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1 . Find the equation of the tangent line to y 6 at. y 9 y y 9 y Ans: A Difficulty: Moderate Section:.. Find an equation of the tangent line to y = f() at =. f y = y = y = y = + Ans: D Difficulty: Moderate Section:.. Find an equation of the tangent line to y = f() at =. f ( ) y = 6 y = 96 y = 96 y = 96 + Ans: C Difficulty: Moderate Section:.. Find the equation of the tangent line to y at. y y Ans: C Difficulty: Moderate Section:. y y. Find the equation of the tangent line to y + at. y + 0 y + 0 y + 0 y + 0 Ans: B Difficulty: Moderate Section:. 6. Compute the slopes of the secant lines between the point at = and points close to it (such as = 0, =, = 0.9, =.) and use these results to estimate the slope of the tangent line at =. Round to two decimal places Ans: D Difficulty: Moderate Section:. Page 90

2 . Compute the slope of the secant line between the points =.9 and =. Round your answer to the thousandths place. f ( ) sin( ) Ans: B Difficulty: Easy Section:. 8. List the points A, B, C, D, and E in order of increasing slope of the tangent line. B, C, E, D, A A, E, D, C, B E, A, D, B, C A, B, C, D, E Ans: B Difficulty: Easy Section:. 9. Use the position function seconds. s t ( ).9t 8 meters to find the velocity at time t. m/sec 9.8 m/sec.8 m/sec.9 m/sec Ans: B Difficulty: Moderate Section:. 0. Use the position function s( t) t meters to find the velocity at time t seconds. m/sec m/sec m/sec m/sec Ans: D Difficulty: Moderate Section:.. Find the average velocity for an object between t = sec and t =. sec if f(t) = 6t + 00t + 0 represents its position in feet.. ft/s 6 ft/s.8 ft/s 6 ft/s Ans: A Difficulty: Moderate Section:. Page 9

3 . Find the average velocity for an object between t = sec and t = 0.9 sec if f(t) = sin(t) + represents its position in feet. (Round to the nearest thousandth.) Ans: C Difficulty: Moderate Section:.. Estimate the slope of the tangent line to the curve at =. 0 Ans: B Difficulty: Easy Section:.. Estimate the slope of the tangent line to the curve at =. Ans: D Difficulty: Easy Section:. Page 9

4 . 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 0 6 C 8 8 m ; The temperature is increasing C per foot at the point feet from the specified point. m 0.6; The temperature is decreasing 0.6 C per foot at the point feet from the specified point. m.; The temperature is decreasing. C per foot at the point feet from the specified point. m 8; The temperature is increasing 8 C per foot at the point feet from the specified point. Ans: C Difficulty: Moderate Section:. 6. 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 0 mph. The speed during the first hours is 60 mph while the speed from 8 to 0 hours is about 0 mph. Between and hours the car was stopped. Difficulty: Moderate Section:.. Compute f() for the function f ( ). 8 8 Ans: B Difficulty: Moderate Section:. Page 9

5 8. Compute f() for the function f( ) Ans: D Difficulty: Moderate Section:. 9. Compute the derivative function f() of f( ). f( ) f( ) ( ) ( ) f( ) f( ) ( ) ( ) Ans: A Difficulty: Moderate Section:. 0. Compute the derivative function f() of f ( ). f( ) f( ) Ans: B Difficulty: Moderate Section:. f( ) f( ) Page 9

6 . Below is a graph of f( ). Sketch a graph of f ( ). Ans: 9+ Difficulty: Moderate Section:. Page 9

7 . Below is a graph of f( ). Sketch a graph of f ( ). Ans: Difficulty: Difficult Section:. Page 96

8 . Below is a graph of f ( ). Sketch a plausible graph of a continuous function f( ). Ans: Answers may vary. Below is one possible answer. Difficulty: Moderate Section:. Page 9

