Download Fundamental Theorem of Calculus

AP Calculus:
®
Fundamental Theorem
of Calculus
2008
Curriculum Module
F ( x)
b
a
x
a
f (x) dx
f (t ) dt
F(b)
F(a)
f.
1 x
4
2
x 5
a
a
b
a
f (x) dx
a
F1 ( x)
1
3( 1 0)
3(0 0)
b
f (x) dx,
c
f (x) dx
F1 (x)
2
3( 2 0)
a
f (x) dx
a b
b
0,
c
b
f (x) dx
c
a
f (x) dx
3
3(1 0)
3(2 0)
3(3 0)
3( x 0)
3(2 1)
3(3 1)
3( x 1)
F2 (x) 3
F2 ( x )
2
3( 2 1)
1
3( 1 1)
F1 ( x)
F2 ( x)
F1 ( x)
F2 ( x)
3(0 1)
3(1 1)
G1 (x)
x
G1 ( x)
2
4( 2 0)
2
x
1
2( 1 0)
2
2
0(0 0)
2
2(1 0)
2
1
G 2 ( x)
G2 (x)
2x
G1 ( x)
G 2 ( x)
G1 ( x)
G 2 ( x)
2x
4(2 0)
2
6(3 0)
2
2 x( x 0)
2
u
H (u)
2
1
1
3
6
3
6( 3 0)
2
1
2( 1 0)
2
0
(u, H (u))
1
2(1 0)
2
6(3 0) 12(6 0)
2
2
2u (u 0)
2
f (t ) 3
2
[ 2,3]
1
F1 ( x)
( x, F1 ( x))
F1 ( x)
F1 ( x)
F1 (x)?
f (t ) 3
2
F2 ( x )
[ 2,3]
1
( x, F2 ( x))
F2 ( x)
F2 ( x)
F2 ( x) ?
F1 ( x)
g (t ) 2t
2
F2 ( x)
[ 2,3]
1
G1 ( x)
( x, G1 ( x))
G1 ( x)
G1 ( x)
G1 ( x) ?
F1 ( x)
x
2
1
G 2 ( x)
( x, G2 ( x))
G 2 ( x)
G 2 ( x)
G2 ( x) ?
G1 ( x)
f (t) 2t
G 2 ( x)
G1 ( x)
x
2
u
6
1
1
3
1
3
1
H (u)
(u, H (u))
H (u)
H (u)
H (u)
H (x)
H (x)
F(x)
F (x) 2 2x
F (x) 3
F (x) 3
F ( x) 2 x
F ( x) 2 x
F ( x) 2 2x
F ( x) 2 2x
F ( x) sin x
F ( x) sin x
F ( x)
2(sin x) cos x
x
1
2 2t dt
2t t 2
x
2x
1
x2 1
F (x)
F(x)
F(x)
F ( x)
F(x)
F(x)
x
3dt
F(x)
2t dt
F(x)
(2 2t )dt
F ( x)
0
x
0
x
0
x
0
sin t dt
3x
0
2t dt
F(x)
F(x)
x
1
x
2t dt
1
x
1
(2 2t )dt
x
1
3dt
sin t dt
sin x
0
2t dt
G(x)
F(x) C
x
a
a
a
b
a
b
a
f (t) dt
F(x) C
f (t) dt
F(a) C
f (t) dt
F(b) C
f (t) dt
F(b)
h(t )
5
0
h(t) dt
v(t )
10
3
v(t) dt
b(t )
6
2
b(t) dt
0,
F(a)
C
F(b)
F (a)
F(a)
b
a
f (t) dt
v(t )
x(t )
t
2
t 10
t
t
0)
2
t 10
x2
d
dx
0
sin t 3 dt
cos(x 6 )
r (t )
0 t
3.514
1.572
t3
4t 2
6
8
8
r (t )dt
0
2.667
r (t )dt
0
3.514
r (t )dt
1.572
t
ln(1 2 t )
a (t )
2.667
0
r (t )dt
0
t 1
t
0.462
r (t )dt
2
1.609
2.555
g(x)
x
0
2.886
sin t 2 dt
1 x 3
1 x
0
0
x 1.772
1.253 x
2.171
1.772 x 2.507
2.802 x 3
dy
dx
y
3x 2
3x 2
y 2
4x 5
1, find y 3 .
4x 5 dx
y 3
y
3x 2
y
x 3 2x 2
4x 5 dx
5x C
1 8 8 10 C
7
C
x 3 2x 2
y
y 3
27 18 15 7
b
a
3
2
y dx
y 3
y 3
y 3
5x 7
1
f x dx
y 3
y 2
3
2
23
3x 2
f b
y 2
3
2
y dx
4x 5 dx
1 (x3 2x 2 5x)
3
2
f a
y 3
1
27 18 15
y 3
y 3
f x
sin x 2 and f 2
8 8 10
23
23
5. Find f 1 .
2
1
f x dx
f 1
f 2
f 1
5
f 1
f 2
2
1
2
1
f 1
f x dx
sin x 2 dx
5.495
f
f
f 0
f 0
f
2
5
f 6
f 2
2
0
2
f x dx
5
1
2 4
2
9
f
f 2
f
2
f 6
f
2
2
2
6
2
f x dx
5
1
4 4
2
13
f x dx
5
1
4 4
2
1
2
22
13 2
f
Area = 4
Area = 2
f 3
f 0
3
f 3
0
f 0
5
f 7
f 9
Area = 9
f x dx
5 4 1
f
f 7
f 3
f 9
f 3
7
3
9
3
f x dx
5 9
4
f x dx
5 9 2
2
0,1 , 3, 5 , 7,
0
1.5
x
5
4 , and 9, 2
f
x 1.5
f
5 x 8
8 x 9
f
f
r t
95
5
0
6e
0.1t
dt
71.392 C
6e
0.1t
C
y
1
2
and y 1
x2
f x
6. Find y 3 .
cos 2x and f 0
3. Find f
dW
dt
f x
cos x3 and f 0
f x
e
x2
and f 5
4
.
1
dW
600 20t t 2 , where
75
dt
2. Find f 1 .
1. Find f 2 .
x t
v t
5sin t 2 .
F t
2t
t
v t
t
1 t2
0
s 0
5.
f
f x
1 ex
x2
f 3.1
x2
1 x5
f 1
5
f 4
In Problems 11–13, use the Fundamental Theorem of Calculus and the given graph. Each
tick mark on the axes below represents one unit.
f
4
1
f x dx
6.2 and f 1
3. Find f 4 .
f
f
4 given that f 4
7.
f
f
2
f 1
f 4
f 8
5
32
3
7
2
7
8