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-We approximate a function f(x) by getting a polynomial (I don`t know how we get it, and I don`t know what characteristics it should have, and I`d like to know please)

-when we need more accuracy we add a higher derivatives, but why is adding a higher derivative gives more accuracy? I tried to imagine what a second derivative represents on a graph, and I came out with a result that may be true, If a second derivative is the rate of change of a first derivative then I should draw a new graph where the independent variable is the function that we defferentiated for once, is that true? I think understanding what a second, third, etc... derivatives represent on a graph can make it clear for me to understand how a higher derivative gives more accuracy.

So, to sum up the questions:

1- how do we find the polynomial when we approximate a function?

2- what does a second, third, etc... derivative represent on a graph?

3- why adding higher derivatives add more accuracy to the "function"? -in other words: precisely, whats the relationship between taking higher derivatives and the accuracy of the result of the polynomial?-

Thank you,