If Archimedes were to quote Taylor’s theorem, he would have said, “*Give me the value of the function and the value of all (first, second, and so on) its derivatives at a single point, and I can give you the value of the function at any other point*”.

It is very important to note that the Taylor’s theorem is *not* asking for the expression of the function and its derivatives, just the value of the function and its derivatives at a *single point*.

Now the *fine print*: Yes, all the derivatives have to exist and be continuous between *x *and *x+h, *the point where you are wanting to calculate the function at. However, if you want to calculate the function approximately by using the *n*^{th} order Taylor polynomial, then 1^{st}, 2^{nd},….,* n*^{th} derivatives need to exist and be continuous in the closed interval [*x,x+h*], while the (*n+1*)^{th} derivative needs to exist and be continuous in the open interval (*x,x+h*).

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