FDWK_3ed_Ch05_pp262-319

Section 5.4 Fundamental Theorem of Calculus 295
We can therefore continue our proof, letting 1h xh
x
f t dt f c,
d F
lim 1 dx
h→0
xh
h f t dt
x
lim f c, where c lies between x and x h.
h→0
What happens to c as h goes to zero As x h gets closer to x, it carries c along with it like
a bead on a wire, forcing c to approach x. Since f is continuous, this means that f c
approaches f x:
lim f c f x.
h→0
Putting it all together,
d F
lim Fx h Fx
Definition of derivatives
dx
h→0 h
lim
h→0
xh
x
f t dt
h
Rules for integrals
lim
h→0
f c for some c between x and x h.
f x.
Because f is continuous
This concludes the proof.
It is difficult to overestimate the power of the equation
■
x
d
d
x
a
f t dt f x. (1)
It says that every continuous function f is the derivative of some other function, namely
a
x
f t dt. It says that every continuous function has an antiderivative. And it says that
the processes of integration and differentiation are inverses of one another. If any equation
deserves to be called the Fundamental Theorem of Calculus, this equation is surely
the one.
EXAMPLE 1
Find
Applying the Fundamental Theorem
d
d
x
x
by using the Fundamental Theorem.
d
cos tdt and d x
x
0
1
dt 1
t 2
SOLUTION
x
x
d
d
x
d
d
x
0
cos tdt cos x
Eq. 1 with ft cos t
1 1
1
dt . Eq. 1 with ft
1 1 1
t 2
x 2
t 2
Now try Exercise 3.