# 3 線性逼近

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## 3線性逼近

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「貼近」這個曲線的直線。

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### 線性逼近

L(x) = y = f(a) + f(a)(x – a)

f(x)  f(a) + f(a)(x – a)

(linear approximation) ，上述的切線函數我們則稱為 f(x) 在 x= a 附近的線性部分/線性化 (linearization) 。

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## 範例一

f(x) = (x + 3)–1/2

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## 範例一 / 解

L(x) = f(1) + f(1)(x – 1)

= 2 + (x – 1)

cont’d

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cont’d

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cont’d

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L(x) 實際函數值

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cont’d

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## 範例二 / 解

cont’d

–2.6 < x < 8.6 的範圍內，估計值的精確度在 0.5 以內。

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cont’d

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aT = –g sin 

aT = –g 

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## 線性逼近在物理上的應用

sin’(x) = cos(x) cos(0) = 1

sin x  0 + 1*(x – 0) = x

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dy = f(x) dx

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## 微分

y = f(x + x) – f(x)

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## 範例三

(a) 計算

f(2) = 23 + 22 – 2(2) + 1

= 9

f(2.05) = (2.05)3 + (2.05)2 – 2(2.05) + 1

= 9.717625

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## 範例三 / 解

y = f(2.05) – f(2)

= 0.717625

dy = f(x) dx

= (3x2 + 2x – 2) dx 當 x = 2, dx = x = 0.05 ，有

dy = [3(2)2 + 2(2) – 2]0.05

cont’d

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## 範例三 / 解

(b) f(2.01) = (2.01)3 + (2.01)2 – 2(2.01) + 1

= 9.140701

y = f(2.01) – f(2)

= 0.140701 當 dx = x = 0.01,

dy = [3(2)2 + 2(2) – 2]0.01

= 0.14

cont’d

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dV = 4r2 dr

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 277

cont’d

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## 微分

(relative error) ，也就是誤差除以整體體積的比例：

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## 微分

dr/r = 0.05/21  0.0024

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## References

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