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http://zh.wikipedia.org/wiki/%E6%B3%B0%E5%8B%92%E7...

泰勒級數的公式請參考這邊

n就是你要展開的次數

用f(^n)表示f的微分次數

f(^0) = f

f(^1) = f'

f(^2) = f''

f(x)的1次泰勒展開 = f(^0)(a)/0!．(x - a)^0 + f(^1)(a)/1!．(x - a)^1

f(x)的2次泰勒展開 = f(^0)(a)/0!．(x - a)^0 + f(^1)(a)/1!．(x - a)^1 + f(^2)(a)/2!．(x - a)^2

a) f(x)=√ x , a=1

f(^0)(x) = f(x)=√ x

=> f(^0)(a) = 1

f(^1)(x) = f'(x) = 1/2．1/√x

=> f(^1)(a) = 1/2

f(^2)(x) = f''(x) = -1/2．1/2．1/(√x)^3

=> f(^2)(a) = -1/4

√ x的1次泰勒展開

= f(^0)(a)/0!．(x - a)^0 + f(^1)(a)/1!．(x - a)^1

= 1/1．1 + (1/2)/ 1．(x-1)

= 1 + 1/2．(x - 1)

√ x的2次泰勒展開

= f(^0)(a)/0!．(x - a)^0 + f(^1)(a)/1!．(x - a)^1 + f(^2)(a)/2!．(x - a)^2

= 1/1．1 + (1/2)/ 1．(x-1) + (-1/4)/2．(x-1)^2

= 1 + 1/2．(x - 1) - 1/8．(x-1)^2

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b) f(x)=e^cos(x), a = 0

cos(0) = 1, sin(0) = 0

f(^0)(x) = f(x)=e^cos(x)

=> f(^0)(a) = e^cos(0) = e

f(^1)(x) = f'(x) =e^cos(x)．sin(x)

=> f(^1)(a) = e^cos(0)．sin(0) = 0

f(^2)(x) = f''(x) =e^cos(x)．[sin(x)]^2 + e^cos(x)．cos(x)

=> f(^2)(a) = e^cos(0)．[sin(0)]^2 + e^cos(0)．cos(0) = e

√ x的1次泰勒展開

= f(^0)(a)/0!．(x - a)^0 + f(^1)(a)/1!．(x - a)^1

= e/1 + 0/1．(x-0)

= e

√ x的2次泰勒展開

= f(^0)(a)/0!．(x - a)^0 + f(^1)(a)/1!．(x - a)^1 + f(^2)(a)/2!．(x - a)^2

= e/1 + 0/1．(x-0) + e/2．(x-0)^2

= e + ex^2/2

圖自己畫看看，利用微分的方法來畫

2013-09-07 02:37:07 補充：

https://www.dropbox.com/s/gsu0vq75arqj7tc/20130906...