All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

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1 AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. What is the definition of a derivative?. What is the alternative definition of a derivative?. Identif or sketch each of the quantities on the figure to the right. a) f () and f (4) b) f (4) f () c) f 4 f f 4 d) Insert the proper inequalit smbol (< or >) between the given quantities. i) f 4 f f 4 f 4 4 ii) f 4 f 4 f ' 4. The figure to the right shows the graph of g '. a) g '0 The graph of g ' b) g ' 8 c) What can ou conclude about the graph of g knowing that g '? 7 d) What can ou conclude about the graph of g knowing that g '4? e) Is g 6 g 4 positive or negative? Eplain. f) Is it possible to find g () from the graph? Eplain. 5. Assume that f ' c. Find f ' c given the following conditions: a) f is an odd function. b) f is an even function.

2 6. Use the graph of f at the right to answer each question. a) Between which two consecutive points is the average rate of change of the function greatest? b) Is the average rate of change of the function between A and B greater than or less than the instantaneous rate of change of B? C c) Sketch a tangent line to the graph between the points B and C such that the slope of the tangent line is the same as the average rate of change of the function between B and C. d) Give an sets of consecutive points for which the average rates of change of the function are approimatel equal. A B D E 7. Sketch a function whose derivative is ALWAYS negative. 8. Sketch a function whose derivative is ALWAYS positive. 9. Use the slope field below to sketch at least two possible graphs of the function f. The equation for this slope field is d cos d =. What do ou think the original function was? 0. Use the grid to the right. a) Draw the slope field for d. d b) If the point (0, ) is on the graph of, draw the graph of.. Complete the following questions from the tetbook: pages 05 08: # 8, 7,, 6,, 44

3 AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. If f ( ) = + + for all values of, then the value of the derivative f '( ) at = is A) B) 0 C) D) E) noneistent. The graph of f is shown in the figure below. f () Which of the following could be the graph of f '? A B C D E. The graph of the function f shown in the figure below has a vertical tangent at the point (, 0) and horizontal tangents at the points (, ) and (, ). For what values of, < < 4, is f not differentiable? A) 0 onl B) 0 and onl C) and onl D) 0,, and onl E) 0,,, and

4 ( ) ( ) f f 4. If f is a function such that lim = 0, which of the following must be true? A) The limit of f () as approaches does not eist. B) f is not defined at =. C) The derivative of f at = is 0. D) f is continuous at = 0. E) f () = 0 ( ) ( ) f + h f 5. Let f be a function such that lim = 5. Which of the following must be true? h 0 h I. f is continuous at =. II. f is differentiable at =. III. The derivative of f is continuous at =. A) I onl B) II onl C) I and II onl D) I and III onl E) II and III onl 6. The graph of the derivative of f is shown below. f ' ( ) Which of the following could be the graph of f? A) B) C) D) E)

5 7. Let f be a function that is differentiable on the open interval (0, 0). If f () = 5, and f (5) = 5, and f (9) = 5, which of the following must be true? I. f has at least zeros. II. The graph of f has at least one horizontal tangent line. III. For some c, < c < 5, f (c) =. A) none B) I and II onl C) I onl D) I and III onl E) I, II, and III 8. The function f is defined on the closed interval [0, 8]. The graph of its derivative f ' is shown below The point (, 5) is on the graph of f (). An equation of the tangent line to the graph of f at (, 5) is A) = B) 5 = C) 5 ( ) = D) + 5 = ( ) E) + 5 = ( + ) 9. The graph of f is shown below. a b Which of the following could be the graph of the derivative of f? A. B. C. a b a b a b D. E. a b a b 0. Complete the following questions from the tetbook: pages 4 5: #5 0, 6 (calculator), 9. Complete the Worksheet: Graphs of f, f ', f '', and F

6 AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. Solve for a and b in order for f () to be both continuous and differentiable at =. ; f a b ;. Solve for a and b in order for g () to be both continuous and differentiable at = 0. a b ; 0 g ; 0. For a d, find f ' given the following information: g g' h h' 4 a) f g h b) f 4 h c) f gh g d) f h

7 4 4. Find all points where the graph of 5 0 has a horizontal tangent line. 5. Find the equation of the tangent line to the graph of f at the point (, ). 6. Let f ( ) ( 4 )( 4 5) = +. Find f ' without using the product rule first, then using the product rule. 7. An equation of the line tangent to the graph of at the point (, 5) is A) = 8 B) + = 8 C) = 64 D) + = 66 E) + = 8. When = 8, the rate at which is increasing is k times the rate at which is increasing. What is the value of k? A) B) 4 C) 6 D) 8 E) 9. Let f. If the rate of change of f at = c is twice its rate of change at =, then c = A) 4 B) C) 4 D) E)

