This preview shows pages 1–11. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon () Math 16b Autumn 2010 1 / 44 The derivative as the rst Taylor polynomial If f ( x ) is di erentiable at a , then the function p ( x ) = b + m ( x a ) where b = f ( ) and m = f ( a ) is the best linear approximation to f near a . For x a we have f ( x ) p ( x ) . Note that f ( a ) = b = p ( a ) and f ( a ) = m = p ( a ) . Thomas Scanlon () Math 16b Autumn 2010 2 / 44 Linear approximation in an example If f ( x ) = sin ( x ) , then f ( ) = sin ( ) = and f ( ) = cos ( ) = 1 So L ( x ) = + 1 x = x is the best linear approximation to sin ( x ) near zero. Thomas Scanlon () Math 16b Autumn 2010 3 / 44 Linear approximation in an example at another point If f ( x ) = sin ( x ) , then f ( / 4 ) = sin ( / 4 ) = 2 / 2 and f ( / 4 ) = cos ( / 4 ) = 2 / 2 So ( x ) = 2 / 2 + 2 / 2 ( x / 4 ) is the best linear approximation to sin ( x ) near / 4 . Thomas Scanlon () Math 16b Autumn 2010 4 / 44 Higher degree Taylor polynomials De nition If f ( x ) is a function which is n times di erentiable at a , then the n th Taylor polynomial of f at a is the polynomial P ( x ) of degree (at most n ) for which f ( i ) ( a ) = p ( i ) ( a ) for all i n . For n = 0, P is constant and P f ( a ) . For n = 1, P is a linear polynomial. Write P ( x ) = b + m ( x a ) . Then f ( a ) = f ( ) ( a ) = P ( ) ( a ) = P ( ) = b and f ( a ) = f ( 1 ) ( a ) = P ( 1 ) ( a ) = P ( a ) = m . The n th Taylor polynomial of f at a is the best approximation to f near a using polynomials of degree at most n . Thomas Scanlon () Math 16b Autumn 2010 5 / 44 Example Compute the third Taylor polynomial of f ( x ) = e x at a = 0. Write P ( x ) = c + c 1 x + c 2 x 2 + c 3 x 3 We need to nd c , c 1 , c 2 , and c 3 so that P ( i ) ( ) = f ( i ) ( ) for i = 0, 1, 2, and 3 In our case f ( i ) ( x ) = e x for all i 0 and e = 1. So, f ( i ) ( ) = 1 for all i . Thomas Scanlon () Math 16b Autumn 2010 6 / 44 Solution, continued P ( x ) = c + c 1 x + c 2 x 2 + c 3 x 3 We compute P ( x ) = c 1 + 2 c 2 x + 3 c 3 x 2 P 00 ( x ) = 2 c 2 + 6 c 3 x P 000 ( x ) = 6 c 3 Thus, 1 = f ( ) ( ) = P ( ) ( ) = c , 1 = f ( 1 ) ( ) = P ( 1 ) ( ) = c 1 , 1 = f ( 2 ) ( ) = P ( 2 ) ( ) = 2 c 2 so that c 2 = 1 2 , and 1 = f ( 3 ) ( ) = P ( 3 ) ( ) = 6 c 3 so that c 3 = 1 6 . Thus, the third Taylor polynomial of f ( x ) = e x at a = 0 is P ( x ) = 1 6 x 3 + 1 2 x 2 + x + 1 Thomas Scanlon () Math 16b Autumn 2010 7 / 44 Graphs of Taylor polynomials of e x Thomas Scanlon () Math 16b Autumn 2010 8 / 44 Another example Problem Find the third Taylor polynomial of f ( x ) = ln ( x ) at a = 1. As before, we write P ( x ) = c + c 1 x + c 2 x 2 + c 3 x 3 We nd P ( x ) = c 1 + 2 c 2 x + 3 c 3 x 2 P 00 ( x ) = 2 c 2 + 6 c 3 x P 000 ( x ) = 6 c 3 Thomas Scanlon () Math 16b Autumn 2010 9 / 44 Solution, continued Di erentiating, f ( x ) = 1 x = x 1 , f 00 ( x ) = x 2 , and f 000 ( x ) = 2 x 3 ....
View
Full
Document
This note was uploaded on 02/08/2011 for the course MATH 16B taught by Professor Sarason during the Fall '06 term at University of California, Berkeley.
 Fall '06
 Sarason
 Calculus, Derivative

Click to edit the document details