Barrons AP Calculus

(a)

(b)

BC  ONLY

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 __

Find    the Taylor  polynomials P 1 ,   P 3 ,   P 5 ,   and P 7     at  x   =   0   for f(x)    =   sin x.

Graph   f   and all four    polynomials in  [−2π,2π]    ×   [−2,2].