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HH_Sec_10_1

HH_Sec_10_1 - 1 Math 10B Lecture Examples Section 10.1 p 2...

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(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
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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 Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/: ‡ Section 10.6: Examples 1–3, 4a ‡ The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course....
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HH_Sec_10_1 - 1 Math 10B Lecture Examples Section 10.1 p 2...

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