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16. AB Calculus – Step-by-Step Name ________________________________

x 2 4 6 8

f x( ) 38 12 9 18

! f x( ) 0 9 -5 12

g x( ) 6 8 2 4

! g x( ) -3 -4 6 3 The functions f and g are continuous and differentiable for all values of x. The continuous function h is given by

h x( ) = f g x( )( ) ! x2. The table above gives values of

f x( ), ! f x( ), g x( ), and ! g x( ) at selected values of x.

a. Explain why there must be a value t for 2 < t < 8 such that

h t( ) = 0.

b. Show that the lines tangent to h at x = 2 and x = 8 are parallel.

c. Show that there must be a value of x, 4 < x < 6 such that

! h x( ) = 0.


17. AB Calculus – Step-by-Step (Calculator allowed) Name ________________________________ Let f be the function given by

f x( ) = 4sin x + 4cos x . As shown in the figure to the right, the graph crosses the y-axis at point P and the x-axis at point Q.

a. Write an equation for the horizontal tangent line to

f x( ) . Justify your answer.

b. Write an equation for the line tangent to point Q.

c. Use the line found in part b) to approximate

f 2.3( ).

d. Find the x-coordinate of the point on the graph of f that satisfies the Mean-Value Theorem between points P and Q.


18. AB Calculus – Step-by-Step Name ________________________________

x 0 1 2 3 4 5 6 7

f x( ) 3 5 13 33 71 133 225 353

! f x( ) 1 4 13 28 48 76 109 148 In the chart above, selected values of x are given along with the values of the differentiable function

f x( ) as well as ! f x( ) .

a. Find the value of x closest to the result of the Mean-Value Theorem for f on [0, 7]. Show your reasoning.

b. If

f!1 is the inverse function of f , find the derivative of

f!1 at x = 5. That is, find

f!15( )[ ]" .

c. If the table above is modeled using

f x( ) = x3 + x + 3, show that the derivative of

f!1 at x = 5 gives the

same result as your answer b) above.

d. Write an equation for the line tangent to

f!1x( ) at x = 5.


19. AB Calculus – Step-by-Step Name ________________________________ Jen is running on a treadmill. The number of calories C she burns over t minutes is a function of the number of minutes she has been on the treadmill

T t( ) and the angle of the ramp

R t( ) . The angle of the ramp is a number r from 1 to 10 where r = 1 is flat and

r = 10 is very steep. Thus:

C = T t( ) + R t( ) where

T t( ) = 8t and

R t( ) =r2t

2.

a. If she runs for 20 minutes at ramp angle 2, how many calories does she burn?

b. If a is a constant and the ramp angle is 2, find the average rate of calories burned in calories/min from

t = 0 minutes to t = a minutes.

c. At t = 10 minutes, find the instantaneous rate of change of calories burned when the ramp angle is 5.

d. At t = 10 minutes, the treadmill is at ramp angle 6 and the angle number is changing at

1

2minute . Find

the instantaneous rate of change of the calories burned in calories/min.

e. At t = 10 minutes, the treadmill is at ramp angle 6. If the instantaneous rate of change of the calories burned is not changing at that moment, find the rate of change of the angle number of the ramp.


20. AB Calculus – Step-by-Step Name ________________________________ A gravel plant crushes gravel into sand. The sand falls from a spout and forms a right circular cone whose radius is always twice its height. The radius of the base is increasing at the rate of 6 inches per minute. The

volume of a cone with radius r and height h is given by

V =1

3!r

2h .

a. Find the rates in which the circumference of the base and the area of the base are changing when the

radius is 8 inches. Specify units. b. Find the rate at which the volume of the sand cone is changing when its height is 5 feet. Specify units.

c. When the height of the sand cone reaches 5 feet, the rate at which its volume changes remains constant.

A suction device then switches on, takes the sand away and puts it on train cars. The rate at which the

sand is suctioned off the cone is given by

S t( ) = 4800! t23

in

3

min. Find the time t in minutes since the

suction device started when the volume of the sand pile is not changing.


21. AB Calculus – Step-by-Step (Calculator allowed) Name ________________________________ A particle moves along a series of curves starting at the origin and moving so that its x-coordinate increases at the rate of 2 units per second.

a. The particle moves along the curve

y = ex!1. How does the

area of right triangle ABC change when x = 2?

b. The particle moves along the curve

y = sin x . At what value of x,

0 < x < ! , will the area of the right triangle ABC not be changing?

c. Two particles move along the curves

y = ex!1 and y = !sin x . For

both particles, the x-coordinate increases at the rate of 2 units per

second. How does the area of triangle ABC change when

x =!

2 ?


22. AB Calculus – Step-by-Step Name ________________________________ A particle moves along the x-axis so that at any time t its position is given by

x t( ) = !t " sin2!t .

a. Find the velocity at time t.

b. Find the acceleration at time t.

c. Determine if the particle is speeding up or slowing down at time

t =2

3. Justify your answer.

d. What are all values of t, 0 ≤ t ≤ 1, for which the particle is at rest? e. How far does the particle travel between the times it is at rest for 0 ≤ t ≤ 1? Justify your answer.


23. AB Calculus – Step-by-Step Name ________________________________ A railroad engine is being positioned in a train yard over straight track. Its velocity is shown in the graph below in 5-second intervals as well as in a table of values. The graph is linear between t = 0 and t = 15 and has horizontal tangent lines at t = 25, t = 35, t =40, and t = 50. At t = 0, the engine is in front of a control tower.

a. At what values of t does the engine have no acceleration? b. Write an expression for the speed of the engine from 0 ≤ t ≤ 15.

c. Give an approximation for the acceleration for the engine at t = 30. Specify units. d. For what values of t is the engine speeding up? Explain your reasoning. e. At what value of t is the engine the farthest from the control tower? Explain your answer.

t (seconds)

v t( ) ft per second 0 12 5 6

10 0 15 -6 20 -10 25 -12 30 -9 35 -2 40 -12 45 0 50 4 55 0 60 -2

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