R dollars 50 dollars 50 dollars dollars E) dollars. Find the marginal profit for producing units. (The profit is measured in dollars.) P 15 dollars dollars dollars dollars E) dollars 5. The cost C (in dollars) of producing units of a product is given by C Find the additional cost when the production increases from 9 t o10. $0.58 $0.6 $0.6 $0.1 E) $ The profit (in dollars) from selling units of calculus tetbooks is given by p Find the additional profit when the sales increase from 15 to 16 units. Round your answer to two decimal places. $5.5 $0.00 $5.55 $11.00 E) $ The profit (in dollars) from selling units of calculus tetbooks is given by p Find the marginal profit when 18. Round your answer to two decimal places. $.80 $86.80 $5.0 $0.00 E) $ Ans: C Larson, Calculus: An Applied Approach (+Brief), 9e Page 68

11 8. The population P ( in thousands) of Japan from 1980 through 010 can be modeled by P15.56t 80.1t 11, 001 where t is the year, with t =0 corresponding to Determine the population growth rate, dp dt. dp dt 1.1t 80.1 dp dt 1.1t 80.1 dp dt 1.1t 80.1 dp dt 1.1t 80.1 E) dp dt t 9. When the price of a glass of lemonade at a lemonade stand was $1.5, 00 glasses were sold. When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function is linear and that the marginal and fied costs are $0.10 and $ 5, respectively. Find the profit P as a function of, the number of glasses of lemonade sold. P P P P E) P When the price of a glass of lemonade at a lemonade stand was $1.5, 00 glasses were sold. When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function is linear and that the marginal and fied costs are $0.10 and $ 5, respectively. Find the marginal profit when 00 glasses of lemonade are sold and when 00 glasses of lemonade are sold. P001.15, P P000.85, P P001.15, P P000.85, P E) P001.15, P Use the product Rule to find the derivative of the function f f f f f E) 1 f 1. Larson, Calculus: An Applied Approach (+Brief), 9e Page 69

12 . Find the derivative of the function f f 6 f E) f f. Find the derivative of the function rule(s) you used to find the derivative. 1, Product Rule. 1, Quotient Rule 5, Product Rule. 5, Quotient Rule E) +, Product Rule. f f State which differentiation. Find the point(s), if any, at which the graph of f has a horizontal tangent line. f 1 0,0,, 0,, 0,,0,,0 0,,,0 E) 0,0,, 5. A population of bacteria is introduced into a culture. The number of bacteria P can be t modeled by P where t is the time (in hours). Find the rate of change t of the population when t = bacteria/hr 9.15 bacteria/hr.65 bacteria/hr.5 bacteria/hr E) 0.5 bacteria/hr Larson, Calculus: An Applied Approach (+Brief), 9e Page 0

13 6. Use the given information to find f of the function f gh g and g, h1 and h f 1 f 11 f 1 f 9 E) f 1.. Find an equation of the tangent line to the graph of f at the given point. f s s s y 8s 1 y s 8 y 8s y 8 s+ 1 E) y 8 + 1s () ( )( ), at 1,6 8. Find an equation of the tangent line to the graph of f at the given point. f s s s y 9s 1 y s 9 y 9s y 9 s+ 1 E) y 9 + 1s () ( 5)( 6), at, 6 9. p Use the demand function 51 to find the rate of change in the demand 5 p for the given price p $.00. Round your answer to two decimal places. 6.5 units per dollar 6.6 units per dollar 6.6 units per dollar 6.11 units per dollar E) 6.5 units per dollar Ans: E Larson, Calculus: An Applied Approach (+Brief), 9e Page 1

14 0. A population of bacteria is introduced into a culture. The number of bacteria P can be t modeled by P 51 5 where t is the time (in hours). Find the rate of change t of the population when t units per dollar 1.5 units per dollar.01 units per dollar.6 units per dollar E).6 units per dollar Ans: C 1. Find dy du, du d, and dy d of the functions y u, u. dy du u, du d, and dy d 56 dy du u, du d, and dy d 16 9 dy du u, du d, and dy d 56 dy du u, du d, and dy d 56 E) dy du u, du d, and dy d dy Find d of y u, u. 1 1 E) none of these choices Ans: C Larson, Calculus: An Applied Approach (+Brief), 9e Page