9 . Below is a graph of f ( ). Sketch a plausible graph of a continuous function f( ). Ans: Answers may vary. Below is one possible answer. Difficulty: Difficult Section:. f ( h) f (0). Compute the right-hand derivative D f(0) lim and the left-hand h0 h f ( h) f (0) derivative D f(0) lim. h0 h 8 9 if 0 f( ) 0 9 if 0 D f(0) 0, Df (0) 8 Df (0) 9, Df (0) 9 Df (0) 8, Df (0) 0 D f(0), D f(0) Ans: A Difficulty: Moderate Section:. Page 98

10 6. 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) 0 s(t) (miles) Ans: The velocity is the change in distance traveled divided by the elapsed time. From hour to the average velocity is (0 80)/( ) = 0 mph. Likewise, the velocity between hour and hour is about 0 mph. Difficulty: Easy Section:.. Use the distances f(t) to estimate the velocity at t =.. (Round to decimal places.) t f(t) Ans: C Difficulty: Easy Section:. 8. For + if 0 f( ) a b if 0 find all real numbers a and b such that f (0) eists. a 8, b any real number a, b any real number a, b 0 a, b 0 Ans: D Difficulty: Moderate Section:. Page 99

11 9. Sketch the graph of a function with the following properties: f (0) 0, f (), f (), f (0), f () 0, and f (). y y y Page 00

12 y Ans: B Difficulty: Moderate Section:. 0. Suppose a sprinter reaches the following distances in the given times. Estimate the velocity of the sprinter at the 6 second mark. Round to the nearest integer. t sec f() t ft ft/sec 6 ft/sec 6 ft/sec 8 ft/sec Ans: A Difficulty: Moderate Section:.. Give the units for the derivative function. ct represents the amount of a chemical present, in milligrams, at time t seconds. seconds per milligram milligrams per second seconds per milligram squared milligrams per second squared Ans: C Difficulty: Easy Section:.. ( h) ( h) 0 lim equals f ( a) for some function f( ) and some constant a. h0 h Determine which of the following could be the function f( ) and the constant a. f ( ) and a f ( ) and a Ans: D Difficulty: Moderate Section:. f a ( ) 0 and f a ( ) and Page 0

13 . ( h ) lim equals f ( a) for some function f( ) and some constant a. Determine h0 h which of the following could be the function f( ) and the constant a. f ( ) and a f ( ) and a f ( ) and a f ( ) and a Ans: A Difficulty: Moderate Section:.. Find the derivative of f() = Ans: C Difficulty: Easy Section:.. Differentiate the function. f ( t) 8t 6 t f ( t) t t f ( t) t Ans: C Difficulty: Moderate Section:. / t f() t t t f() t t 6. Find the derivative of f ( ) + +. f( ) + f( ) + Ans: B Difficulty: Easy Section:. f( ) + f ( ) + 0 Page 0

14 . Differentiate the function. f ( s) s s / / / s f() s / 6s / / s s f() s 6 Ans: D Difficulty: Moderate Section:. s s f() s 6 / 6 s 0 f() s / 6s / / 8. Find the derivative of + f( ). 0 f( ) f( ) + 0 f( ) f( ) + Ans: C Difficulty: Moderate Section:. 9. Find the derivative of + + f( ). 9 f( ) + + f( ) Ans: A Difficulty: Moderate Section:. 9 f( ) + f ( ) Differentiate the function. f ( ) 6 6 f ( ) 8 9 / f( ) f ( ) Ans: A Difficulty: Moderate Section:. f ( ) Page 0

15 . Find the third derivative of f ( ) +. f ( ) 0 f ( ) Ans: D Difficulty: Moderate Section:. f ( ) 0 f ( ) 0 +. Find the second derivative of y 6 +. d y d y d y 6+ d d d Ans: B Difficulty: Moderate Section:. d y d. Using the position function s t t t ( ), find the acceleration function. a( t) t a( t) 0t a( t) 0t a( t) 0 t Ans: C Difficulty: Moderate Section:.. Using the position function s( t) t +, find the velocity function. t vt ( ) + t t vt ( ) t t vt ( ) t t vt ( ) t t Ans: B Difficulty: Moderate Section:.. Using the position function s( t) t + t +, find the velocity function. t v( t) t + t t v( t) 9 t + 8 t t Ans: A Difficulty: Moderate Section:. v( t) t + t + t v( t) t t t Page 0

