Ask a question from expert

Ask now

the Fundamental Theorem of Calculus PDF

5 Pages408 Words193 Views
   

Added on  2021-11-15

the Fundamental Theorem of Calculus PDF

   Added on 2021-11-15

BookmarkShareRelated Documents
Solution Task 1:
(a)
Consider 1 2lim , 1Rt
tLteR

Since 1R, let’s consider a particular case
3
2R, then



1 2
31 22
1
22
2
lim
lim
lim lim
1lim lim
Rt
t
t
t
t
tt
t
tt
Lte
te
te
et











Simplify further,
1 2lim
1
0
Rt
t
Lte
e







Hence 1 2lim Rt
tte

(b)
Consider the limit 2 3lim Rtt
tLttee

Since 0e
So,



2 3
2
2
lim
0
0
Rtt
t
Rt
Rt
Lttee
ttee
tte







Hence 2 3lim 0Rtt
tttee

Solution Task 2:
(a)
Consider the definite integral
0
1
F
aFxsxKeedx where ,sa and F are positive
constants.
Now,




0
0
0
0
1
1
1
F
aFxsx
F
sxaFax
F
sxaFax
F
saxaFsx
Keedx
eeedx
eeedx
eeedx












Simplify further,





0 0
00
1
11 1
1 1 1
FF
saxaFsx
aFFFsaxsx
aFsaFsF
sFaFsF
Keedxedx
eeesas
eeesas
eeesas









Simplify further,
sFseK

aFsFsese



1 1
sF
sFaF
aesa
ssa
aese
ssa





the Fundamental Theorem of Calculus PDF_1
Hence


1 1sFaFaeseKssa

(b)
To show 1 1sFaFssaKaese
Last result from part (a)



1 1
1 1
sFaF
sFaF
aeseKssa
ssaKaese




This completes the proof.
Solution Task 3:
Consider the system of equation,
1 2 3 3
2 1 1 4
0 1 1 2
1 1 0
x
y
za

(a)
The augmented matrix is,
1 2 3 3
2 1 1 4
0 1 1 2
1 1 0 a

Now let’s perform elementary row operations.
Perform 2 2 1 3 3 12 ,RRRRRR
1 2 3 3
0 5 5 10
0 1 1 2
0 3 3 3a


2
2 5
RR
1 2 3 3
0 1 1 2
0 1 1 2
0 3 3 3a


3 3 2 4 4 2, 3RRRRRR
1 2 3 3
0 1 1 2
0 0 0 0
0 0 0 3a


(a)
Since system has four equations and three unknown that is system has either infinitely
many solutions or no solution.
Now,
If 3 0a that is for 3a system has infinitely many solutions and if 3a0 that is
if a3 then the system is inconsistent
(b)
If 3a then system has infinitely many solutions that is
2 3 3
2
xyz
yz


Suppose zt then
1 , 2 and xtytzt Where tR
Solution Task 4:
Given the matrices,
0 0 4 3 4 1
3 0 0 and 1 2 1
0 1 2 3
tsAB
ts

(a)
the Fundamental Theorem of Calculus PDF_2

End of preview

Want to access all the pages? Upload your documents or become a member.

Related Documents
Solution1: Given Let’s simplify.
|4
|205
|26

Solved problems on calculus, mechanics, differential equations, linear algebra and matrix
|6
|610
|369

Differentiation and Integration Examples with Solutions
|6
|591
|378

Assignment Maths Problem and Solutions
|8
|685
|19

Solution 1: To prove for an integer.
|2
|274
|42

Deriving Posterior Distribution for Poisson Distribution with Gamma Prior
|9
|702
|455