Class notes

Fundamental Theorem of Calculus lecture notes

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Course
Math 112

Institution
University Of The Fraser Valley (UFV)

Comprehensive Class notes on the fundamental theorem of calculus antiderivatives/integration with examples Based on CLP 2 textbook

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Uploaded on January 13, 2023
Number of pages 3
Written in 2022/2023
Type Class notes
Professor(s) Endleman robin
Contains Lecture 4

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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) =

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