SECTION 5.4 Polynomial Approximations 289
Notice that as long asxdoes not move too far away from the pointa,
then the above approximation is pretty good.
Let’s take a second look at the above. Note that in approximating
a functionf by alinear functionLnear the point a, then
(i) The graph of Lwill pass through the point (a,f(a)), i.e., L(a) =
f(a), and(ii) The slope of the liney=L(x) will be the same as the derivative
off atx=a, i.e.,L′(a) =f′(a).That is the say, the “best” linear functionLto use in approximatingf
nearais one whose 0-th and first derivatives atx=aare the same as
forf:
L(a) = f(a) and L′(a) = f′(a).So what if instead of using a straight line to approximatefwe were
to use a quadratic function Q? What, then, would be the natural
requirements? Well, in analogy with what was said above we would
requirefandQto have the same first three derivatives (0-th, first, and
second) atx=a:
Q(a) = f(a), Q′(a) = f′(a), and Q′′(a) = f′′(a).