Lesson 17.pdf - 17 Taylor’s Theorem August 1 2018 If f(x = a0 a1 x a2 x2 we find Taylor theorem a0 = f(0 a1 = f 0(0 a2 = f 00(0 2 In general for a

# Lesson 17.pdf - 17 Taylor’s Theorem August 1 2018 If f(x...

• Notes
• 10

This preview shows page 1 - 4 out of 10 pages.

17. Taylor’s Theorem August 1, 2018 Taylor theorem If f ( x ) = a 0 + a 1 x + a 2 x 2 , we find a 0 = f (0) , a 1 = f 0 (0) , a 2 = f 00 (0) 2! . In general, for a polynomial f ( x ) = a 0 + a 1 x + a 2 x 2 + ... + a n x n , we find a 0 = f (0) , a 1 = f 0 (0) , a 2 = f 00 (0) 2! , ..., a n = f ( n ) (0) n ! . (1) J.L. Lagrange once thought that every continuous function f ( x ) near x = a can be written as a power series n =0 a n ( x - a ) n . We now know that this is true only for functions with good differentiable condition 1 ; in this case, a function f ( x ) equals to its Taylor series : f ( x ) = f ( a ) + f 0 ( a )( x - a ) + f 00 ( a ) 2! ( x - a ) 2 + ... + f ( n ) ( a ) n ! ( x - a ) n + ...... When a = 0, we get f ( x ) = f (0) + f 0 (0) x + f 00 (0) 2! x 2 + ... + f ( n ) (0) n ! x n + ... (2) By comparing (1), these coefficients f ( n ) (0) n ! are expected. Here is one of the theorems in this theory. 1 The condition is: any order of derivative f ( n ) exists, and the corresponding radius of convergence of the Taylor series is positive. 143

Subscribe to view the full document.

Theorem 0.1 (Taylor theorem) Let f and its first n derivatives f 0 , f 00 , f 000 , ..., f ( n ) be con- tinuous on [ a, b ] and differentiable up to order n +1 on ( a, b ) . Let x 0 ( a, b ) . Then for each x ( a, b ) , x 6 = x 0 , there is a number c between x 0 and x such that f ( x ) = f ( x 0 )+ f 0 ( x 0 )( x - x 0 )+ f 00 ( x 0 ) 2! ( x - x 0 ) 2 + ... + f ( n ) ( x 0 ) n ! ( x - x 0 ) n + f ( n +1) ( c ) ( n + 1)! ( x - x 0 ) n +1 . Proof: Fix x [ a, b ] with x 6 = x 0 . Let M be the unique solution of the equation 2 f ( x ) = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) + f 00 ( x 0 ) 2! ( x - x 0 ) 2 + ... + f ( n ) ( x 0 ) n ! ( x - x 0 ) n + M ( x - x 0 ) n +1 . We want to show that there exists c between x 0 and x such that M = f ( n +1) ( c ) ( n +1)! . Let us define F ( t ) := f ( t ) + f 0 ( t )( x - t ) + ... + f ( n ) ( t ) n ! ( x - t ) n + M ( x - t ) n +1 . We observe that F ( x ) = f ( x ) and F ( x 0 ) = f ( x ). Also, F is continuous on [ a, b ] and differentiable on ( a, b ). Then by Roll’s theorem, there exists c between x and x 0 such that F 0 ( c ) = 0 . By the definition of F , we calculate F 0 ( t ) = f 0 ( t ) + [ f 00 ( t )( x - t ) - f 0 ( t )] + ... + f ( n +1) ( t ) n ! ( x - t ) n - f ( n ) ( t ) n ! n ( x - t ) n - 1 + M ( n + 1)( x - t ) n , and hence F 0 ( c ) = f ( n +1) ( c ) n ! ( x - c ) n - M ( n + 1)( x - c ) n . Then M = f ( n +1) ( c ) ( n +1)! as desired. Remarks: 1. The polynomial p n ( x ) = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) + ...... + f ( n ) ( x 0 ) n ! ( x - x 0 ) n is called the Taylor polynomial for f of degree n in powers of x - x 0 . 2 If a = b + cx with c 6 = 0, then x = a - b c is the unique solution. 144
2. The power series f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) + ...... + f ( n ) ( x 0 ) n ! ( x - x 0 ) n + ... is called the Taylor series for f in powers of x - x 0 .

Subscribe to view the full document.

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes