1. BOÄ MOÂN TOAÙN ÖÙNG DUÏNG - ÑHBK
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TOAÙN 1
GIAÛI TÍCH HAØM MOÄT BIEÁN
• BAØI 6: KHAI TRIEÅN TAYLOR
• TS. NGUYEÃN QUOÁC LAÂN (12/2007)

2. CAÙC ÑÒNH LYÙ TRUNG BÌNH
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Cöïc trò taïi x0:   > 0 :  x  (x0 – , x0 + )  f(x)
 f(x0)
Fermat: f ñaït cöïc trò taïi x0  (a,b) & khaû vi taïi x0
 f’(x0) = 0
Minh hoaï hình
hoïc:

3. ÑÒNH LYÙ ROLL
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Haøm f(x) lieân tuïc treân [a,b], khaû vi trong (a,
b), f(a) = f(b)
  x0(a, b): f’(x0) = 0
Minh hoaï hình
hoïc:
VD: Chöùng minh
phöông trình 4ax3
+ 3bx2 + 2cx – (a + b
+ c) = 0 coù ít
nhaát 1 nghieäm
thöïc trong
khoaûng (0,haøm
Giaûi: Xeùt 1)

4. ÑÒNH LYÙ (SOÁ GIA) LAGRANGE
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Haøm f(x) lieân tuïc treân [a,b], khaû vi trong
(a,b)
  c  (a, b): f(b) – f(a) = f’(c)(b – a)
Aùp duïng: Khaûo
saùt tính ñôn
ñieäu cuûa haøm y
= f(x) baèng ñaïo
haøm
VD: CMinh BÑThöùc
arctgx  arctgy  x  y

5. KHAI TRIEÅN TAYLOR
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Haøm y = f(x) coù ñaïo haøm taïi x0  f(x)  f(x0) +
f’(x0)(xthöùc Taylor: f coù ñaïo haøm caáp n+1 treân
Coâng – x0)
(a,b); x0 , x(a, b)
f ' '  x0  f  n   x0 
f  f  x0   f '  x0  x  x0    x  x0 2  ...   x  x0 n  ?
2! n!
n
f  k   x0  f ( n 1) c 
 f x    x  x0 k   x  x0 n1 , c   x0 , x 
k 0 k! (n  1)!
   
Rn  x  : Phaàn dö
Lagrange
CT Taylor (phaàn dö Peano): f coù ñhaøm ñeán caáp
n treân (a,b)
f k   x0 
f ( x)  
n
k!

 x  x0 k  o  x  x0 n , x  x0 
k 0

6. KHAI TRIEÅN MAC – LAURINT
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x0 = 0: Khai trieån Mac – Laurint (phoå
bieán)
f  n  0  n n
f  k  0  k
f ( x)  f 0   f ' 0 x    x  Rn  x    x  Rn ( x)
n! k 0 k!
f ( n1) c  n1
Phaàn dö Rn ( x)  x , c  c x   0, x 
(n  1)!
Lagrange:
Phaàn dö Rn ( x)  o x n1 , x  0  
Peano: trieån Mac – Laurint cuûa haøm
VD: Khai a/ ex
b/ cosx
e x  1  x      ox n1  , x  0
x2 xn
Keát 2! n!
 ox 2 n 1  , x  0
x2 x4 x 2n
cos x  1       1
n
quaû: 2! 4! 2n !

7. MINH HOAÏ KHAI TRIEÅN MAC – LAURINT
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Minh hoaï hình hoïc khai trieån Mac - Laurint haøm
f(x) = sinx
p1 ( x)  x
x3
p2 ( x )  x 
6
x3 x5
p3 ( x)  x  
6 120
Chuù yù: Ñoà
thò ña thöùc
xaáp xæ tieán
daàn veà ñoà
thò haøm ñöôïc

8. KHAI TRIEÅN MAC – LAURINT HAØM CÔ BAÛN
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Haøm löôïng giaùc: sinx, cosx. Haøm tgx (chæ
ñeán caáp ba)
x3 x5
sin x  x     
 1n1 x 2n1  o x 2n , x  0  
3! 5! (2n  1)!
x2 x4  1n x 2n  o x 2n1 , x  0  
cos x  1     
2! 4! (2n)!
x3
 
tgx  x   o x 4 , x  0
3
Khai trieån ex: taùch muõ chaün, leû & ñan daáu. cos
chaün  muõ chaün; sin leû  muõ leû; tg leû 
muõ leû. K0 ñan daáu  shx, chx