Bài giảng chương 2: Some expended problems of Functions of one variable | Môn toán cao cấp

The function f(x) is assumed to possess derivatives to the nth order. The Taylor series expansion of the function f(x) in a neighborhood of the value x0 in the remainder formula is. The function f(x) is assumed to possess derivatives to the nth order. Tài liệu giúp bạn tham khảo, ôn tập và đạt kết quả cao. Mời bạn đọc đón xem !

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Môn: Toán Cao Cấp (KTHCM)

Trường: Đại học Kinh tế Thành phố Hồ Chí Minh

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Tác giả:

Profile Ma Thu Hương
lOMoARcPSD| 47207194
Department of Mathemacs, Faculty of Basic
Science, Foreign Trade University
-------------------------------------------------------------------------------------
Part II, Calculus
Chapter 2:
Some expended problems of Funcons
of one variable
lOMoARcPSD| 47207194
Instructor Dr. Son Lam
CONTENTS
---------------------------------------------------------------------------------------------------------------------------
I – Taylor Series Formula
II – L’Hôpital’s Rule
lOMoARcPSD| 47207194
I – Taylor Series Formula Theorem
lOMoARcPSD| 47207194
The funcon f(x) is assumed to possess derivaves to the nth
order.
The Taylor series expansion of the funcon f(x) in a neighborhood
of the value x0 in the remainder formula is:
f (x0)(x x0) f (x0)(x x0)2 ... f ( )n (x0)(x x0)n Rn f x( )
f x(
0
)
1! 2! n! f ( )n ( )
n
where R
n
(x x
0
)
n!
and lies between x and x0
R
lOMoARcPSD| 47207194
Note: x xlim 0 x x n0 n 0
lOMoARcPSD| 47207194
I – Taylor Series Formula Theorem
The funcon f(x) is assumed to possess derivaves to the nth order.
The Maclaurin series expansion of the funcon f(x) in a
neighborhood of the value x0 in the remainder formula is:
f (0) f (0) 2 ... f ( )n
(0) xn Rn
f x( ) f (0) x x
1! 2! n!
f (n 1)( ) n 1 (Lagrange
residual)
where R
n
x
lOMoARcPSD| 47207194
(n 1)!
and lies between x and 0.
R
Note:
x
lim
0
n
n
0 (Peano
residual) x
lOMoARcPSD| 47207194
lOMoARcPSD| 47207194
I – Taylor Series Formula
Example
Find the Maclaurin series expansion for the funcon :
f x( ) e
x
x2
f
( )k
( )x e
x
f
( )k
(0) 1,k 1,2,...
lOMoARcPSD| 47207194
I – Taylor Series Formula
ex 1 x x2
...xn Rn
1! 2! n!
Example
Find the Maclaurin series expansion for the funcon :
f x( ) e
x
x
1 x x2 ... xn R
n
lOMoARcPSD| 47207194
I – Taylor Series Formula
e
1! 2! n!
ex 1 x x2!2 ... xnn!
... xnn!
1! n 1
lOMoARcPSD| 47207194
I – Taylor Series Formula
e 1
1 1 1 1 1
2,716666
1! 2! 3! 4! 5!
Example
Find the Maclaurin series expansion for the funcon :
f x( ) e
x
x
1 x x2 ... xn R
n
lOMoARcPSD| 47207194
I – Taylor Series Formula
e
1! 2! n!
e
x
1
x
R
1
1!
ex 1 x x2 R2 1! 2!
Example
lOMoARcPSD| 47207194
I – Taylor Series Formula
Find the Taylor series expansion for the funcon :
a. f x( ) sin x
b. g x( ) cosx
Example
Find the Taylor series expansion for the following funcon at the
point x0=1.
P x( ) x
5
2x
4
x
3
x
2
2x 1
lOMoARcPSD| 47207194
I – Taylor Series Formula
P x( ) x
5
2x
4
x
3
x
2
2x 1
lOMoARcPSD| 47207194
II – L’Hôpital’s Rule
Theorem
lOMoARcPSD| 47207194
II – L’Hôpital’s Rule
f ( )x
lim L lim
f x( )
L x x
0
g x ( ) x x
0
g x( )
This rule, which is more an applicaon of dierenaon than a rule
for nding derivaves, is useful for nding the liming value of the
rao of funcons at a point (x = x0) where that value is undened,