8 0. What is the instantaneous rate of change at = of the function f given b f? A) B) 6 C) D) E) 6 4. Which of the following is an equation of the tangent line to f at the point where f '? A) = 8 5 B) = + 7 C) = +.76 D) =. E) =.46. At what point on the graph of is the tangent line parallel to the line 4 =? A), B), C), D), E), 8 4. If u, v, and w are nonzero differentiable functions, then the derivative of uv w is uv ' u ' v A) w' u ' v ' w uvw' B) w uvw' uv ' w u ' vw C) w u ' vw uv ' w uvw' D) w uv ' w u ' vw uvw' E) w 4. Let f be a differentiable function such that f () = and f ' ( ) = 5. If the tangent line to the graph of f at = is used to find an approimation to a zero of f, that approimation is A) 0.4 B) 0.5 C).6 D).4 E) Complete the following questions from the tetbook: pages 4 6 # 9 (odd), 4, 7, 8, 0,,, 6, 7, 0, 7 40, 4, 47, 56

9 AP Calculus.4 Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. Bole s Law states that if the temperature of a gas remains constant, its pressure is inversel proportional to its volume. Show that the rate of change of the pressure is inversel proportional to the square of the volume.. Once again tring to blow up earth because it interferes with his view of Venus, Marvin the Martian lands on the moon. Bugs Bunn, as alwas, interferes with his plan. Chasing Bugs, Marvin fires a warning shot straight up into the air with his Acme Disintegration Pistol. The height (in feet) after t seconds of the shot is given b s t =.66t + 5t+. ( ) a) Find the velocit and acceleration as functions of time. (What is the meaning of the acceleration function?) b) How long will it take for Marvin s shot to reach its maimum height? c) What is the maimum height for Marvin s shot?. A bug begins to crawl up a vertical wire at time t = 0. The velocit, v, of the bug at time t, 0 t 8 is given b the function whose graph is shown below. v (t) t At what value of t does the bug change direction? A) B) 4 C) 6 D) 7 E) 8 4. If the position of a particle on the ais at time t is 5t, then the average velocit of the particle for 0 t is A) 45 B) 0 C) 5 D) 0 E) 5 5. A particle moves along the ais so that its position at time t is given b t t 6t 5 the particle zero?. For what value of t is the velocit of A) B) C) D) 4 E) 5

10 6. Rocket A has a positive velocit v (t) after being launched upward from an initial height of 0 feet at time t = 0 seconds. The velocit of the rocket is recorded for selected values of t over the interval 0 t 80 seconds as shown in the table below. T (sec) v (t) (ft/sec) a) Find the average acceleration of Rocket A over the time interval 0 t 80 seconds. Indicate units of measure. b) Using the data, find an estimate for v ' 5. Indicate units of measure. 7. A particle moves along the ais so that its position at an time t > 0 is given b the function t t t measured in feet and t is measured in seconds. a) Find the displacement during the first seconds., where is b) Find the average velocit during the first seconds. c) Find the instantaneous velocit when t = seconds. d) Find the acceleration of the particle when t = seconds. e) When is the particle moving left? f) At what value or values of t does the particle change directions? g) When is the particle speeding up? 8. The cost involved in maintaining annual inventor for a certain manufacturer is given b C Where is the number of items stored. Find the marginal cost of storing the 5 st item.,008, Complete the following questions from the tetbook: pages 5 40 # 5, 8, 8, 9,, 5, 7, 7

11 AP Calculus.5 Worksheet All work must be shown in this course for full credit. Unsupported answers ma receive NO credit.. A spring is bobbing up and down so that its position at an time t > 0 is given b st 4sin a) What is the initial position of the spring? t. b) Which wa is the particle moving to start? Justif our response. c) At t = 5, is the spring moving up or down? Justif our response. 4 d) Is the spring speeding up or slowing down at t = 5? Justif our response. 4. If sec, find d d. 4. If f sin, find f ', f '', f ''', and f. What do ou think the function f 00 is? 4. ( ) sin + h sin lim h 0 h = A) 0 B) C) sin D) cos E) noneistent 5. An equation of the line tangent to the graph of = + cos at the point (0, ) is A) = + B) = + C) = D) = E) = 0 6. If = tan cot, then d d = A) sec csc B) sec csc C) sec + csc D) sec csc E) sec csc

12 tan 7. If f ( ) =, then '( ) f π = 4 A) B) C) + π D) π E) π 8. [Calculator] A particle moves along a line so that at time t, 0 t What is the velocit of the particle when its acceleration is zero? t, its position is given b st 4cost 0. A) 5.9 B) 0.74 C). D).55 E) If f ( ) = sin, then '( ) f π = A) B) C) D) E) 0. [Calculator] A bod is moving in simple harmonic motion (up/down) with position st cos a) Find the velocit, v (t), of the object at an time t. t, where 0 < t < π. b) Find the zeros of v (t). c) Find the acceleration, a (t), of the object at an time t. d) Find the zeros of a (t). e) When is the object stopped? Justif our response. f) When does the object change direction? Justif our response. g) When does the object speed up? Justif our response.. Complete the following questions from the tetbook: pages #, 4, 5, 9, 6,,, 7, 0,, 6, 7, 40, 4 ou should also begin reviewing for our chapter test..5: Review: # 4, 4, 5, 57, 59 6, 7, 7 This isn t due until the da of our eam.