Page 263 - Numerical Analysis Using MATLAB and Excel

Chapter 6 Fourier, Taylor, and Maclaurin Series

Example 6.13
Express the function

y = f t() = e t (6.131)
in a Maclaurin’s series.

Solution:

A Maclaurin’s series has the form of (6.132), that is,
n ()
0 () n
------------x +
fx() = f0() + f' 0()x + f'' 0() 2 … + f -----------------x (6.132)
2! n!
t
e
For this function, we have ft() = e t and thus f0() = 1 . Since all derivatives are , then,
f' 0() = f'' 0() = f''' 0() = … = 1 and therefore,
3
2
t
t
f t() = 1 ++ ----- + ----- + … (6.133)
t
n
2!
3!
MATLAB displays the same result.
t=sym('t'); fn=taylor(exp(t)); pretty(fn)
2 3 4 5
1 + t + 1/2 t + 1/6 t + 1/24 t + 1/120 t

Example 6.14

In a semiconductor diode , the instantaneous current i D and voltage v D are related as
D
v ⁄ nV
i ( D v ) D = I e D T (6.134)
D
where I D is the DC (average) component of the current, the constant has a value between 1
n
and depending on the material and physical structure of the diode, and V T is the thermal volt-
2
age which depends on the temperature, and its value at room temperature is approximately

25 mV .
Expand this relation into a power series that can be used to compute the current when the volt-
age is small and varies about v D = . 0

Solution:

Since the voltage is small and varies about v D = 0 , we can use the following Maclaurin’s series.

6−46 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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