( a) use the formula for Taylor Polynomials, find the Taylor Polynomial of degree 2 centered at a = 8 for the function f ( x ) = 35x. (b ) Approximate the value of 3 9 using your degree 2 polynomial from part ( a ) . ( C) Using Taylor's inequality, find an upper bound for the error in your approximation in part (b ).11.2.1 Taylor Series Definition Now that we have discussed the concept of series, we can return to our discussion on Taylor polynomials of infinite degree. Recall that a Taylor polynomial for f(x) at a matches all derivatives of f at a. If we take the limit as the number of terms in the polynomial approaches infinity, we obtain a special type of power series, called Taylor Series where the Cn's are given by on = f ( n ) ( a ) n! The Taylor Series of f at a is given by E f ( n) ( a ) ( x - a) n = f(a) (x - a )0 + f'( a ) f"(a) n! 11 ( x - a)1 f"(a) 2! (2 - a ) 2 + 3! (x - a) 3+... n=0 In the special case that a = 0, this series is known as the Maclaurin Series of f f (n) (0) an = f (0 ) + f'(0) f"(0) 202 f''(0) + + ... n! 1! 2! 3! n= 0 Recall that a power series for function f at a is of the form Ecn(20 - a)n n=012.0.1 Approximating Functions with Taylor Series We will use the nth degree Taylor polynomial, Tn(m), of a function f (2:) to approximate the function itself. This is especially useful for approximating diicult function values, denite integrals or limits. We would like to get an estimate for the remainder Rn($), where Rn($) = f0?) TN?) Then, if we approximate f(;r0) by Tn(:co) for 330 near a, we would like to gure out how big is an($0)| = \"(9:0) Tn(;r0)|? If the remainder is close to 0, then the Taylor polynomial will be a good approximation for the function itself. Taylor's Inequality Suppose that f is n + 1 times differentiable and let Rn(;r) denote the difference between f ((12) and the Taylor polynomial of degree n for f(ac) centered at (L. Then if | f (\"+1)(c)| S M for all 0 between a and x, the error RH (29) = f(m) i Tn(1?) satises the inequality M n+1 (n + 1)! I IRWB)' S |ma To use the Taylor's Inequality to bound the error in the approximation an) m Tn(:1;), we do the following: 1. Find an upper bound M on the absolute value of the 'n + lst derivative of f between a (the center) and m (the value you want to sub into your function and approximate). M (71+ 1)! 2. The error IRnr)' is at most |:E * 04\"\