15 . Find the derivative of the function. f () t (1 ) t 1 f () t (1 ) t 1 f () t (1 ) t 1 f () t (1 ) t f () t (1 ) t 1 f () t (1 ) t Ans: E E). Differentiate the given function. 9 y / / 1 5 1/ / E) 9 / Find the derivative of the function. 8 f ( ) ( 6 ) f ( ) ( 6 ) 56 f ( ) 6 8 (6 ) 56 f ( ) ( 6 ) 56 f ( ) ( 6 ) 56 E) f ( ) (6 ) 56 6 Larson, Calculus: An Applied Approach (+Brief), 9e Page

16 6. Find the derivative of the given function. Simplify and epress the answer using positive eponents only. c ( ) E) Find the derivative of the function. 8 f ( ) 6 f( ) 6 f( ) f( ) 6 f( ) E) f( ) Larson, Calculus: An Applied Approach (+Brief), 9e Page

17 8. Find the derivative of the function. 5 5 g ( ) g( ) g( ) g( ) g( ) 5 E) Ans: E g( ) You deposit $ 000 in an account with an annual interest rate of change r (in decimal form) compounded monthly. At the end of years, the balance is Find the rates of change of A with respect to r when r , E) 6, r A The value V of a machinet years after it is purchased is inversely proportional to the square root of t 5. The initial value of the machine is$ 10,000. Find the rate of depreciation when t. Round your answer to two decimal places per year per year 16. per year.1 per year E) 10.6 per year Larson, Calculus: An Applied Approach (+Brief), 9e Page 5

18 51. Find the second derivative of the function. f ( ) f ( ) f ( ) f ( ) f ( ) 169 E) None of the above 5. Find the third derivative of the function E) 0 6 f. 5. Find the 6 f E) 1 1 of f Determine whether the statement is true or false. If it is false, eplain why or give an eample that shows it is false. If y f g, then y f g True False. The product rule is f g f gg f Ans: B Larson, Calculus: An Applied Approach (+Brief), 9e Page 6

19 55. Find the third derivative. y E) 0 8 Ans: E 56. Find the value g() for the function,08 0,080 1,185 0,081 E),0,6 8 6 gt () t 6t Find the indicated derivative. () Find y if 8 y E) 1680 Ans: E 58. Find the second derivative for the function equation f( ) E) 0 f( ) and solve the Larson, Calculus: An Applied Approach (+Brief), 9e Page

20 59. 5 Find the second derivative for the function f( ) and solve the equation 5 + f ''( ) 0. 0 no solution E) 1 Ans: C 60. A brick becomes dislodged from the Empire State Building (at a height of 105 feet) and falls to the sidewalk below. Write the position s(t), velocity v(t), and acceleration a(t) as functions of time. st ( ) 16t 105 ; vt () t; at () st ( ) 16t 105 ; vt () t; at () st ( ) 16t 105 ; vt () t; at () st ( ) 16t 105 ; vt () t; at () E) st ( ) 16t 105 ; vt () ; at () t Ans: C 61. Find y implicitly for 9 6 y 9 y 9 y y y 8 y 8 y y 8 6 E) 8 y 8 6y Ans: C y. Larson, Calculus: An Applied Approach (+Brief), 9e Page 8

21 6. dy Find d for the equation 9 y. 5 6y dy 11 d 1 dy 9 d 1 dy 11 d 1 dy 9 d 1 E) dy d Ans: C 6. Find the slope of the graph at the given point. 0 5 E) Larson, Calculus: An Applied Approach (+Brief), 9e Page 9

22 6. Find the slope of the graph at the given point. 0 1 E) Find the rate of change of with respect to p. p p p p p E) p Larson, Calculus: An Applied Approach (+Brief), 9e Page 80

23 66. Find the rate of change of with respect to p. 00 p, 0 00 p p 1 p p 1 p 1 p 1 E) p p 1 6. Find dy d implicitly and eplicitly(the eplicit functions are shown on the graph) and show that the results are equivalent. Use the graph to estimate the slope of the tangent line at the labeled point. Then verify your result analytically by evaluating dy d at the point. 1 1, y 1 1, y 1 1, y 1 1, y E) 1 1, Larson, Calculus: An Applied Approach (+Brief), 9e Page 81

