Taylor Polynomials A graphical introduction

Best first order (linear) approximation at x=0. OZ calls this straight line function P 1 (x). Note: f (0)=P 1 (0) and f ’ (0) = P ’ 1 (0). Approximating

Best second order (quadratic) approximation at x=0. OZ calls this quadratic function P 2 (x). Note: f (0)=P 2 (0), f ’ (0) = P ’ 2 (0), and f ‘’ (0) = P ’’ 2 (0). Approximating

Best third order (cubic) approximation at x=0. OZ calls this cubic function P 3 (x). Note: f (0)=P 3 (0), f ’ (0) = P ’ 3 (0), f ’ (0) = P ’’ 3 (0), and f ‘’’ (0) = P ’'’ 3 (0). Approximating

Best sixth order approximation at x=0. OZ calls this function P 6 (x). P 6 “matches” the value of f and its first six derivatives at x = 0. Approximating

Best eighth order approximation at x=0. OZ calls this function P 8 (x). P 8 “matches” the value of f and its first eight derivatives at x = 0. Approximating

Best tenth order approximation at x=0. This function is P 10 (x). Approximating

Best hundredth order approximation at x=0. This function is P 100 (x). Notice that we cannot see any difference between f and P 100 on the interval [-3,3]. Approximating

Best hundredth order approximation at x=0. This function is P 100 (x). But what about [-6,6]? Approximating

Best hundredth order approximation at x=0. This function is P 100 (x). But what about [-6,6]? Approximating

Compare Different Centers Third order approximation at x=0 Third order approximation at x = -1

Taylor Polynomial Approximations Derivative matching as a means to good and better approximation. Can we find (Taylor) polynomials that do what we want? Approximating closely related functions by similarly related polynomials. Three Themes