mean value theorem for integrals
DESCRIPTION
Mean Value Theorem for Integrals. If f(x) is continuous on [a, b], then there exists a number x=c in [a, b] such that. For continuous functions, there must be some average value such that the area under the original curve is equal to the average value multiplied by the length from x=a to x=b. - PowerPoint PPT PresentationTRANSCRIPT
Mean Value Theorem for Integrals
b
a
f x dx f c b a
If f(x) is continuous on [a, b], then there exists a number x=c in [a, b] such that
a bc
f(c)
For continuous functions, there must be some average value such that the area under the original curve is equal to the average value multiplied by the length from x=a to x=b.
Average Value of a function, f(x)
1 b
a
Avg f c f xb a
Examples: Find the point(s) on the graph that represents the average value of the function.2( ) 3 2 [1, 4]f x x x on 1
. 4
2
1
13 2
4 1f c x x dx
43 2
1
1
3f c x x
164 16 1 1
3f c 1
48 163
2.
3
91, 3f x on
x
3
3
1
19
2f c x dx
3
21
1 9
2 2x
23 2 16x x 23 2 16 0x x
3 8 2 0x x 8, 23
x 8,163
Find the point(s) on the graph that represents the average value of the function.
1 9 9
2 18 2
14 2
2
3
92
x
3 9
2x
39
1.652
x
1.65, 2
b
a
f x dx Area under curve
x
a
f t dt Accumulation functionFunction that finds area under curve at any value of x.
0
cosx
t dt0sinxt
sin sin 0x
sin x
Second Fundamental Theorem of Calculus (SFTC)
x
a
df t dt f x
dx
If f(x) is continuous on an interval containing x=a, then
or
u x
a
df t dt f u x u x
dx
4.2
0
1xdt dt
dx
2 1x
5.
3
sinx
x
dt dt
dx
3cos cosd
x xdx
2 33 sin sinx x x