warm up. 6.4 fundamental theorem of calculus if you were being sent to a desert island and could...
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Warm Up
6.4 Fundamental Theorem of Calculus
If you were being sent to a desert island and could take only one equation with you,
x
a
df t dt f x
dx
might well be your choice.
Quote from CALCULUS by Ross L. Finney and George B. Thomas, Jr., ©1990.
x
a
df t dt f x
dx
If f is continuous on [a,b], then
1. Derivative of an integral.
Fundamental Theorem of Calculus (FTC) Part One
a
xdf t dt
xf x
d
2. Derivative matches upper limit of integration.
1. Derivative of an integral.
Fundamental Theorem of Calculus
a
xdf t dt f x
dx
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
Fundamental Theorem of Calculus
x
a
df t dt f x
dx
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
New variable.
Fundamental Theorem of Calculus
What’s the significance?
• Every continuous f is the derivative of some other function, namely
• Every continuous function has an antiderivative.
• The processes of integration and differentiation are inverses of each other!
x
a
dttf )(
20
1
1+t
xddt
dx 2
1
1 x
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
Example 1
You try!
Find dy/dx x
dtty1
2 )4(
4)(' 2 xxF
2
0cos
xdt dt
dx
2 2cosd
x xdx
2cos 2x x 22 cosx x
The upper limit of integration does not match the derivative, but we could use the chain rule.
Example 2
53 sin
x
dt t dt
dxThe lower limit of integration is not a constant, but the upper limit is.
53 sin xdt t dt
dx
3 sinx x
We can change the sign of the integral and reverse the limits.
Example 3
2
2
1
2
x
tx
ddt
dx eNeither limit of integration is a constant.
2 0
0 2
1 1
2 2
x
t tx
ddt dt
dx e e
It does not matter what constant we use!
We split the integral into two parts.
Example 4
2 2
0 0
1 1
2 2
x x
t t
ddt dt
dx e e
2 2
1 12 2
22xx
xee
(Limits are reversed.)
(Chain rule is used.)
2 2
2 2
22xx
x
ee
Homework 6.4A
Derivatives Quiz coming soon!