II. Where will the formulas come from ?
When we used the trapezoidal rule, we
approximated a given function f(x) by degree-one polynomials (straight
lines) which went through two points on a subinterval. When we used
Simpson's rule, we approximated f(x) by degree-two polynomials
(parabolas) which went through three points on a subinterval (of size
(2)(x)). We could proceed in this
fashion (which may be what is being done in your calculator) and get
polynomials of high degree which approximate a given function quite
well. But the idea of Taylor series takes a different tack:
We will (Section 10) concentrate on one
point, say x = 0, and try to find polynomials which match the
derivatives of the function at that point. We will call the
polynomials T(x) rather than P(x) since they will be the
beginnings of the Taylor series.
For example, consider again f(x) = ex and suppose we let
T0(x)=1, | T1(x)=1+x, | T2(x)=1+x+(x2/2!), | for all x. |
T0(0) = 1 = f(0) | ||||
T1(0) = 1 = f(0) | and | T'1(0) = 1 = f'(0) | ||
T2(0) = 1 = f(0) | and | T'2(0) = 1 = f'(0) | and | T''2(0)= 1 = f''(0) |
The result is that T0 goes through
the same height as f(x) = ex
when x=0. Then T1(x) not only goes through
the same height, but has the same slope at 0 as ex. And T2(x) not
only has the same height and the same slope, but it has the same
second derivative, so it has the same concavity. The graphs
look like this:
If there is some way to continue on forever, then, in the limit, our
infinitely long ``polynomial'' might equal ex.
(Continued...)