Hi, I have three questions about the application of quadratic approximation, what it is & when to use it. It ties in with a question about linear approximation also, I'll give an example first of what I'm talking about, just for you to evaluate if I'm wrong in the way I see the whole process, I just worry & want to get the logic right you know(adsbygoogle = window.adsbygoogle || []).push({});

So, the formula for linear approximation is [tex]f(x) = f'(x_0)(x - x_0) + f(x_0)[/tex]. I see a relationship to the slope formula, [tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex] (via algebra you obtain the linear approximation, & the slope is an infinitesimal which is logical) and this explains to me why the linear approximation will approximate a function about a certain point and not be so accurate as you go further and further from [tex]x_0[/tex]

Also, from my understanding of this concept, it is a method to approximate a function about a point, say you want to approximate [tex]\sqrt{9.1}[/tex] you follow the above framework, setting x = 9 (a known function close by).

[tex]f(x) = \sqrt{x}[/tex]

[tex]f'(x) = \frac{1}{2\sqrt{x}}[/tex]

[tex]f(x) = f'(x_0)(x - x_0) + f(x_0)[/tex]

[tex]f(x) = \frac{1}{2\sqrt{9}}(x - 9) + \sqrt{9}[/tex]

[tex]f(x) = \frac{1}{6}}(x - 9) + 3[/tex]

[tex]f(x) = \frac{1}{6}}x - \frac{9}{6}} + \frac{18}{6}}[/tex]

[tex]f(x) = \frac{1}{6}}x + \frac{9}{6}} [/tex]

1.The calculator matches the answer here very closely an all is good.How would quadratic approximation fit in here?Does quadratic approximation have any relevance to this equation at all? Am I applying it where it doesn't have any meaning.I only heard about it yesterday & my intuition tells me that it's just a means of getting a more exact answer, is this correct?

2.Where does the derivation for the quadratic approximation come from? The way I understand the linear approx is coming from the slope formula & point-slope formula but I see no way a quadratic fits in.

3.What is the deal with the formula [tex](1 + x)^{r} = 1 + rx[/tex]? Using this type of formulation a teacher in an mit lecture finds a linear approximation & then he gets quadratic factors in the answer and then he tells the class tothrow them away.http://www.youtube.com/watch?v=BSAA0akmPEU&feature=SeriesPlayList&p=590CCC2BC5AF3BC1 (27th minute). What I wonder is why use this when the normal linear approximation works fine, why memorize unnecessary baggage?

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# Quadratic Approximation et al.

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