Civil Engineering Reference
In-Depth Information
Evaluate the successive derivatives as follows:
()
==
()
=∴ =
′
()
==
()
=∴ =
′′
()
==−
f
0 0 0 0
01 01 1
02
b
sin
cos
sin
b
0
0
f
b
b
1
1
()
=∴ =
f
00 0
03 01
1
3
b
b
2
2
′′′
()
= −
()
=−∴=−
f
!
b
cos
b
3
3
!
′′ ′′
()
=
()
=∴ =
f
04
!
b
=
s
in
00 0
b
4
4
1
5
′′ ′′ ′
()
==
()
=∴ =
′′′ ′′′
()
=
f
05 01
!
b
cos
b
5
5
!
=−
()
=∴ =
f
06
00 0
07 01
1
7
!
b
sin
b
6
6
′′′ ′′′ ′
()
=
= −
()
=−∴=−
f
!
b
cos
b
7
7
!
′′
()
==
()
=∴ =
f
′′′ ′′′ ′
08 00 0
!
b
sin
b
8
8
1
9
′′′ ′′′ ′′′
()
==
()
=∴ =
f
09 01
!
b
cos
b
9
9
!
The polynomial can then be written with the coefficients.
1
3
1
5
1
7
1
9
=
()
=− +
3
5
7
9
y
sin
x
x
x
x
−
x
+
x
!
!
!
!
The value of sin45°=sin(
p
/4) can be evaluated to check the accuracy of
the approximation.
(
)
=−
(
1
3
)
+
(
1
5
)
−
(
1
7
)
+
(
1
9
)
3
5
7
9
sin
pp p
p
p
p
y
=
4
4
4
4
4
4
!
!
!
!
y
=
0 785398
.
1634 0 0808455122 0 0024903946 0 0000365762
0 00000
−
.
+
.
−
.
+
. 0
0 7071067829
3134
y
= .
The exact solution to the same accuracy is
y=
0
.
7071067812.
Example 3.9.
Taylor series polynomial expansion
Expand
y
=
e
−-x
into a polynomial using Taylor series including up to the
fifth degree term. Check by calculating
e
−-
1
.
==+++++
−
2
3
5
ye bbxbxbx
b x
0
1
2
3
5
