(9/1/08)Math 10B. Lecture Examples.Section 10.1. Taylor polynomials†Example 1 Find the first-, second-, and th ird-degree Taylor polynomial approximations ofy = ln x, ce ntered at x = 1.Answer: P1(x) = x − 1 • P2(x) = (x − 1) −12(x − 1)2• P3(x) = (x − 1) −12(x − 1)2+13(x − 1)3•Figures A1a, A1b and A1c show y = ln x (the heavy curve) and the Taylor polynomials (the finer curves).x2 4y123x2 4y123x2 4y123P1(x) = x − 1 P2(x) = (x − 1) −12(x − 1)2P3= (x − 1) −12(x − 1)2+13(x − 1)3Figure A1a Figure A1b Figure A1cExample 2 (a) Find the fourth-degree Taylor Polynomial approximation P4(x) of f(x) = excentered at x = 0. (b) How accurately does the polynomial P4(x) from part (a)approximate 9 + exat x = 0.1 and x = 4?Answer: (a) P4(x) = 10 + x +12!x2+13!x3+14!x4• (The graphs of y = 9 + exand y = P4(x) are shownin Figure A2.) (b) |(9 + ex) − P4(x)|.= 8.47 × 10−8at x = 0.1 and.= 20.26 at x = 4x−4 −2 2 4 6y204060y = 9 + exy = P4(x)Figure A2†Lecture notes to accompany Section 10.1 of Ca lculus by Hughes-Hallett et al1Math 10B. Lecture Examples. (9/1/08) Section 10.1, p. 2Example 3 (a) Find the second-degree Taylor polynomial approximation P2(x) of f (x) = x2centered at x = 0. (a) Show that in this case P2(x) = f(x).Answer: (a) P2(x) = 1 + 2(x − 1) + (x − 1)2(b) 1 + 2(x − 1) + (x − 1)2= 1 + 2x − 2 + x2− 2x + 1 = x2Interactive ExamplesWork 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 sectionsof the textbook for the