Find a constant given an equation, it's derivative value...

[Colin67]

Some advice on how to proceed please.

The question is:




What I have done is find dy/dx as 2acosaxsinax (it could be written as asin2ax as well). Now I am stuck, how do I solve given a is a separate constant and a part of the angles?

 

[MarkFL]

Let's look at what you've found:

[MATH]\d{y}{x}=a\sin(2ax)[/MATH]
Now, we're told to let:

[MATH]\d{y}{x}=1[/MATH]
[MATH]x=\frac{\pi}{4}[/MATH]
And so we have:

[MATH]1=a\sin\left(2a\frac{\pi}{4}\right)[/MATH]
[MATH]1=a\sin\left(a\frac{\pi}{2}\right)[/MATH]
At this point, we should recognize that if \(a\in\mathbb{Z}\) then \(a\) must be odd (if it's even the sine function is zero), so let:

[MATH]a=2k+1[/MATH] where \(k\in\mathbb{Z}\)

[MATH]1=(2k+1)\sin\left((2k+1)\frac{\pi}{2}\right)[/MATH]
Now, the sine function is only going to be \(\pm1\) and so the coefficient must be the same value. What can we then conclude about \(a\)?

 

[Colin67]

It must be +1 or -1

 

[MarkFL]

It must be +1 or -1

That was my conclusion as well, although there may be a more satisfying analysis that will be posted.

 

[Colin67]

Thank you so much

 

[topsquark]

What I have done is find dy/dx as 2acosaxsinax (it could be written as asin2ax as well). Now I am stuck, how do I solve given a is a separate constant and a part of the angles?

2acosaxsinax really needs to be 2a cos(ax) sin(ax). Please use parentheses and spaces for clarity,

-Dan