Part II Determine whether Rolle's Theorem c

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AP Calculus AB Unit 5 Homework

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Applications of the Derivative - Part II Determine whether Rolle’s Theorem can be applied to the function on the indicated interval. If Rolle’s Theorem can be applied, find all values of c that satisfy the theorem. 1. f (x) = x2 − 4x on the interval [0, 4] 2. f (x) = (x + 4)2 (x − 3) on the interval [−4, 3] 3. f (x) = 4 − |x − 2| on the interval [−3, 7] 4. f (x) = sin x on the interval [0, 2π] 5. f (x) = cos 2x on the interval π3 , 2π 3 Determine whether the Mean Value Theorem can be applied to the function on the indicated interval. If the Mean Value Theorem can be applied, find all values of c that satisfy the theorem. 6. f (x) = x3 − x2 − 2x on [−1, 1] √ 7. f (x) = x − 3 on [3, 7] 8. f (x) = 9.

x+2 on [ 12 , 2] x

h(x) = 2 cos x + cos 2x on [0, π]

Use the graph of the function f (x) pictured below to determine if Rolle’s Theorem or the Mean Value Theorem, whichever is indicated, can be applied or not. Give reasons for your answer.

10. Rolle’s Theorem on [−5, −1] 11. Rolle’s Theorem on [−2, 8] 12. Mean Value Theorem on [−1, 8]

13.

Determine if the Mean Value Theorem holds true for f (x) = −2 + 21 |x − 3| on the interval 0 < c < 5. Give a reason for your answer. If it does hold true, find the guaranteed value of c.

14.

Determine if the Mean Value Theorem holds true for g(x) = 2x + sin2 x on the interval 0 < c < 5. Give a reason for your answer. If it does hold true, find the guaranteed value of c.

15.

For t ≥ 0, the temperature of a cup of coffee in degrees Fahrenheit t minutes after it is poured is modeled by the function F (t) = 68 + 93(0.91)t . Use your calculator to find the value of F 0 (4). Using correct units of measure, explain what this value means in the context of the problem.

16.

Administrators at a hospital believe that the number of beds in use is given by the function t B(t) = 20 sin + 50 10 where t is measured in days. (a) Use your calculator to find the value of B 0 (7). Using correct units of measure, explain what this value means in the context of the problem. (b) For 12 ≤ t ≤ 20, what is the maximum number of beds in use?

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AP Calculus AB - Unit 5 Homework

Unit 5: Applications of Derivatives - Part II

Use the table given below which represents values of a differentiable function g on the interval 0 ≤ x ≤ 6. Be sure to completely justify your reasoning, citing appropriate theorems where applicable. 17. Estimate the value of g 0 (2.5) 18. If one exists, on what interval is there guaranteed to be a value of c such that g(c) = −1? Justify your reasoning. x g(x)

0 −3

2 1

3 5

4 2

6 1

19. If one exists, on what interval is there guaranteed to be a value of c such that g 0 (c) = 0? Justify your reasoning. 20. If one exists, on what interval is there guaranteed to be a value of c such that g 0 (c) = 4? Justify your reasoning.

21.

A particle moves along the x-axis such that its position, for t > 0, is given by the function p(t) = e2t − 5t. (a) What are the values of p0 (2) and p00 (2)? Explain what each value represents. (b) Based on the values found in part (a), what can be concluded about the speed of the particle at t = 2? Give a reason for your answer. (c) On what intervals of t is the particle moving to the left? To the right? Justify your answer. (d) Does the particle ever change direction? Justify your answer.

22.

The position of a particle is given by the function p(t) = (2t − 3)e2−t for t > 0. (a) What is the average velocity from t = 1 to t = 3? (b) Find an equation for v(t), the velocity of the particle. (c) For what values of t will v(t) = 0?

23. The graph of v(t), the velocity of a moving particle, is given below. What conclusions can be made about the movement of the particle along the x-axis and the acceleration, a(t), of the particle for t > 0? Give reasons for your answers.

24.

A test plane flies in a straight line with positive velocity v(t), in miles per minute at time t minutes, where v is a differentiable function of g. Selected values of v(t) for 0 ≤ t ≤ 40 are shown in the table below: t (min) v(t) (miles per min)

0 7.0

5 9.2

10 9.5

15 7.0

20 4.5

25 2.4

30 2.4

35 4.3

40 7.3

(a) Find the average acceleration on the interval 5 ≤ t ≤ 20. Express your answer using correct units of measure. (b) Based on the values in the table, on what intervals is the acceleration of the plane guaranteed to equal zero on the open interval 0 < t < 40? Justify your answer. (c) Does the data represent velocity values of the plane moving away from its point of origin or returning to its point of origin? Give a reason for your answer. t 7t (d) The function f , defined by f (t) = 6 + cos( 10 ) + 3 sin( 10 ), is used to model the velocity of the plane, in miles per minute, for 0 ≤ t ≤ 40. According to this model, what is the acceleration of the plane at t = 23? What does this value indicate about the velocity at t = 23? Justif your answer, indicating units of measure.

