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Mathematics
Calculus
Is a function better approximated by a line in some regions?
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[QUOTE="fresh_42, post: 5535010, member: 572553"] If you consider a linear approximation, i.e. by the first derivative, then you have the Taylor series (expanded at a point ##x_0##) written as ##f(x) = f(x_0) + (x-x_0) f'(x_0) + R(x,x_0)## where the term ##R(x,x_0)## is the remainder, that makes the difference. So we have to consider this term. One way to write it, is ##R(x,x_0) = \frac{1}{2} (x-x_0)^2 f''(\zeta)## for a point ##\zeta \in ]x,x_0[##. Now you can choose a small interval around your points ##x_0 = B## and ##x_0 = - \ln A## and calculate the deviation ##|R(x,x_0)|## from linearity. [/QUOTE]
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Forums
Mathematics
Calculus
Is a function better approximated by a line in some regions?
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