290 CHAPTER 5 Series and Differential Equations
Such a quadratic function is actually very easy to build: the result
would be that
Q(x) = f(a) +f′(a)(x−a) +f′′(a)
2!(x−a)^2.(The reader should pause here to verify that the above quadratic func-
tion really does have the same first three derivatives asf atx=a.)
This “second-order” approximation is depicted here. Notice the im-
provement over the linear approximation.
In general, we may approximate a function with a polynomialPn(x)
of degreen by insisting that this polynomial have all of its firstn+ 1
derivatives atx=aequal those off:
Pn(a) =f(a), Pn′(a) =f′(a), Pn′′(a) =f′′(a), ···, Pn(n)(a) =f(n)(a),
where, in general,f(k)(x) denotes the k-th derivative off at x. It is
easy to see that the following gives a recipe forPn(x):
Pn(x) =f(a)+f′(a)(x−a)+
f′′(a)
2!(x−a)+f′′′(a)
3!(x−a)^3 +···+f(n)(a)(x−a)n