Gods Gift To Calculators: The Taylor Series

Part I of Essay
It is incredible how far calculators have come since my parents were in college, which was when the square root key came out. Calculators since then have evolved into machines that can take natural logarithms, sines, cosines, arcsines, and so on. The funny thing is that ....

Part II of Essay
.... the graph below. We'll start out trying to approximate function values near x=0. To do this we start out using the lowest order polynomial, f0(x)=a0, that passes through the y-intercept of the graph (0,f(0)). So f(0)=ao. Next, we see that the graph of f1(x)= a0 + a1x will also pass through x= 0, and will have the same slope as f(x) if we let a0=f1(0). Now, if we want to get a better polynomial approximation for this function, which we do of course, we must make a few generalizations. First, we let the polynomial fn(x)= a0 + a1x + ....