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( f(x+h) - f(x) ) / h

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·¥­È¡Gªø«× a ªº½u¬q¤À¦¨¨â¬q§@¬°¯x§Îªº¨âÃä¡A¦p¦ó¤Á¤~¨Ï¯x§Î³Ì¤j¡HFermat ªº·Qªk¡A³Ì¤j®É¡Aµy¬°Åܰʤ@¤U¡A¨ä­È¤£ÅÜ¡C¨ãÅé§@ªk¬O¡G°²³] f(x) ªº·¥­È¥X²{¦b x0¡A«h ³] f(x0) = f(x0+h)¡A¤Æ²¦¹¦¡¡A¨Ã§â§t h ªº¶µ¥á±¼¡A¥i±o x0¡C

­±¿n¡G¥H y = xp ¬°¨Ò¡C

 

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¬°¤°»ò»Ý­n δ - ε ¤è¦¡ªº©w¸q¡H

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¥Ñ©ó§Ú­ÌµLªk³B²zµL­­¤p¡A¦]¦¹­n°jÁרϥΥ¦¡A¦Ó¥²¶·±Ä¥Î¤U­± δ - ε ªº©w¸q¤è¦¡¡C

 

 

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Epsilon-Delta Definition

http://mathworld.wolfram.com/Limit.html

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Continuous Function

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Derivative

 

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http://zh.wikipedia.org/zh-tw/«D¼Ð·Ç¤ÀªR

 

 

®õ°Ç®i¶}¦¡»Pªñ¦ü

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http://mathworld.wolfram.com/TaylorSeries.html

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ÃÒ©ú

°²³] f(x) = A + B (x-a) + C (x-a)2 + D (x-a)3 + E (x-a)4 + ..... , ·L¤À¤@¦¸±o¡]³oùئ³§Q¥Î¨ì d/dx xn = n xn-1¡AÃÒ©ú¦p¤U¡^

f'(x) = B + 2 C (x-a) + 3 D (x-a)2 + 4 E (x-a)3 + ...

f''(x) = 2 C + 2*3 D (x-a) + 3*4 E (x-a)2 + ...

:

©ó¤W¦C«íµ¥¦¡²Õ¥þ¥N¤J x = a¡A«h :

f(a) = A

f'(a) = B

f''(a) = C / 2

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·í | x-a | < 1 ®É (x - a)n ¶V¤p¡A °ª¦¸¶µ¥i©¿²¤

 

¥H¦³­­ªº®õ°Ç®i¶}¶µ¼Æ¨Óªñ¦ü¤@¨Ç¨ç¼Æªº¹Ï¨Ò
http://yll.loxa.edu.tw/0_gsp/taylor/taylor.htm

 

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sin(x) ªº®õ°Ç®i¶}§@¹Ï¡]¥Î fortran µ{¦¡¤Î pgplot ø¹Ï¡^

 

 

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d/dx xn = n xn-1

ÃÒ©ú¡G¥Î°ò¥»©w¸q d/dx xn = limx->0 (1/Δx) . [ (x+Δx)n - xn ] ¡A¨ä¤¤ (x+Δx)n ¥H¦h¶µ¦¡®i¶}¡A¬° xn + nxn-1Δx + O(Δ2)¡A¬G

­ì¦¡ = limx->0 (1/ Δ x) . [ nxn-1Δx + O(Δ2) ] = limx->0 [ nxn-1 + O(Δ) ] = nxn-1 ¡A±oÃÒ¡C

 

«ü¼Æ¨ç¼Æ

d/dx ex = ex

ÃÒ©ú¡G ­º¥ý¥H¯Å¼Æ®i¶} ex = ∑n=0 xn/n! ¡]³o¬O¤@­Ó­È±o°O¦íªº ex ªº©w¸q¡A³s x ¬O¤@­Ó¯x°}®É³£·|¹ï¡^

³v¶µ·L¤À¤§ ∑n=0n xn-1/n! = ∑n=1 xn-1/(n-1)! ¡Aµo²{­ì¤½¦¡¤£ÅÜ¡]¥u­n­«©w m = n-1¡^¡A±oÃÒ¡C

 

Euler ³Ì³ßÅwªº¤½¦¡

e - 1 = 0

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eix = cos x + i sin x

³o¬O¦³¦Wªº Euler ¤½¦¡ (¦è¤¸1748¦~) ¦p¦óµo²{ªº¥i¯à¹Lµ{

http://en.wikipedia.org/wiki/Euler%27s_formula

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¤T¨¤¨ç¼Æ

§Q¥Î sin x = (eix - e-ix) / 2i ¤Î cos x = (eix + e-ix) / 2 ªºÃö«Y¡A¥i±oª¾

d/dx sin x = cos x

d/dx cos x = sin x

 

¹ï¼Æ¨ç¼Æ

d/dx ln x = 1/x

d/dx ln f(x) = f'(x)/f(x)

§ó¦hªº¨Ò¤l¥i¨£ http://www.amath.nchu.edu.tw/~tdoc/4_4.htm

 

 

·L¤À¹Bºâ

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d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

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¦X¦¨¨ç¼Æªº·L¤À¡AÃìÂê«ß (Chain Rule)

¦X¦¨¨ç¼Æ f(g(x)) = f¡Cg(x)

f¡Cg' (x) = f'(g(x)) g'(x)

 

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Euler ºtºâªk

 

 

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