Many students have asked me why it is important to for x to be close to zero for the Maclaurin’s expansion to be a good approximation. So here, I plot it in the 4 curves:
y =e^x which is our actual curve.
y = 1+x which is the estimation of e^x, up to and including x.
y = 1+x+\frac{x^2}{w!} which is the estimation of e^x, up to and including x^2.
y = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!} which is the estimation of e^x, up to and including x^3.

Curves and its estimation
Curves and its estimation

We can observe from the graphs, that as we increase the degree of order, the estimated curves become more like that of e^x, although it still tend to deviate a lot. The idea of maclaurin’s is that it provides us a way to interpolate and write the humble e^x equation out in a polynomial that actually never ends. So as we continue to consider more terms, our estimation will get close and closer to the actual curve.

And clearly, we see that when x=0, we found the actual value. This is simply because the maclaurin’s expansion is centred about zero. 🙂

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