Calc AB 2011-2012
AP Review Quiz
Round 1
1. Given that f ( x ) = cos ( 2x ) write two different limits that represent f ′ (π 3) . Find
the value of each limit.
2. The table below gives values of f ( x ) and its continuous derivative, f ′ ( x ) , over the
interval −2 ≤ x ≤ 5 . Show all set-ups leading to your conclusions.
-2
0
1
3
5
x
f ( x)
-4
1
3
4
6
f ′( x)
6
2
-1
2
4
a. Approximate f ′ ( −1) .
b. Find the average value of f ′ ( x ) on the interval −2 ≤ x ≤ 3 .
c. Approximate f ( 0.9 ) .
d. Does the equation f ′ ( x ) = 1 have any solutions on the interval 1 ≤ x ≤ 5 ? Explain.
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3. The graphs below represent f ( x ) , f ′ ( x ) , and f ′′ ( x ) . Determine which is which and
write it on the graph.
4. A particle travels along the y-axis such that y = 8 when t = 5 and v ( t ) , its velocity at
( )
time t, is given by et cos t 2 . Write a function that gives the position, y ( t ) , of the
particle at time t.
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Calc AB 2011-2012
AP Review Quiz
Round 2
1. Let R be the region bounded by the curves y = 3x + 1 and y = x + 1 .
a. Find the volume of the solid formed by rotating the region about the line x = −1 .
b. Find the volume of the solid formed having base R and cross sections perpendicular
to the x-axis that are semicircles.
2. The table below contains values for the differentiable function g ( x ) .
2
5
7
10
14
x
g( x)
6
4
5
7
10
Approximate the average value of g ( x ) on the interval 2 ≤ x ≤ 14 using a trapezoidal
approximation with four trapezoids.
3. Approximate the distance traveled on the interval 1 ≤ t ≤ 3 by a particle moving along
the x-axis with velocity v ( t ) = 3 t 3 − 3t + 1 . Show all work/set up.
x2
dx , completely rewrite the integral using the
4. Given the definite integral ∫
0 1− x
substitution x = sin (θ ) . Do NOT evaluate.
1
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5. A water tank has the shape of an inverted circular cone (point facing down) with base
radius 2 meters and height 4 meters. If water is pumped into the tank at a rate of
2 m 3 min , find the rate at which the water level is rising when the water is 3 meters
deep.
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Calc AB 2011-2012
AP Review Quiz
Round 3
( )
1. Given that f ′ ( x ) = ( t − 3) sin t 2 with f ( 8 ) = 4 , find f (1) to three decimal places.
2. Given that f ′ ( t ) = 6t 2 + 1 , f ( 0 ) = 3 , and g ( t ) = f −1 ( t ) , find g′ ( −3) to three decimal
places. (That’s three decimal places based on storing intermediate values!)
3. The table below shows values of a decreasing, twice-differentiable function f ( x ) and
the function g ( x ) . Use the table to answer the questions.
2
4
6
8
10
x
10
6
5
3
0
f ( x)
f ′( x)
g( x)
g′ ( x )
-1
-3
-2
-4
-1
6
8
8
4
2
3
5
4
1
3
a. Write the equation of the line tangent to the function h ( x ) = f ( x ) ⋅ g ( x ) − 2 at x = 4 .
b. Let r ( x ) = f −1 ( x ) . Find the equation of the line tangent to r ( x ) at x = 6 .
c. Find the slope of the curve m ( x ) = ( f  g ) ( x ) at x = 6 .
d. True or false: f ′′ ( x ) > 0 for all x, 2 ≤ x ≤ 10 . Justify.
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(
)
4. Given that f ( x ) = ln cos ( x ) + e x with 0 ≤ x ≤ 4 , state the intervals on which f ( x )
is concave up. All intervals should be correct to three decimal places.
5. The average rate of change of the function f ( x ) on the interval 2 ≤ x ≤ a is given by
the formula
5a 2 − 20
. Find f ′ ( 2 ) .
a−2
6. Given the function f ( x ) = ln ( sin ( 2x ) + 2 ) . If the MVT applies, find any value of c
guaranteed on the interval 0 ≤ x ≤ 3 . Show all work. Answer to three decimal
places, of course.
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Calc AB 2011-2012
AP Review Quiz
1. Evaluate each of the following.
2
a. ∫ ( 3x + 4 ) e3x +8 x−2 dx
c.
⎛ 1
⎞
∫ ⎜⎝ 2x + cos (π x ) − sec ( x + 3)⎟⎠ dx
2
Round 4
b.
∫3
d.
∫ 1+ 9x
2 x+7
dx
x
4
dx
2. A population is growing at a rate of 400e4t /5 people per year. In 1990 (t = 0), there
were 3000 people. How many people are predicted by the model in the year 2012?
