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January 17th, 2008, 09:08 PM   #1
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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!
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January 18th, 2008, 05:12 AM   #2
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You wrote (1) + 2c(1) = p'(1); did you mean b + 2c(1) = p'(1)?
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