16 6 6. Using the position function st ( ), find the acceleration function. t 9 at ( ) at () at ( ) t t t Ans: D Difficulty: Moderate Section:. 9 at () t. The height of an object at time t is given by velocity at t =. h t t t ( ) Determine the object's Ans: C Difficulty: Easy Section:. 8. The height of an object at time t is given by acceleration at t =. h( t) t + t. Determine the object's 0 9 Ans: B Difficulty: Easy Section:. 9. Find an equation of the line tangent to f ( ) + 9at = 6. g( ) 9 g( ) 9 g( ) g( ) Ans: A Difficulty: Easy Section:. 0. Find an equation of the line tangent to f ( ) + 8 at =. 8 g( ) g( ) + + Ans: D Difficulty: Moderate Section:. + g( ) g( ) + 6 Page 0

17 . Use the graph of f( ) below to sketch the graph of f( ) on the same aes. (Hint: sketch f ( ) first.) y y y Page 06

18 9 8 y y Ans: A Difficulty: Difficult Section:.. Determine the real value(s) of for which the line tangent to horizontal. f ( ) 9 is , 0 = 0 Ans: C Difficulty: Easy Section:.. Determine the real value(s) of for which the line tangent to horizontal. f ( ) 6 8 is =, = = 0, =, = = 0 = 0, = Ans: B Difficulty: Easy Section:.. Determine the value(s) of, if there are any, for which the slope of the tangent line to f ( ) + 0 does not eist. 6., 0 0, The slope eists for all values of. Ans: C Difficulty: Moderate Section:. Page 0

19 . Find the second-degree polynomial (of the form a + b + c) such that f(0) = 0, f '(0) =, and f ''(0) =. Ans: A Difficulty: Moderate Section:. 6. Find a formula for the nth derivative ( n f ) ( ) of f( ). + 0 ( n) n 0 n! f ( ) ( ) n ( + 0) ( n) n n! f ( ) ( ) n ( + 0) Ans: D Difficulty: Difficult Section:. ( n) n 0 n! f ( ) ( ) ( + 0) ( n) n n! f ( ) ( ) ( + 0) n n. Find a function with the given derivative. 8 f ( ) f ( ) 6 f ( ) f ( ) 6 Ans: B Difficulty: Moderate Section:. f ( ) 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 (00). t f() t $00 $000 $00 $0 f (00) ; The rate at which the average monthly salary is increasing each year in 00 is increasing by $ per year. f (00) ; The average monthly salary is increasing by $ per year in 00. f (00) 0; The rate at which the average monthly salary is increasing each year in 00 is increasing by $0 per year. f (00) 0; The average monthly salary is increasing by $0 per year in 00. Ans: A Difficulty: Moderate Section:. Page 08

20 9. Find the derivative of f ( ) 9. / f ( ) + / / f ( ) + + / / f ( ) + / / 0 f ( ) / Ans: B Difficulty: Moderate Section:. 60. Find the derivative of 6 + f( ) (9 ) (9 ) Ans: D Difficulty: Moderate Section: Find the derivative of f( ) (6 ) (6 ) Ans: A Difficulty: Moderate Section:. Page 09

21 6. Find the derivative of the function. 8 / 8 8 / 8 8 / / Ans: B Difficulty: Moderate Section:. 6. Find the derivative of ( ) 8 + f. f ( ) + f ( ) f ( ) f ( ) + 0 Ans: C Difficulty: Moderate Section:. 6. Find the equation of the tangent line to the graph of y = f () at =. + f y y + Ans: A Difficulty: Moderate Section:. 9 y y + 6. Find an equation of the line tangent to h( ) f ( ) g( ) at if f ( ), f ( ), g( ), and g( ). y y y 9 y 9 + Ans: C Difficulty: Moderate Section:. Page 0

