AP Calculus - College Board...AP ® Calculus: Fundamental Theorem of Calculus 2008 Curriculum Module...
28
AP ® Calculus: Fundamental Theorem of Calculus 2008 Curriculum Module
Transcript of AP Calculus - College Board...AP ® Calculus: Fundamental Theorem of Calculus 2008 Curriculum Module...
AP® Calculus:Fundamental Theorem of Calculus
2008Curriculum Module
( ) ( )x
aF x f t dt
f (x) dx
a
bF(b) F(a) .f
41 x
52 x
f (x) dxa
a0,
f (x)dx
a
bf (x)dx
b
a,
a b c
f (x)dx
a
bf (x)dx
b
cf (x)dx
a
c
F
1(x) 3
2 1
)(1 xF )02(3 )01(3 )00(3 )01(3 )02(3 )03(3 )0(3 x
F
2(x) 3
2 1
2 ( )F x )12(3 )11(3 )10(3 )11(3 )12(3 )13(3 )1(3 x
)(1 xF )(2 xF
)(1 xF )(2 xF
G
1(x) 2x
x 2 1
)(1 xG2
)02(4
2
)01(2
2
)00(0
2
)01(2
2
)02(4
2
)03(6
2
)0(2 xx
x 2 1
)(2 xG
G
2(x) 2x
)(1 xG )(2 xG
)(1 xG )(2 xG
2 1 3
1
u 6 3 1 1
)(uH 2
)03(6
2
)01(2
0 2
)01(2
2
)03(6
2
)06(12
2
)0(2 uu
))(,( uHu
3)(tf ]3,2[
2 1
)(1 xF
))(,( 1 xFx
)(1 xF
)(1 xF
F
1(x)?
3)(tf ]3,2[
2 1
2 ( )F x
))(,( 2 xFx )(1 xF
)(2 xF
)(2 xF
2 ( )?F x
)(1 xF )(2 xF
( ) 2g t t ]3,2[
2 1
)(1 xG
))(,( 1 xGx
)(1 xG
1( )G x
1 ( )?G x
x 2 1
)(2 xG
))(,( 2 xGx )(1 xG
)(2 xG
)(2 xG
2 ( )?G x
)(1 xG )(2 xG
f (t) 2t
x 2 1 3
1
3
1
u 6 1
)(uH
))(,( uHu
)(uH
)(uH
)(uH
)(xH
)(xH
F(x) 2 2t dt
1
x
2t t2
1
x
2x x2 1
F (x) 2 2x
3)(xF
3)(xF
xxF 2)(
xxF 2)(
xxF 22)(
xxF 22)(
xxF sin)(
xxF sin)(
xxxF cos)(sin2)(
)(xF
F(x) 3dt
0
x
F(x) 3dt
1
x
F(x) 2t dt
0
x
F(x) 2t dt
1
x
0( ) (2 2 )
xF x t dt
1( ) (2 2 )
xF x t dt
F(x) sin t dt
0
x
F(x) sin t dt
1
x
F(x) 2t dt
0
3x
F(x) 2t dt
0
sin x
G(x) F(x) C
f (t)dt
a
xF(x) C
f (t) dt
a
aF(a) C 0, )(aFC
f (t) dt
a
bF(b) C
f (t)dt
a
bF(b) F(a)
F(b) F(a) f (t)dt
a
b
)(th
h(t) dt
0
5
)(tv
v(t) dt
3
10
)(tb
b(t)
2
6dt
)(tv
)(tx 2t
10t
2t 10t
)0t
d
dxsin t3 dt
0
x2
)cos( 6x
64)( 23 tttr
80 t
3.514
1.572( )r t dt
8
0( )r t dt
2.667
0( )r t dt
3.514
1.572( )r t dt
2.667
0( )r t dt
0t
)21ln()( tta 1t
2t
462.0 609.1 555.2 886.2
g(x) sin t2 dt
0
x
31 x
01 x 772.10 x 171.2253.1 x 507.2772.1 x 3802.2 x
dy
dx3x2 4x 5
y 2 1, find y 3 .
y 3x2 4x 5 dx
y 3
y 3x2 4x 5 dx
y x3 2x2 5x C
1 8 8 10 C
7 C
y x3 2x2 5x 7
y 3 27 18 15 7 23
f x dx
a
bf b f a
y dx
2
3y 3 y 2
y 3 y 2 y dx
2
3
y 3 1 3x2 4x 5 dx
2
3
y 3 1 (x3 2x2 5x)
2
3
y 3 1 27 18 15 8 8 10
y 3 23
y 3 23
f x sin x2 and f 2 5. Find f 1 .
f x dx
1
2f 2 f 1
f 1 f 2 f x dx
1
2
f 1 5 sin x2
1
2dx
f 1 5.495
f
f 2 5
f 0
f 2
f 6
f 0 f 2 f x dx
2
05
1
22 4 9
f
f 2 f 2 f x dx
2
25
1
24 4 13
f 6 f 2 f x dx
2
65
1
24 4
1
222 13 2
f
f 3 5
f 0
f 7
f 9
f 0 f 3 f x dx
0
35 4 1
f
f 7 f 3 f x dx
3
75 9 4
f 9 f 3 f x dx
3
95 9 2 2
0,1 , 3, 5 , 7, 4 , and 9, 2
0 x 1.5 f
1.5 x 5 f 5 x 8 f
8 x 9 f
Area = 4 Area = 2
Area = 9
r t 6e 0.1t C
95 6e 0.1tdt
0
571.392 C
y 21
x2 and y 1 6. Find y 3 .
f x cos 2x and f 0 3. Find f4
.
dW
dt
1
75600 20t t2 , where
dW
dt
f x cos x3 and f 0 2. Find f 1 .
f x e x2
and f 5 1. Find f 2 .
x t
v t 5sin t2 .
F t
2t
t 0
v tt
1 t2 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
f x dx
1
46.2 and f 1 3. Find f 4 .
f
f 4 given that f 4 7.
f
f 2 5
f 1
f 4
f 8
32
3
7
2
7 8
