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 nth order Taylor polynomial, then 1st, 2nd,…., nth 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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Author: Autar Kaw
Autar Kaw (http://autarkaw.com) is a Professor of Mechanical Engineering at the University of South Florida. He has been at USF since 1987, the same year in which he received his Ph. D. in Engineering Mechanics from Clemson University. He is a recipient of the 2012 U.S. Professor of the Year Award. With major funding from NSF, he is the principal and managing contributor in developing the multiple award-winning online open courseware for an undergraduate course in Numerical Methods. The OpenCourseWare (nm.MathForCollege.com) annually receives 1,000,000+ page views, 1,000,000+ views of the YouTube audiovisual lectures, and 150,000+ page views at the NumericalMethodsGuy blog. His current research interests include engineering education research methods, adaptive learning, open courseware, massive open online courses, flipped classrooms, and learning strategies. He has written four textbooks and 80 refereed technical papers, and his opinion editorials have appeared in the St. Petersburg Times and Tampa Tribune.
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The Numerical Methods Guy 
its was a very good example but only one will not satisfy the student…..sorry but in a way what i wanna say is that i want you guys to post some more examples so that we students can get an idea about how the theorem works….thank you
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plz help me to solve the example! use taylor’s theorem with n=2 to approximate (1+x)3,x>-1
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