Michelle Krummel

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25. If the position of a particle is defined by the function x(t) = t3 − 9t2 + 24t for t > 0, is the speed of the particle increasing or decreasing when t = 2.5? Justify your answer. 26. The function whose graph is pictured below represents the velocity v(t) of a particle moving along the x-axis from time t = 0 to t = 9 seconds. (a) On what intervals is the particle moving to the right? Left? Justify your answer. (b) On what intervals is the particle slowing down? Speeding up? Justify your answer. (c) At what values of t is the particle momentarily stopped and changing direction? Justify your answer. (d) On what interval of time is the acceleration zero? Justify your answer. 27.

A particle moves along the x-axis so that its position at any time t ≥ 0 is tiven by the function p(t) = t3 − 4t2 − 3t + 1, where p is measured in feet and t is measured in seconds. (a) Find the average velocity on the interval t = 1 to t = 2 seconds. Give your answer using correct units. (b) On what intervals of time is the particle moving to the left? Justify your answer. (c) Using appropriate units, find the value of p0 (3) and p00 (3). Based on these values, describe the motion of the particle at t = 3 seconds. Give a reason for your answer. (d) What is the maximum velocity on the interval from t = 0 to t = 3 seconds? Show the analysis that leads to your conclusion. (e) Find the total distance the particle moves on the interval [1, 5].

28.

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A particle moves along a line so that at time t, where 0 ≤ t ≤ π, its position is given by s(t) = −4 cos t− t2 +10. What is the velocity of the particle when its acceleration is zero?

29. The graph below represents the position p(t) of a particle that is moving along the x-axis, where p(t) is measured in centimeters and t is measured in seconds. (a) For what intervals of time is the particle moving to the right? Justify your answer. (b) For what intervals of time is the particle moving to the left? Justify your answer. (c) Express the velocity v(t) as a piecewise-defined function on the interval (0, 6). (d) At what values of t is the velocity undefined on the interval (1, 6)? Graphically justify your answer. (e) Find the average velocity of the particle on the interval [1, 6]. 30.

A particle moves along the x-axis so that at any time 0 ≤ t ≤ 5, the velocity, in meters per second, is given by the function v(t) = (t − 2)2 cos 2t. (a) Use your calculator to determine how many times on the interval 0 ≤ t ≤ 5 the particle changes direction. Give a reason for your answer. (b) Using appropriate units, find the value of v 0 (2). Describe the motion of the particle at this time. Justify your answer. (c) Using appropriate units, what is the average acceleration between t = 1 and t = 3.5 seconds?

Michelle Krummel

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31. Jeff leaves his house riding his bicycle toward school. His velocity v(t), measured in feet per minute, on the interval 0 ≤ t ≤ 15, for t minutes, is shown in the graph below. (a) Find the value of v 0 (4). Explain, using appropriate units, what this value represents. (b) On the interval 0 ≤ t ≤ 5, is there any interval of time at which a(t) = 0? Explain how you know. (c) On the interval 0 ≤ t ≤ 5, does Rolle’s Theorem guarantee that there will be a value of t such that a(t) = 0? Justify your answer. (d) At some point, Jeff realizes that he forgot something at home and has to turn around. After how many minutes does he turn around? Give a reason for your answer. 32.

A particle moves along the x-axis so that its velocity at time t is given by 2 t v(t) = −(t + 1) sin 2 (a) Find the acceleration of the particle at t = 2. Is the speed of the particle increasing at t = 2? Explain why or why not. (b) Find all times in the open interval 0 < t < 3 when the particle changes direction. Justify your answer.

33.

The graph of the velocity v(t), in feet per second, of a car traveling on a straight road, for 0 ≤ t ≤ 50 is shown below. A table of values for v(t), at 5 second intervals of time, is also shown.

(a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer. (b) Find the average acceleration of the car over the interval 0 ≤ t ≤ 50. Indicate units of measure. (c) Find one approximation for the acceleration of the car at t = 40. Show the computations you used to arrive at your answer. Include units of measure. (d) Is the speed of the car increasing or decreasing at t = 40? Give a reason for your answer.

Michelle Krummel

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AP Calculus AB - Unit 5 Homework 34.


Two runners, A and B, run on a straight racetrack for 0 ≤ t ≤ 10 seconds. The graph below, which consists of two line seqments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per second, 24t . of Runner B is given by the function v defined by v(t) = 2t+3

(a) Find the velocity of Runner A and the velocity of Runner B at t = 2 seconds. Indicate units of measure. (b) Find the acceleration of Runner A and the acceleration of Runner B at time t = 2 seconds. Indicate units of measure.

35. An object moves along the x-axis with initial position x(0) = 2. The velocity of the object at time t ≥ 0 is given by the function v(t) = sin( π3 t). (a) What is the acceleration of the object at time t = 4? (b) Consider the following two statements: Statement I: For 3 < t < 4.5, the velocity of the object is decreasing. Statement II: For 3 < t < 4.5, the speed of the object is decreasing. Are either or both of these statements correct? For each statement, provide a reason why it is correct or incorrect.

Michelle Krummel

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