(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find f'(c) and the error estimate for:

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

2. Relevant equations

The error is given by:

[tex] E(\Delta x) = \frac {1}{2}M \Delta x [/tex]

and

[tex] f''(c) \leq M [/tex]

3. The attempt at a solution

So the first derivative is:

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

and for the second derivative I got:

[tex] f''(x) = (x^{2}+1)^{-3/2} [/tex]

So for f'(c) I got:

[tex] f'(c) = \frac {2}{\sqrt{5}} [/tex]

which is correct in the back of the book but for the error i got:

[tex] f''(c) = 5^{-3/2} [/tex]

so

[tex] E(\Delta x) = \frac {1}{2} (5)^{-3/2} \Delta x \; \approx \; .04472 \Delta x[/tex]

but according to the book it is:

[tex] E(\Delta x) = \frac {1}{4\sqrt{2}} \Delta x \; \approx \; .17678 \Delta x [/tex]

I have done the second derivative over now a couple of times and still get the same answer but here is how i got f''(x):

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

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

which then boils down to:

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

Next lets combine the terms on the top to get something like:

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

so x^2 will cancel on the top fraction to get:

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

and that can be rewritten as:

[tex] \frac {1}{\sqrt{x^{2}+1}} * \frac {1}{x^{2}+1} [/tex]

which can further be rewritten as:

[tex] (x^{2}+1)^{-1/2} * (x^{2}+1)^{-1} = (x^{2}+1)^{-3/2} [/tex]

so:

[tex] f''(x) = (x^{2}+1)^{-3/2} [/tex]

where did I go wrong?

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# Homework Help: Derivatives and Increments -help, again

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