22 f( ) 66. Find an equation of the line tangent to h ( ) at if g ( ) f ( ), f ( ), g( ), and g( ). y + y + y y Ans: B Difficulty: Moderate Section:. 6. A small company sold 000 widgets this year at a price of $0 each. If the price increases at rate of $. per year and the quantity sold increases at a rate of 0 widgets per year, at what rate will revenue increase? $./year $0/year $0/year $06./year Ans: B Difficulty: Moderate Section:. 68. Find the derivative of ( + ) f( ). 6 f ( ) ( + ) f ( ) ( + ) Ans: C Difficulty: Moderate Section:. f ( ) ( + ) f ( ) ( + ) Find the derivative of f ( ). f( ) f( ) Ans: D Difficulty: Moderate Section:. 0. Differentiate the function. f t t t ( ) f( ) f( ) t t f() t t t f() t t Ans: C Difficulty: Difficult Section:. t t f() t f() t 9 t 8 t t Page

23 . Find the derivative of f( ). + Ans: B Difficulty: Moderate Section:.. Find the derivative of f( ). + 9 f( ) 0 ( + 9) f( ) 0 ( + 9) Ans: A Difficulty: Moderate Section:.. Differentiate the function. f( ) f( ) f( ) f( ) f ( ) Ans: A Difficulty: Difficult Section:. f( ) f( ) 0 ( + 9) 8 ( + 9) Page

24 . Find an equation of the line tangent to f( ) at =. y = + y = y = + y = + Ans: D Difficulty: Moderate Section:.. Use the position function s t ( ) t 6 meters to find the velocity at t = seconds. 9 m/s 9 m/s 9 m/s 9 m/s Ans: B Difficulty: Moderate Section:. 6. Compute the derivative of h( ) f g( ) at = 8 where f ( 8) 8, g( 8), f ( 8), f (), g( 8) 9, and g(). h( 8) 6 h( 8) h( 8) 6 h( 8) 6 Ans: C Difficulty: Moderate Section:.. Find the derivative where f is an unspecified differentiable function. f(9 ) 6 f (9 ) (6 9 ) f (9 ) Ans: A Difficulty: Moderate Section:. f (6 ) f (6 9 ) 8. Find the derivative where f is an unspecified differentiable function. f f f f f Ans: D Difficulty: Moderate Section:. 9. Find the second derivative of the function. f ( ) 00 f f 00 f( ) (00 ) / 00 f( ) / (00 ) Ans: C Difficulty: Moderate Section:. 00 f( ) (00 ) / 00 f( ) (00 ) / Page

25 80. Find a function g ( ) such that g( ) f ( ). f ( ) + ( ) + g( ) + (6 ) Ans: D Difficulty: Moderate Section:. g( ) + g ( ) + 8. Use the table of values to estimate the derivative of h( ) f g( ) at = f() 6 g() h(6) h(6) h(6) h(6) Ans: A Difficulty: Moderate Section:. 8. Find the derivative of f ( ) sin( ) cos( ). f ( ) cos + 9sin f ( ) cos 9sin f ( ) cos + sin f ( ) cos sin Ans: A Difficulty: Easy Section:.6 8. Find the derivative of f ( ) sin + 9. f ( ) sin cos + 8 f ( ) sin + 8 f ( ) sin cos + 9 f ( ) sin cos + 8 Ans: D Difficulty: Easy Section:.6 8. Find the derivative of 9cos f( ). 8( sin cos ) f( ) 8( sin cos ) f( ) Ans: C Difficulty: Moderate Section:.6 8( sin cos ) f( ) 8( sin cos ) f( ) Page

