Post on 18-Jan-2016
5.3 – The Fundamental Theorem of Calculus
Examples
Use WolframAlpha to determine g. Then determine g'(x).
1
1. cosx
g x t dt 21
13.
1
xg x dt
t
Question So what does g represent here?
3
32.
xg x t dt
integral ( ), variable, ,b
af x dx f x a b
First Fundamental Theorem of Calculus – Part 1
If f is continuous on [a, b], then the function g is defined by
x
ag x f t dt
is continuous on [a, b] and differentiable on (a, b), then g(x) is the _______________
of f(t) in terms of x. That is, g'(x) = _____.
First Fundamental Theorem of Calculus – Part 1 (Alternate Definition)
If f is continuous on [a, b], then
______x
a
df t dt
dx
Examples
11. ln
xdt dt
dx 23
32.
5
xddt
dx t t
Evaluate the following.
2
13. cos
udt dt
du 24. 9csc
x
dt dt
dx
Examples
Use WolframAlpha to determine the following:
sin 5
15.
xdr dr
dx 4 3
36. cos
xdt dt
dx
integral{ (deri ), variable, , }vative ,b
af xf x x a xbd
First Fundamental Theorem of Calculus – Part 1 (Alternate Definition)
If f is continuous on [a, b] and u is a function of x, then
___ _____u
a
df t dt f
dx
Examples
Determine the derivative of g by hand.
2
27. sin
xedt t dt
dx
1 5
ln8. sec
x
dt t dt
dx
First Fundamental Theorem of Calculus – Part 2
If f is continuous on [a, b], then the function g is defined by
( )b b b
aaaf x d F b F ax F x F x
where F is the general antiderivative of f, that is, a function such that F ' = f.
Examples
Evaluate:
8 3 2
1
/ 4
0
1
0
2
22
1. 2
2. sec tan
3. 10
2 2 04. where
4 0 2
x
x dx
t t dt
dx
if xf x dx f x
x if x