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2
dy ( x + 5 )
=
3. Given
with y (1) = 2 , find an explicit formula for y.
dx
y
dW
W ⎞
⎛
= 8W ⎜ 1−
, with W = 300 when t = 6 . Write the equation the
⎝ 400 ⎟⎠
dt
line tangent to W ( t ) at t = 6 and use it to approximate W ( 6.2 ) .
4. Given that
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Calc AB 2011-2012
AP Review Quiz
Round 5
⎧⎪
3x 2 − 2,
x ≤1
1. Given the function f ( x ) = ⎨
⎪⎩ 3x − 3sin (π x ) − 2 x > 1
a. Show that f ( x ) is continuous at x = 1 by using the definition of continuity.
b. Show that f ( x ) is not differentiable at x = 1 .
2. Find a formula for the slope of the curve defined by x 2 y + y 2 x = 0 at the point ( x, y ) .
3x + 2 − 5 x
3. Find the horizontal asymptotes of the function y =
.
6x + 7 + 8 x
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4. The function g is defined by g ( x ) =
f ( x ) is shown below.
x
x
+ ∫ f ( t ) dt , −3 ≤ x ≤ 7 , where the graph of
2 1
a. Write the equation of the line tangent to g ( x ) at x = 5 .
b. State the intervals on which g ( x ) is increasing. Show all work (possibly including a
little Algebra I-style work).
c. State the intervals on which g ( x ) is concave down. Justify your response.
d. True or false: g ( x ) > 0 for all x, −3 ≤ x ≤ 7 . Justify.
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Calc AB 2011-2012
AP Review Quiz
Round 6
1. The function f ′ ( x ) , the derivative of f ( x ) , is defined on −4 ≤ x ≤ 8.75 and has
horizontal tangent lines at x = −3 , x = 0.9 , x = 4.7 , and x = 8 .
a. For what values of x, −4 < x < 8.75 , does f ( x ) have a relative minimum?
Justify.
b. State the absolute minimum of the function g ( x ) , where g ( x ) = 2 − 3 f ′ ( x ) .
Make a convincing argument in support of your response.
c. Given that f ( −1) = 0 , true or false: f ( −4 ) < 0 . Justify.
2. Give an example of a derivative, in factored form, of a function that has a critical
point at which it does not have a relative extrema.
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3. Given that the position of a particle moving along the line x = −3 is given by
t2
, for 0 ≤ t ≤ 2 . Find the maximum speed of the particle on
y ( t ) = cos ( 3t ) −
2 + 3t 2
the interval. Show work/set up justifying your response.
4. Let R be the region bounded by the curves f ( x ) = −x 2 + 5x and g ( x ) = x ⋅sin ( x ) .
a. Find the volume of the solid whose base is the region R and whose cross sections
perpendicular to the x-axis are squares.
b. Find the area of the region R.
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Calc AB 2011-2012
AP Review Quiz
Round 7
1. The function f is twice-differentiable and increasing. Some values of the function and
its derivative at selected x-values are shown in the table below.
1
3
6
7
9
x
f ( x)
-3
1
2
5
7
f ′( x)
4
3
1
2
5
a. Approximate f ′ ( 8 ) . Show the calculation that leads to your approximation.
b. Use a trapezoidal sum with four partitions to approximate the average value of
f ( x ) on the interval 1 ≤ x ≤ 9 . Show all work.
c. Is the data in the table consistent with the claim that f ( x ) has a point of
inflection on on the interval 1 ≤ x ≤ 9 ? Why or why not?
d. Let g ( x ) = f −1 ( x ) . Write the equation of the line tangent to g ( x ) at x = 7 .
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2. Let f ( x ) = ∫ v ( t ) dt and g ( x ) = ∫ v ( t ) dt . Graphically, what is the difference
x
x
3
8
between f ( x ) and g ( x ) ? Be specific.
3. Given that f ( x ) is even and
∫ f ( x ) dx = 4 , find ∫ f ( 2x − 3) dx .
5
1
1
−1
4. The function g ( x ) is an odd, differentiable function with g ( −3) = 5 and
g′ ( −3) = −2 . Approximate g ( 3.1) with a tangent line approximation at x = 3 .
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Calc AB 2011-2012
AP Review Quiz
Round 8
1. Given that f ′ ( x ) = ( 3x + 1) ( 2x − 5 ) , state all intervals on which f ( x ) is increasing
and concave down. Show work.
2
2. Evaluate each limit.
csc ( x + h ) − csc ( x )
a. lim
h→0
h
c.
tan ( x ) − 1
4 x−π 4
lim
x→π
1− cos 2 ( 2x )
e. lim
x→0
x2
3
b. lim
( x + h )7 − x 7
h→0
d.
f.
lim
x→−∞
h
−3x
7x 2 − 3x + 2
e5 x+2 − e17
x→2
x−3
lim
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3. Find
4
dy
for y = 5e2 x + ∫
3t 2 + 2 dt .
sin
x
(
)
dx
4. Find the average value of f ( x ) = ∫
x
0
t + 1 dt on the interval 1 ≤ x ≤ 4 .
5. Define a critical number.
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