The Fundamental Theorem of CalculusThe Fundamental Theorem of Calculus

This One’s a Two-Parter!•FTC Part Two First•Then FTC Part One

FTC: Part Two…FirstFTC: Part Two…First

If f is continuous on [a, b], and if F is any antiderivative of f on [a, b], then

But we already knew that now didn’t we?!

b

a

f t dt F b F a

FTC: Part OneFTC: Part One

If f is continuous on [a, b], then the function

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

x

a

F x f t dt

x

a

dF df t dt f x

dx dx

FTC: Part One…One More TimeFTC: Part One…One More Time

Isn’t ?

And, since F(a) is just a constant w/a derivative of zero, then to take the derivative we just take the derivative of F(x)…which is f(x)!

x

a

f t dt F x F a

x

a

dF df t dt f x

dx dx

FTC: Part One…An ExampleFTC: Part One…An Example

Find

Find

Find

cosxd

t dtdx

cos x

20

1

1

xddt

dx t 2

1

1 x

2

3

xdt dt

dx2x

FTC: Part One…With a Twist…or a Kink!FTC: Part One…With a Twist…or a Kink!

Apply the second FTC on the integral…

Now take the derivative of (don’t forget the chain rule!)

2

0

cosxd

t dtdx

2

0

cosx

t dt 2sin sin 0x 2

0sin

xt

2sin sin 0x

FTC: Part One…Look at the Kinkage! FTC: Part One…Look at the Kinkage!

So…

because of the…CHAIN RULE!!!

2sin sin 0d

xdx

2xcos(x2) ZERO!!!

2

0

cosxd

t dtdx 22 cosx x

Some More Kinky Problems…Some More Kinky Problems…

Find

Find

Find 2

0

tanxd

t dtdx

2 tan 2x

3

0

xdtdt

dx 2 33x x

5

3 sinx

dt t dt

dx

3 sinx x 5

3 sinxdt t dt

dx

Using Graphs and the FTCUsing Graphs and the FTC

Find h(1).

1

x

h x f t dt

1

x

h x f t dt 1

1

1h f t dt 0

Using Graphs and the FTCUsing Graphs and the FTC

Is h(0) positive or negative? JYA

1

x

h x f t dt

1

x

h x f t dt 0

1

0h f t dt 0

Using Graphs and the FTCUsing Graphs and the FTC

Find the value of x for which h(x) is a maximum.

1

x

h x f t dt

1

x

h x f t dt h x f x Where does f(x) change from positive to negative?

@ 4 4 is a maximumx f

Using Graphs and the FTCUsing Graphs and the FTC

Find the x-values of the inflection points of h(x).

1

x

h x f t dt

1

x

h x f t dt h x f x Where does the slope of f(x) change?@ 1, 3, and 6x x x

This is a graph of f(t) NOT h(x)!

AssignmentAssignment

p. 302 #1-19 oddp. 303 #572002 AB-2 Handout