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.
Notice that ex = f(x) = f'(x) = f''(x) = ...   We then have :

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...)