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© 2012 www.mastermathmentor.com - 16 - Stu Schwartz
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.
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© 2012 www.mastermathmentor.com - 17 - Stu Schwartz
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.
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© 2012 www.mastermathmentor.com - 18 - Stu Schwartz
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.
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© 2012 www.mastermathmentor.com - 19 - Stu Schwartz
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.
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© 2012 www.mastermathmentor.com - 20 - Stu Schwartz
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.
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© 2012 www.mastermathmentor.com - 21 - Stu Schwartz
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 ?
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© 2012 www.mastermathmentor.com - 22 - Stu Schwartz
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.
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© 2012 www.mastermathmentor.com - 23 - Stu Schwartz
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
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