Let f(x)=e ^ x

Compare e^x and the line tangent to e^x at x=0. Near 0, they look alike, but they don't stay close for very long.

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The tangent line happens to be the 1st degree polynomial which agrees with e^x at x=0, and whose first derivative also agrees with e^x at x=0.

What if we create a 2nd degree polynomial which agrees with e^x at x=0, and whose two first two derivatives match the first two derivatives of e^x at x=0?

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Now look at how well a 6th degree polynomial whose 6 derivatives match the first 6 derivatives of exp(x) at x=0 matches the curve!

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