Thread: Find if function is increasing or decreasing (Find the derivative)

1. Find if function is increasing or decreasing (Find the derivative)

$ne^{-\frac{n^2}{8}}$

$ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}}$

with quotient rule

$\frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}$

$= \frac{(e^{\frac{n^2}{8}})-(n)(\frac{n}{4})}{(e^{\frac{n^2}{8}})^2}$

$= \frac{1 - \frac{n^2}{4}}{e^\frac{n^2}{8}}$

What can I do next? Or can I assume the function is decreasing now?

2. Re: Find if function is increasing or decreasing (Find the derivative)

Originally Posted by moonman
$ne^{-\frac{n^2}{8}}$

$ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}}$

with quotient rule

$\frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}$

$= \frac{(e^{\frac{n^2}{8}})-(n)(\frac{n}{4})}{(e^{\frac{n^2}{8}})^2}$

$= \frac{1 - \frac{n^2}{4}}{e^\frac{n^2}{8}}$

What can I do next? Or can I assume the function is decreasing now?
... what if $|n| < 2$ ?

3. Re: Find if function is increasing or decreasing (Find the derivative)

Originally Posted by skeeter
... what if $|n| < 2$ ?
If $|n| < 2$ then the function would be positive...so then then the function is positive?

4. Re: Find if function is increasing or decreasing (Find the derivative)

no ... what does $f' > 0$ tell you about $f$ ?

5. Re: Find if function is increasing or decreasing (Find the derivative)

Originally Posted by skeeter
no ... what does $f' > 0$ tell you about $f$ ?
That tells me that f is increasing?

6. Re: Find if function is increasing or decreasing (Find the derivative)

The derivative of the function vanishes at n = 2, -2? This function cannot be monotonic. Looks to me that it has points of extrema. It can only be locally increasing or decreasing.

7. Re: Find if function is increasing or decreasing (Find the derivative)

So can the Integral Test be performed on this if it were a series?

8. Re: Find if function is increasing or decreasing (Find the derivative)

Ok, I found an error in differentiation.

9. Re: Find if function is increasing or decreasing (Find the derivative)

$ne^{-\frac{n^2}{8}}$

$ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}}$

with quotient rule

$\frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}$

$= \frac{(e^{\frac{n^2}{8}})-(n)(e^\frac{n^2}{8})(\frac{n}{4})}{(e^{\frac{n^2}{ 8}})^2}$

$=\frac{1-\frac{n^2}{4}}{1}$ always negative
Correct? Can someone please check the derivative?

10. Re: Find if function is increasing or decreasing (Find the derivative)

you had it correct the first time ...

$\frac{d}{dx} \left[x \cdot e^{-\frac{x^2}{8}}\right]$

$x\left(-\frac{x}{4}\right)e^{-\frac{x^2}{8}} + e^{-\frac{x^2}{8}}$

$e^{-\frac{x^2}{8}}\left[1 - \frac{x^2}{4}\right]$

the derivative is negative for $|x| > 2$ and positive for $|x| < 2$