Page 277 - Numerical Analysis Using MATLAB and Excel

Chapter 6 Fourier, Taylor, and Maclaurin Series

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(
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f'' x() = – sin x , f'' a() = f'' – π 4 = 22 , f''' x() = – cos x ,
(
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⁄
)
f''' a() = f''' – π 4 = – 22 , and so on. Therefore,
⁄
⁄
⁄
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)
(
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)
)
(
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)
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(
f x() = – 22 + ( 22 x + π 4 + ( 24 x + π 4 ) 2 – ( 212 x + π 4 ) 3 + …
n
MATLAB displays the same result.
x=sym('x'); y=sin(x); z=taylor(y,4,−pi/4); pretty(z)
1/2 1/2 1/2 2
- 1/2 2 + 1/2 2 (x + 1/4 pi) + 1/4 2 (x + 1/4 pi)
1/2 3
- 1/12 2 (x + 1/4 pi)
9.
v ⎞
⎛
iv() = k1 + ---- 1.5
⎝ V ⎠
The Taylor series for this relation is
i'' v ( ) 2 i''' v ) ( 3
0
0
iv() = iv ( 0 ) i' v ) ( + 0 ( v – v ) 0 -------------- vv ) ( + – 0 --------------- vv ) ( + – 0 + …
2!
3!
Since the voltage is small, and varies about v = 0 , we expand this relation about v = 0 and
v
the series reduces to the Maclaurin series below.
iv() = i0() + i' 0()v + i'' 0() 2 … (1)
------------v +
2!
By substitution of v = 0 into the given relation we get
i0() = k
The first and second derivatives of are
i
⁄
3k ⎛ v ⎞ 12 3k
i' v() = ------- 1 + ---- i' 0() = -------
2V ⎝ V ⎠ 2V
⁄
3k ⎛ v ⎞ – 12 3k
i'' v() = ---------- 1 + ---- ⎠ i'' 0() = ----------
2⎝
4V V 4V 2
and by substitution into (1)

3k 3k 2 ⎛ 3 3 2 ⎞
iv() = k + -------v + ----------v + … = k1 + -------v + ----------v + …
2V 8V 2 ⎝ 2V 8V 2 ⎠

6−60 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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