section 5.4a fundamental theorem of calculus. deriving the theorem let apply the definition of the...

Section 5.4a

FUNDAMENTAL THEOREM OF CALCULUS

Deriving the Theorem

x

aF x f t dtLet

Apply the definition of the derivative:

0

limh

F x h F xdF

dx h

0

lim

x h x

a a

h

f t dt f t dt

h

0

lim

x h

x

h

f t dt

h

0

1lim

x h

xhf t dt

h

Rule for Integrals!


This is average value of f fromx to x + h. Assuming that f iscontinuous, it takes on its averagevalue at least once in the interval…

0

1lim

x h

xhf t dt

h

1 x h

xf t dt f c

h

For some c between x and x + h

Back to the proof…


0

1lim

x h

xh

dFf t dt

dx h

0limh

f c

What happens to c as h goes to zero???

As x + h gets closer to x, it forces c to approach x…

Since f is continuous, this means that f(c) approaches f(x):

0

limh

f c f x

Putting it all together…


0

limh

F x h F xdF

dx h

0

lim

x h

x

h

f t dt

h

Definition of Derivative Rule for Integrals

0

limh

f c

f xFor some c between

x and x + h

Because f iscontinuous

The Fundamental Theorem ofCalculus, Part 1If is continuous on [a, b], then the function

x

aF x f t dt

has a derivative at every point x in [a, b], and

x

a

dF df t dt f x

dx dx

…the definite integral of a continuous function is adifferentiable function of its upper limit of integration…

f

The Fundamental Theorem ofCalculus, Part 1

x

a

df t dt f x

dx

• Every continuous function is the derivative of some other function.

• The processes of integration and differentiation are inverses of one another.

• Every continuous function has an antiderivative.

A Powerful Theorem Indeed!!!


cosxd

t dtdx

Evaluate each of the following, using the FTC.

cos x

20

1

1

xddt

dx t 2

1

1 x


2

1cos

xy t dtFind dy/dx if

1cos

uy t dt 2u x

The upper limit of integration is not x y is a composite of:

and

Apply the Chain Rule to find dy/dx:

dy dy du

dx du dx

1cos

ud dut dt

du dx

cosdu

udx

2 2cos 2 2 cosx x x x


53 sin

xy t t dtFind dy/dx if

53 sin

x

dt t dt

dx 53 sin

xdt t dt

dx

53 sin

xdt t dt

dx 3 sinx x


2

2

x t

xy e dtFind dy/dx if

2

2

x t

x

de dt

dx2 2

0 0

x xt tde dt e dt

dx

2 2 2 2x xd de x e xdx dx

ChainRule

2 22 2x xxe e

Deriving More of the Theorem

Let x

aG x f t dt

If F is any antiderivative of f, then F(x) = G(x) + C for someconstant C.

F b F a G b C G a C

Let’s evaluate F (b) – F(a):

G b G a

b a

a af t dt f t dt

b

af t dt

0

The Fundamental Theorem ofCalculus, Part 2If is continuous at every point of [a, b], and if F is anyantiderivative of on [a, b], then

b

af x dx F b F a

This part of the Fundamental Theorem is also called theIntegral Evaluation Theorem.

ff


b

af x dx F b F a

• Any definite integral of any continuous function can be calculated without taking limits, without calculating Riemann sums, and often without major effort all we need is an antiderivative of !!!

Another Very Powerful Theorem!!!

f

f


b

aF x

The usual notation for F(b) – F(a) is

A “note” on notation:

b

aF x or

depending on whether F has one or more terms…


Evaluate the given integral using an antiderivative.

3 3

11x dx

Antiderivative:

4

4

xx

34

14

xx

81 1

3 14 4

24How can we support thisanswer numerically???



1

23x dx

Antiderivative:

13

ln3x

267.889

3ln3

1

2

13

ln3x

1 213 3

ln 3

1 19

ln3 3



5 3 2

0x dx

Antiderivative:

5 22

5x

10 5 22.361

55 2

0

2

5x

5 225 0

5



01 cos x dx

Antiderivative:

sinx x3.142

0sinx x

sin 0 sin 0

0 0 0



3

04sec tanx x dx

Antiderivative:

sec x

4

3

04 sec x

4 sec 3 sec0

4 2 1



4

0

1 udu

u

Antiderivative:

1 22u u

4 1 2

01u du

0

41 2

02u u

2 4 4 0

section 5.4a fundamental theorem of calculus. deriving the theorem let apply the definition of the...

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