Derivatives and Graphs

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October 5th 2008, 04:00 PM
Pandora

Derivatives and Graphs

Hi.
I would appreciate some help with a couple of calculus questions I'm battling with. For the first one:

Why is that a positive value at a turning point indicates a minimum for thr second derivative test? An example would be most helpful, as I don't seem to have much information on this.

Also ( and this is a biggie), how would I go about finding any asymptotes, intercepts or anything of interest in this graph

x^3/(x^2-2)

I've sketched the graph, and this confused me even more. Are there 2 vertical asymptotes?

Thanks in advance
October 5th 2008, 04:19 PM
icemanfan

If the second derivative is positive at a point, then the graph is concave up at that point, which means that if the first derivative is zero at the same point, then that point must be a local minimum.

As for $f(x) = \frac{x^3}{x^2 - 2}$ you are correct that there are two vertical asymptotes, $x = -\sqrt{2}$ and $x = \sqrt{2}$ . Also, $y = x$ is an asymptote for this graph.

To find critical points, you have to calculate the derivative, which is given by the quotient rule:

$f'(x) = \frac{(x^2 - 2)(3x^2) - x^3(2x)}{(x^2 - 2)^2}$

$f'(x) = \frac{3x^4 - 6x^2 - 2x^4}{(x^2 - 2)^2}$

$f'(x) = \frac{x^4 - 6x^2}{(x^2 - 2)^2}$

$f'(x) = 0 \implies x^4 - 6x^2 = 0$

$x^4 = 6x^2$

So x = 0 is a solution; if x is not 0, then we have

$x^2 = 6$

$x = \pm \sqrt{6}$

Hence there are three possible critical points.
October 5th 2008, 04:24 PM
iz1hp

and the quotient rule is:

$ \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} $

where

$ f(x) = x^3 $

and

$ g(x) = x^2 - 2 $
October 5th 2008, 04:37 PM
Pandora

.

Wow, thank you for such a clear and fast reply. Brilliant.

I am now faced with some more problems though, which I did not think were too bad until I actually got to them.

I want to row reduce a matrix which I am fine with, but to get it I need to create a system of linear equations of a parabola that goes through 3 different points. How would I go about that?

And finally, I am given 3 points and asked to find an angle between the two lines that they form. Doe this involve a cross product or am I way off base?

Unfortunately for me, you can't do finance without going through calculus first..
October 5th 2008, 04:43 PM
icemanfan

Quote:

Originally Posted by Pandora

I want to row reduce a matrix which I am fine with, but to get it I need to create a system of linear equations of a parabola that goes through 3 different points. How would I go about that?

You need to know what direction the parabola is facing. If you don't know the answer to that, then there could be an infinite number of parabolas passing through those 3 points.