Civil Engineering Reference
In-Depth Information
3.7
tAYLOR SERiES POLYnOMiAL EXPAnSiOn
The differentiation of a continuous function is used to find slopes, cur-
vatures, and values for a function but can also be used to find other
relationships of functions. Often in structural engineering, there is a
need to find differential relationships. One simple way to easily evalu-
ate transcendental equations is to use polynomial expansion developed
for the Taylor series. This is often referred to as the power series. The
general Taylor series polynomial expansion of a function is as follows:
=
()
=+++++
0
2
3
n
yf xbbx bx bx
bx
n
1
2
3
Successive derivatives of the function evaluated at zero can yield the coef-
ficients,
b
.
()
=
′
()
=
′′
()
=
()
′′′
()
=
()
= !
f
0
01
012
0123 3
b
0
f
b
1
f
b
2
f
b
b
3
3
()
=
()
…=
f
i
0123
()
i bi b
!
i
i
(
)
=
()
…=
f
n
0
123()
nb nb
n
!
n
By taking successive derivatives of the function then evaluating them,
the coefficients of the polynomial may be found. This is how most digi-
tal equipment like computers and calculators find values for transcendental
equations.
Example 3.8
Taylor series polynomial expansion
Expand
y=sin
(
x
) into a polynomial using Taylor series including up to the
ninth degree term. Check by calculating sin45°.
=
()
=+++++
2
3
9
y
sin
x bbxbxbx
b x
0
1
2
3
9