such as 0 or
lOMoARcPSD| 47207194
II – L’Hôpital’s Rule
0
lOMoARcPSD| 47207194
II – L’Hôpital’s Rule
Example
lOMoARcPSD| 47207194
II – L’Hôpital’s Rule
2
x
- x
2
2
x
- 2
2
x
2
2
lim x 2lim x
- 2 x 2lim x - 22
x 2 x - 2
2
2
ln2 4
2
x
- x
2
0 lim
0
x 2 x -
2
2
x
- x
2
2
x
ln2 - 2x
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lOMoAR cPSD| 47207194
Department of Mathematics, Faculty of Basic
Science, Foreign Trade University
------------------------------------------------------------------------------------- Part II, Calculus Chapter 2:
Some expended problems of Functions of one variable lOMoAR cPSD| 47207194 Instructor Dr. Son Lam CONTENTS
--------------------------------------------------------------------------------------------------------------------------- I – Taylor Series Formula II – L’Hôpital’s Rule lOMoAR cPSD| 47207194
I – Taylor Series Formula Theorem lOMoAR cPSD| 47207194
The function f(x) is assumed to possess derivatives to the nth order.
The Taylor series expansion of the function f(x) in a neighborhood
of the value x0 in the remainder formula is: f (x0)(x x0) f
(x0)(x x0)2 ... f ( )n (x0)(x x0)n Rn f x( ) f x( 0) 1! 2! n! f ( )n ( ) n where Rn (x x0) n! and lies between x and x0 R lOMoAR cPSD| 47207194 Note: x xlim 0 x x n0 n 0 lOMoAR cPSD| 47207194
I – Taylor Series Formula Theorem
The function f(x) is assumed to possess derivatives to the nth order.
The Maclaurin series expansion of the function f(x) in a
neighborhood of the value x0 in the remainder formula is: f (0) f (0) 2 ... f ( )n (0) xn Rn f x( ) f (0) x x 1! 2! n! f (n 1)( ) n 1 (Lagrange residual) where Rn x lOMoAR cPSD| 47207194 (n 1)! and lies between x and 0. R lim Note: n x 0 n 0 (Peano residual) x lOMoAR cPSD| 47207194 lOMoAR cPSD| 47207194 I – Taylor Series Formula Example
Find the Maclaurin series expansion for the function : f x( ) ex x2
f ( )k ( )x ex f ( )k (0) 1,k 1,2,... lOMoAR cPSD| 47207194 I – Taylor Series Formula e x 1 x x2 ...xn Rn 1! 2! n! Example
Find the Maclaurin series expansion for the function : f x( ) ex x 1 x x2 ... xn Rn lOMoAR cPSD| 47207194 I – Taylor Series Formula e 1! 2! n! ex 1 x x2!2 ... xnn! ... xnn! 1! n 1 lOMoAR cPSD| 47207194 I – Taylor Series Formula 1 1 1 1 1 e 1 2,716666 1! 2! 3! 4! 5! Example
Find the Maclaurin series expansion for the function : f x( ) ex x 1 x x2 ... xn Rn lOMoAR cPSD| 47207194 I – Taylor Series Formula e 1! 2! n! x ex1 R1 1! ex 1 x x2 R2 1! 2! Example lOMoAR cPSD| 47207194 I – Taylor Series Formula
Find the Taylor series expansion for the function : a. f x( ) sin x b. g x( ) cosx Example
Find the Taylor series expansion for the following function at the point x0=1. P x( ) x5 2x4 x3 x2 2x 1 lOMoAR cPSD| 47207194 I – Taylor Series Formula P x( ) x5 2x4 x3 x2 2x 1 lOMoAR cPSD| 47207194 II – L’Hôpital’s Rule Theorem lOMoAR cPSD| 47207194 II – L’Hôpital’s Rule f ( )x f x( ) lim L lim L x x 0 g x ( ) x x g x( ) 0
This rule, which is more an application of differentiation than a rule
for finding derivatives, is useful for finding the limiting value of the
ratio of functions at a point (x = x0) where that value is undefined, such as 0 or lOMoAR cPSD| 47207194 II – L’Hôpital’s Rule 0 lOMoAR cPSD| 47207194 II – L’Hôpital’s Rule Example lOMoAR cPSD| 47207194 II – L’Hôpital’s Rule 2x - x2 2x - 22 x2 2 lim x 2lim x - 2 x 2lim x - 22 x 2 x - 2 22ln2 4 2x - x2 0 lim 0 x 2 x - 2 2x - x2 2xln2 - 2x