AP Calculus
FTC - Connecting F, F’, F”
1)
Mathematician:
11/19/13
This is an AP Exam question from 1999.
x
The graph of f, consisting of three line segments, is given below. Let g (x )  f (t )dt .

A)
1
Compute g(4).
4
B)
Compute g(-2).
3
2
C)
Find the instantaneous rate of change of g,
With respect to x, at x = 1.
1
-2
-1
1
2
3
-1
2)
-2
D)
Find the absolute minimum of g on the closed
interval [-2, 4]. Justify your answer.
E)
The second derivative of g is not defined at x = 1 and x = 2. How many of these
values are x-coordinates of points of inflection of the graph of g? Justify your
answer.
Find the linearization of f (x ) at x = 2 when f (x )  4 
x 1
 sec(t )dt .
3
4
3)
Let F (x ) 
x

3t 2  1 dt . Find F (2) , F '(2) , F "(2) .
2
4)
The figure to the right shows the graph off , whose domain is the closed interval [-2, 6].
x
Let F (x )  f (t )dt .

1
A)
Find F ( 2).
B)
Find F (6).
2
f
1
-2
-1
1
2
3
4
5
6
t
-1
C)
For what value(s) of x does F (x )  0 ?
D)
For what value(s) of x is F increasing?
5)
Let H (x )  f (t )dt where f is the continuous function with the domain [0, 12]
-2
x

0
graph of f
graphed here:
A)
Find H (0) .
B)
On what intervals is H increasing? Why?
2
C)
On what intervals is H concave up? Why?
D)
Is H (12) positive or negative? Why?
E)
Where does H achieve its maximum?
4
6
8
10
12
AP Calculus
§6.4 Warm-up
1)
Mathematician:
If f is continuous on an open interval I containing a, then F defined by
TRUE OR FALSE:
x
F ( x)   f (t ) dt is continuous on I. Justify your answer.
a
If b > a, then
b
2
d
e x dx is positive. Justify your answer.

dx a
2)
TRUE or FALSE:
3)
Let f ( x )   ln(2  sin t ) dt . If f(3) = 4, then f(5) =
x
Hint: calculator friendly
a
A) 0.040
4)
What is
A) 0
B) 0.272
1
lim
h0 h
C) 0.961
D) 4.555
E) 6.667
xh
B)

f (t ) dt
?
x
1
C)
f '( x )
D)
f ( x)
E) Nonexistant
x
5) At x = π, the linearization of f ( x ) 
A)
6)
y  1
B)
y  x
 cos
C)
3
t dt is
y 
D)
y  x 
The area of the region enclosed between the graph of y  1  x
A) 0.886
B) 1.253
C) 1.414
4
E)
y  x
and the x-axis is
D) 1.571
E) 1.748