This approximation of e−0.25 is correct to four places.In Figure N10–2 we see the graphs of f (x) and of the Taylor polynomials:
FIGURE N10–2
Notice how closely P 4 (x) hugs f (x) even as x approaches 1. Since the series can be shown to
converge for x > 0 by the Alternating Series Test, the error in P 4 (x) is less than the magnitude of the
first omitted term, at x = 1. In fact, P 4 (1) = 0.375 to three decimal places, close to e−1 ≈
0.368.
EXAMPLE 45
(a) Find the Taylor polynomials P 1 , P 3 , P 5 , and P 7 at x = 0 for f (x) = sin x.
(b) Graph f and all four polynomials in [−2π,2π] × [−2,2].
(c) Approximate sin using each of the four polynomials.
SOLUTIONS:
(a) The derivatives of the sine function at 0 are given by the following table: