In this discussion, we saw that increasing a made the tangent a worse approximate of the curve f(x). Furthermore, we saw that increasing a will increase f''(0) but not f'(0). Hence, the reason larger a worsened the tangent's approximation of f(x) is because even though our tangent accounts for f', it does not account for f''.

We used the instantaneous slope to approximate nearby points with tangent lines, but we know slopes change. Moreover, since we have learned concavity, we know how to study how the slope of a curve is changing at a point. If we can capture how the slope will change as x moves away from 0, our approximation will surely improve.

First, we approximate f' with the line tangent to the curve at x=0:

y=f^{\prime}\left(0\right)+f^{\prime\prime}\left(0\right)x.

In our original tangent line, we simply approximated slope as f'(0). We integrate our improved slope approximation to yield a better line of approximation L for f near x=0:

L\left(x\right)=\int y\,\mathrm{d}x=\int_{}^{}\left(f^{\prime}\left(0\right)+f^{\prime\prime}\left(0\right)x\right)\differentialD x=f^{\prime}\left(0\right)x+\frac12f^{\prime\prime}\left(0\right)x^2+C.

Finally, we find C by checking L(0):

L\left(0\right)=C.

Hence, C=f(0) because we want L to intersect with f at x=0. Thus, L is given by

L\left(x\right)=f\left(0\right)+f^{\prime}\left(0\right)x+\frac12f^{\prime\prime}\left(0\right)x^2.

As you can see, L(x) is a much closer approximate of f(x) near x=0 than the original tangent.

Maclaurin Series Basics