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AP Calculus BC
Additional Practice 5.3 – 5.4
Name:__________________________
x
 f (t )dt , where f is the function whose graph is shown.
1. Let g(x) =
0
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a) Evaluate g(0), g(1), g(2) and g(6).
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b) On what intervals is g increasing?
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c) Where does g have a maximum value? What is the maximum value?
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d) Where does g have a minimum value? What is the minimum value?
e) Find the instantaneous rate of change of g with respect to x at x = 2.
f) Find the equation for the line tangent to the graph of g at x = 2.
x
2. If F(x)=
 g (t )dt , and g(x) is shown to the right, answer the following.
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0
a) Identify all critical number of F(x).
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b) On what interval(s) is F(x) decreasing?
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c) On what interval(s) is F(x) concave up?
d) Is F(3) positive or negative? Explain.
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x
3. Suppose that 5x3 + 40 =
 f (t )dt .
What is f(x)?
c
4. Given:

 f ( x) dx  
2

2

f ( x) dx  


f ( x) dx  

0

2
3
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2
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4
2

 f ( x) dx  2
0
2


f ( x) dx  

2
Give exact (non-decimal) answers for the following:

0

A.
f ( x ) dx
B.
2 
0


C.
  2 f ( x)  2  dx
D.
f ( x) dx
F.
f ( x) dx
H.
2
 2
0
  f ( x)  1  dx
 3  f ( x) dx
0
2
I.
2



f ( x) dx
2 
2
G.

2
0

2

0
E.
 2 f ( x) dx
2
f ( x) dx
J.
  1  f ( x)  dx

f’(x)
From your 5.3 review on antiderivatives, we know that given f’(x),
we can find f(x). Complete table to the right.
f(x)
3x-2
5  tan x   sec2 x
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General formula for Antiderivatives (polynomial/exponents)
1
If f ‘(x) = axn, then f(x) =
3
B)
5
x3
x2
sin(3x)
I. Find the general antiderivatives of the following functions.
A) 5x3
x
43
x
3
C)
D)
2
x 2
4
General formula for Antiderivatives (trig functions)……using sine as an example, but true for all trig functions
If f ‘(x) = a sin kx, then f(x) = ___________________
II.
Find the general antiderivatives of the following functions.
A) 3sec2 (3x)
B) 4cos x – cos (4x)
C) 4csc (4x) cot (4x)
Recall: Fundmental Theorem of Calculus If f is continuous on [a,b], and F is any antiderivative
b
of f on [a,b], then
 f ( x)dx  F (b)  F (a)
a
III. Evaluate the integral.
16
A)
 x 4 dx
0
1
9
B)
 2
  x
4
2

 x dx
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Fundamental Theorem of Calculus: If f is continuous on [a,b], then
g ( x)

F
f (t )dt , then F '  f ( g ( x)) g '( x)
(a is a constant)
a
g ( x)
and
F

f (t )dt , then F '  f ( g ( x)) g '( x)  f (h( x)) h '( x)
h( x)
I: Find f '( x ) .
2
x
A)
C)
f ( x)   3t 2  1 dt
B)
f ( x )   cos(t )dt
2
x
x3
x2
f ( x)   sin(t )dt
D)
x2
f ( x)   2t 3  7 dt
4
II: Below the x -axis.
Calculate the area of the region between f(x) = 0 and g(x) = x 2 – 4x
III: Above and below the x-axis
Calculate the area between
y  1  x 2 and the x -axis on the interval 0,3 .
IV: Area bounded on top by different functions
Calculate the area shown bounded by the x -axis
and the functions y = x3 and y =
x.
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V: The graph of a continuous function f with domain [0, 8] is shown.
x
Let h be the function defined by
h( x)   f (t ) dt .
1
a) Find h(1).
b) Is h(0) positive or negative? Justify your answer.
c) Find the value of x for which h(x) is a local maximum.
d) Find the value of x for which h(x) is a minimum.
e) Find the x-coordinates of all points of inflection of the graph of h(x).
VI: Given f is the differentiable function whose graph is shown in the given figure. The position at time
t (sec) of a
t
particle moving along a coordinate axis is s 
 f ( x) dx meters. Use the graph to answer the questions.
0
y
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a) What is the particle’s velocity at
t 5?
(3,3)
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b) Is the acceleration of the particle at
or negative?
(1,1)
t  5 positive
x
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c) What is the particle’s position at
t 3?
d) At what time during the first 9 seconds does
s have its largest value?
e) Approximately when is the acceleration zero?
f) When is the particle moving toward the origin? Away from the origin?
g) On which side of the origin does the particle lie at
t 9?
(5,2)
(2,2)
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