HH_Sec_10_1

HH_Sec_10_1 - 1 Math 10B. Lecture Examples. (9/1/08)...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
(9/1/08) Math 10B. Lecture Examples. Section 10.1. Taylor polynomials Example 1 Find the frst-, second-, and third-degree Taylor polynomial approximations o± y = ln x , centered at x = 1. Answer: P 1 ( x ) = x - 1 P 2 ( x ) = ( x - 1) - 1 2 ( x - 1) 2 P 3 ( x ) = ( x - 1) - 1 2 ( x - 1) 2 + 1 3 ( x - 1) 3 Figures A1a, A1b and A1c show y = ln x (the heavy curve) and the Taylor polynomials (the fner curves). x 2 4 y 1 2 3 x 2 4 y 1 2 3 x 2 4 y 1 2 3 P 1 ( x ) = x - 1 P 2 ( x ) = ( x - 1) - 1 2 ( x - 1) 2 P 3 = ( x - 1) - 1 2 ( x - 1) 2 + 1 3 ( x - 1) 3 Figure A1a Figure A1b Figure A1c Example 2 (a) Find the ±ourth-degree Taylor Polynomial approximation P 4 ( x ) o± f ( x ) = e x centered at x = 0. (b) How accurately does the polynomial P 4 ( x ) ±rom part (a) approximate 9 + e x at x = 0 . 1 and x = 4? Answer: (a) P 4 ( x ) = 10 + x + 1 2! x 2 + 1 3! x 3 + 1 4! x 4 (The graphs o± y = 9 + e x and y = P 4 ( x ) are shown in Figure A2.) (b) | (9 + e x ) - P 4 ( x ) | . = 8 . 47 × 10 - 8 at x = 0 . 1 and . = 20 . 26 at x = 4 x - 4 - 2 2 4 6 y 20 40 60 y = 9 + e x y = P 4 ( x ) Figure A2 Lecture notes to accompany Section 10.1 o± Calculus by Hughes-Hallett et al
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Math 10B. Lecture Examples. (9/1/08) Section 10.1, p. 2 Example 3 (a) Find the second-degree Taylor polynomial approximation P 2 ( x ) of f ( x ) = x 2 centered at x = 0. (a) Show that in this case P 2 ( x ) = f ( x ). Answer: (a) P 2 ( x ) = 1 + 2( x-1) + ( x-1) 2 (b) 1 + 2( x-1) + ( x-1) 2 = 1 + 2 x-2 + x 2-2 x + 1 = x 2 Interactive Examples Work the following Interactive Examples on Shenks web page, http//www.math.ucsd.edu/ashenk/: Section 10.6: Examples 13, 4a The chapter and section numbers on Shenks web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course....
View Full Document

Page1 / 2

HH_Sec_10_1 - 1 Math 10B. Lecture Examples. (9/1/08)...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online