Post on 30-Dec-2015
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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