Really need some help with Calculus.

[BrandyD2013]

I am a senior in high school, I'm in calculus, and I have this weekly review worksheet with a few problems I am stuck on.
Please try and help me out!

Determine the intervals from -2(pi) to 2(pi) where the function f(x)=(tanx)/(secx) is increasing.

**I do not know how to type the pi symbol, which is why I put (pi) instead.

So I know that I have to solve the equation (tanx)/(secx) to get critical points. I just do not know how to solve this equation. If i could just get this equation solved, I would know what to do from there.

Please help!

 

[MarkFL]

You want to differentiate the given function, not solve. Just a small point regarding terminology.

We can make our job here a whole lot easier by simplifying the function first:

\(\displaystyle f(x)=\dfrac{\tan(x)}{\sec(x)}=\dfrac{\sin(x)}{\cos(x)}\cdot\cos(x)=\sin(x)\)

Now, what do you get when you differentiate this?

 

[BrandyD2013]

I'm not sure, I am a bit lost. But I see what you are doing there. I just don't know how to get my critical points.

 

[MarkFL]

If \(\displaystyle f(x)=\sin(x)\) then what is the derivative?

 

[Deleted member 4993]

I'm not sure, I am a bit lost. But I see what you are doing there. I just don't know how to get my critical points.

What is the definition of critical points of function?

Do you know how does the function sin(x) look like when it is plotted between -2π to 2π?

 

[BrandyD2013]

The derivative of sinx is cosx.

 

[BrandyD2013]

And yes, critical points are points on the function which equal zero.

 

[Deleted member 4993]

And yes, critical points are points on the function which equal zero.

No....

The following will define critical points of the function:

1.	Any value of x that makes f ' (x) = 0.
2.	Any value of x in which f ' (x) does not exist.
3.	If you are working on a closed interval, then the endpoints of the interval are also critical points.
4.	Any point that is either a vertical asymptote or a removable point of discontinuity.

So what are the points within the given domain where function is not differentiable or f'(x) = 0?

Be careful here - your original function tan(x)/sec(x) has "removable" discontinuities at x = ±(2n+1)*π/2

 

[MarkFL]

Yes, you have the correct derivative, and the critical points are found by equating the derivative to zero, i.e.,

\(\displaystyle f'(x)=\cos(x)=0\)

Now with \(\displaystyle -2\pi\le x\le2\pi\) what values of x do we have?

 

[BrandyD2013]

I think I've got it, thank you!