24 68. Let represent the units of labor and y the capital invested in a manufacturing process. When 15,50 units are produced, the relationship between labor and capital can be modeled by 100 y 15,50. Find the rate of change of y with respect to when 1500 and y 15, E) Find dy/d for the following equation: y 5y9 0. dy 5 d y dy d 5 y dy 1 d 5 y dy 5 d 5 y E) dy 1 d y Ans: B 0. dy Find for the equation 0 d y y by implicit differentiation and evaluate the derivative at the point (50,) E) 0 Ans: B Larson, Calculus: An Applied Approach (+Brief), 9e Page 8

25 1. Assume that and y are differentiable functions of t. Find dy/dt using the given values. y 6 for, d/ dt E) 1. dy Given y 10, find dt dy 60 dt dy 10 dt dy 10 dt dy dt 10 E) dy dt 60 Ans: C d when = 9 and. dt. Assume that and y are differentiable functions of t. Find d/dt given that, y 8, and dy / dt. y E) 1.00 Ans: E Larson, Calculus: An Applied Approach (+Brief), 9e Page 8

26 . Area. The radius, r, of a circle is increasing at a rate of 5 centimeters per minute. Find the rate of change of area, A, when the radius is. da 0 dt da 160 dt da 160 dt da 0 dt E) da 0 dt 5. Volume and radius. Suppose that air is being pumped into a spherical balloon at a rate of in. / min. At what rate is the radius of the balloon increasing when the radius is in.? dr dt 9 dr 1 dt dr 9 dt dr dt E) dr 1 dt 9 Ans: E 6. The radius r of a sphere is increasing at a rate of inches per minute. Find the rate of change of volume when r = 8 inches. Round your answer to one decimal place. 80. cubic inches per minute 1. cubic inches per minute 6.0 cubic inches per minute 1. cubic inches per minute E) 8. cubic inches per minute Larson, Calculus: An Applied Approach (+Brief), 9e Page 8

27 . Profit. Suppose that the monthly revenue and cost (in dollars) for units of a product are R 900 and C At what rate per month is the profit changing if 50 the number of units produced and sold is 100 and is increasing at a rate of 10 units per month? $86,960 per month $8660 per month $8960 per month $60 per month E) $89,960 per month Ans: B 8. The lengths of the edges of a cube are increasing at a rate of 8 ft/min. At what rate is the surface area changing when the edges are 15 ft long? 8 ft /min 10 ft /min 0 ft /min 560 ft /min E) 10 ft /min Ans: B 9. A point is moving along the graph of the function d y 9 such that dt centimeters per second. Find dy/dt for the given values of. (a) (b) 8 E) dy dt dy dt dy 16 dt dy dt dy dt dy dt 16 dy 8 dt dy dt 16 dy 8 dt dy dt Ans: B Larson, Calculus: An Applied Approach (+Brief), 9e Page 85

28 80. A point is moving along the graph of the function y 1 centimeters per second. d such that 5 dt Find dy/dt when. dy dt 5 dy dt 85 dy dt 85 dy dt 5 E) dy dt 1 Ans: B 81. Boat docking. Suppose that a boat is being pulled toward a dock by a winch that is 1 ft above the level of the boat deck. If the winch is pulling the cable at a rate of ft/min, at what rate is the boat approaching the dock when it is 8 ft from the dock? Use the figure below. 8.5 ft/min.00 ft/min 8. ft/min 1.5 ft/min E) 1.80 ft/min 8. An airplane flying at an altitude of 5 miles passes directly over a radar antenna. When the airplane is 5 miles away (s = 5), the radar detects that the distance s is changing at a rate of 50 miles per hour. What is the speed of the airplane? Round your answer to the nearest integer. 55 mi/hr 6 mi/hr 510 mi/hr 18 mi/hr E) 118 mi/hr Larson, Calculus: An Applied Approach (+Brief), 9e Page 86

29 8. A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 0 feet per second is 80 feet from third base. At what rate is the player s distance s from home plate changing? Round your answer to one decimal place. 58. feet/second 0. feet/second 0. feet/second 19.9 feet/second E) 1.9 feet/second 8. A retail sporting goods store estimates that weekly sales and weekly advertising costs are related by the equation S The current weekly advertising costs are $100, and these costs are increasing at a rate of $10 per week. Find the current rate of change of weekly sales. 16,500 dollars per week 16,0 dollars per week 8,0 dollars per week 85,150 dollars per week E) 1,01,50 dollars per week Larson, Calculus: An Applied Approach (+Brief), 9e Page 8