26 8. Find the derivative of f ( ) sin sec. f( ) sec tan f( ) sec tan Ans: B Difficulty: Moderate Section:.6 sec f( ) tan sec tan f( ) tan 86. Find the derivative of the function. f ( w) w sec 0w f ( w) 0wsec (0 w) tan(0 w) f ( w) wsec (0 w) 0w sec (0 w) tan(0 w) f ( w) wsec (0 w) 0w sec(0 w) f ( w) wsec (0 w) 0w sec (0 w) tan (0 w) Ans: B Difficulty: Moderate Section:.6 8. Find the derivative of the function. Ans: f ( ) cos sin f ( ) 6 cos sin 8 sin sin 8 cos Difficulty: Difficult Section: Find an equation of the line tangent to f ( ) tan at. (Round coefficients to decimal places.) y. 8.0 y. 8.0 y y. +. Ans: C Difficulty: Moderate Section: Find an equation of the line tangent to f ( ) sin at. y ( ) y ( ) y ( ) y ( ) Ans: D Difficulty: Moderate Section:.6 Page

27 90. Find an equation of the line tangent to f ( ) cos at. (Round coefficients to decimal places.) y.0.6 y. +.6 y..6 y Ans: B Difficulty: Moderate Section:.6 9. Use the position function s( t) cos t 9t (Round answer to decimal places.) feet to find the velocity at t = seconds. v() =.8 ft/s v() = 6. ft/s v() = 6. ft/s v() =.8 ft/s Ans: A Difficulty: Moderate Section:.6 9. Use the position function s( t) sin( t) 6 meters to find the velocity at t = seconds. (Round answer to decimal places.) v() = 9.9 m/s v() = 0. m/s v() = 8. m/s v() =.9 m/s Ans: D Difficulty: Moderate Section:.6 9. Use the position function to find the velocity at time t t. 0 Assume units of feet and seconds. sin 9t st ( ), t t 9 v( ) 0 ft/sec v( ) ft/sec 9 v( ) ft/sec v( ) ft/sec Ans: C Difficulty: Moderate Section:.6 9. A weight hanging by a spring from the ceiling vibrates up and down. Its vertical position is given by s( t) sin( t). Find the maimum speed of the weight and its position when it reaches maimum speed. speed =, position = 0 speed =, position = speed = 0, position = 0 speed = 0, position = Ans: B Difficulty: Moderate Section:.6 Page 6

28 sin 9. Given that lim, find 0 sin( t). t lim t0 Ans: C Difficulty: Easy Section: Given that cos cost lim 0, find lim. 0 t0 t 0 6 Ans: A Difficulty: Easy Section:.6 9. Given that sin lim, find 0 9t lim t0 sin( t ) Ans: D Difficulty: Easy Section: Given that sin lim, find 0 tan( t). t lim t0 Ans: B Difficulty: Moderate Section: For f ( ) sin, find f () ( ). cos cos sin sin Ans: D Difficulty: Easy Section: The total charge in an electrical circuit is given by Q( t) sin( t) t 8. The current is dq the rate of change of the charge, it (). Determine the current at t (Round dt answer to decimal places.) i() 0. i().6 i() 9.6 i().0 Ans: B Difficulty: Moderate Section:.6 Page

29 0. Compute the slope of the line tangent to 6 slope = 8 slope = Ans: B Difficulty: Moderate Section:. 0. Find the derivative y ( ) implicitly. y y + y + y at (, ). slope = 8 9 slope = 6 y( ) y + y y( ) y Ans: B Difficulty: Moderate Section:. y y( ) y( ) y + 0 y 0. Find the derivative y ( ) implicitly if y y + y. y y y y y y y + y y Ans: C Difficulty: Moderate Section:. 0. Find the derivative y ( ) implicitly if sin y. y y y + y y + y y y( ) cos y y y( ) cos y Ans: D Difficulty: Moderate Section:. cos y y y( ) y y( ) cos y Page 8

