Summary Fundamental Theorem of Calculus

1.3 Fundamental Theorem of Calculus
Theorem. (1.3.1):letto be a continuous function on [a,b]
i) Let
g(x) St fdt,
=

where a is a constant
then f(x)
g'(x) is



ii) Let I
be any antiderivative of of

e. FP-dx
↓




F(b)
=
-
F()

b
Notation:F(x) F(b)
=
-
F(a)
a "evaluated from a to-b




(ii)
eg. of

(Xdx
①
iff(x) x,F(X) x-
=
=




F(b) F(a)
b2 1az
- =
-




② SoexdX
F(X) 2x =




F(0) e 10
F(X)/: F(1) e 1
= - =
-
-
=




physical interpretation of(ii)

(bg'(X)dX g(x)p g(b) =
= -



g(a)
m a
m
integral rate of total change ing
change of 'g' over over [a,b]
[a,b]


a proofofFTC:(just for f(x)20)

i) set
g(x) Sof(t)dt=




img(x+h)- ga
11 we want:
g.(x) = e



,,,iy
=




yy
*
a i

-
A g(4)
=




vonisNcglyta
#
e M.h
m.n g(x
=
+
1) -



g(x) =




where m and M are the min and max values

of on
f
[x,x+h]


(for his)
by dividing by h m
=g(x h) g(x) M
+ -

=




as h ->0 m,M f(x) =