Question 5 In this problem we investigate the error involved in linear approximation. (a) Consider f (3:) = 333. Write the linear approximation of f for as near 2. Let's call this linear approximation L(o:). (b) Dene the error of this linear approximation to be E00) = f (3:) L(:c). Show that E lim ('9') a:>2$2 = 0. (This means that not only is the error small for a: m 2, but that the error is small relative to (:L' 2). It turns out that if f is any function that is differentiable at m = a, and E(m) is the error E in the linear approximation, then lim (:17) xm :1: a, = 0 our concrete example is one case of a far more general fact.) E (c) Calculate lim2 . What is the relationship between this limit and f "(2)? m> (It turns out that if f is twice differentiable at a: = a, then Em _ f\"(a) lim mm (m (1)2 _ 2 Math 140 Written Homework 9: 3.10, 4.1 Page 2 of2 The interpretation is that for at z a, II E(E) m f ((1)0? (1)2 2