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 January 17th, 2008, 10:08 PM #1 Newbie   Joined: Jan 2008 Posts: 1 Thanks: 0 linearization I am approximating functions using taylor series, but I am confused on how to derive it. So say I can use a degree 1 function p(x) to approximate ln(x) at 1. I want p(1) = ln(1) p'(1) = ln'(1) I can get a line by definition of slope p(x) = 0 + (x-1) But for degree 2 it gets complicated for me. p(1) = ln(1) p'(1) = ln'(1) p''(1) = ln''(1) Now I know the formula is p(x) = 0 + (x-1) + 1/2(x-1)^2 but I am not sure how to derive it besides the power series. I want to think about it in terms of all the derivatives at a point equaling the derivatives of the function. So I set up this system a + b(1) + c(1)^2 = p(1) (1) + 2c(1) = p'(1) 2 = p''(1) But this doesn't seem to be working. How can I derive the formula for taylor series without thinking of power series, because I am kind of confused on where the (x-a)^n/n! exactly comes from. Thank you very much!
 January 18th, 2008, 06:12 AM #2 Global Moderator   Joined: Dec 2006 Posts: 17,211 Thanks: 1291 You wrote (1) + 2c(1) = p'(1); did you mean b + 2c(1) = p'(1)?

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