30 0. Find y () implicitly. y + y + y y y y y + 8 y + + y + + y y 8 y + y + y 6 + y y y y y y y y y + 8 y + y Ans: C Difficulty: Difficult Section:. 06. Find an equation of the tangent line at the given point. y 0 at (,) y y y Ans: C Difficulty: Moderate Section:. y 0. Find an equation of the tangent line at the given point. y y at (,) 8 9 Ans: y Difficulty: Moderate Section:. 08. Find the second derivative, y ( ), of y. y y( ) y y y y( ) y y Ans: B Difficulty: Moderate Section:. y( ) y y( ) y y y y y Page 9

31 09. Find the second derivative, y ( ), of + + cos y y. 8 (cos y + )( y) y( ) y sin y ( cos y + 6) y y( ) 6 y+ cos y Ans: D Difficulty: Moderate Section:. ( cos y + ) y y( ) 6 y + sin y ( cos y 6)( y) y( ) 6 y+ sin y 0. Find the location of all horizontal and vertical tangents for y 6. horizontal: none; vertical: (, 0), (, 0) horizontal: (, 0); vertical: (, 0), (, 0) horizontal: (, 0), (, 0); vertical: none horizontal: none; vertical: (, 0) Ans: A Difficulty: Moderate Section:.. Find the location of all horizontal and vertical tangents for y 6 0. horizontal:,,, ; vertical: ( 6, 0) horizontal:,,, ; vertical: (0, 0) horizontal:,,, ; vertical: none horizontal:,,, ; vertical: ( 6, 0) Ans: C Difficulty: Moderate Section:.. 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 ( ) 00, 6, , 0 Rolle's Theorem not satisfied Ans: A Difficulty: Easy Section:.8. Using the Mean Value Theorem, find a value of c that makes the conclusion true for f ( ), in the interval [,]. c.9 One or more hypotheses fail c 0.9 c 0 Ans: C Difficulty: Easy Section:.8 Page 0

32 . Using the Mean Value Theorem, find a value of c that makes the conclusion true for f ( ) cos,, One or more hypotheses fail c 0 Ans: B Difficulty: Easy Section:.8 c c.88. Prove that has eactly one solution. Ans: Let f 8 +. The function f() is continuous and differentiable everywhere. Since f(0) < 0 and f() > 0, f() must have at least one zero. The derivative of f ( ) 8 + is f ( ) +, which is always greater than zero. Therefore f() can only have one zero. Difficulty: Moderate Section:.8 6. Prove that has eactly two solutions. f + 8. The function f () is continuous and differentiable Ans: Let everywhere. Since f ( ) > 0, f (0) < 0 and f () > 0, f () must have at least two zeros. The derivative of f + 8 f ( ) , is which has one zero. Therefore f () can only have two zeros. Difficulty: Difficult Section:.8. Find all functions g such that g( ) f ( ). f ( ) 9 6 g( ) g( ) 8 6 g( ) 6 C, for some constant C 9 8 g( ) C, for some constant C 8 Ans: D Difficulty: Easy Section:.8 8. Find all the functions g( ) such that g( ). 6 g ( ) g( ) c g ( ) 0 g( ) c Ans: D Difficulty: Moderate Section:.8 Page

33 9. Find all the functions g( ) such that g( ) sin. g( ) cos c g( ) cos g( ) cos c g( ) sin c Ans: B Difficulty: Moderate Section:.8 0. Determine if the function f ( ) + is increasing, decreasing, or neither. Increasing Decreasing Neither Ans: A Difficulty: Easy Section:.8. Determine if the function f ( ) + is increasing, decreasing, or neither. Increasing Decreasing Neither Ans: C Difficulty: Easy Section:.8. Eplain 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. f( ),, 6 Ans: The function is not continuous on the specified interval, so the Mean Value Theorem does not apply. Note that f () and f (6), so that f(6) f() ( ). (6) () Also, f( ). ( ) Since f( ) 0 for all in the domain of f, there is no value of c such that f(6) f() f( c). That is, there is no value of c such that f( c). (6) () Difficulty: Moderate Section:.8 Page