differentiation: largest rect. inscribed in semicircle

[Sophie]

I am working on worded problems (optimization problems) and I am making the same mistake again and again with my differentiation. I am getting very frustrated, if someone could explain the following I would be very very greatful.

Question. Find the area of the largest rectangle that can be inscribed in a semicircle of radius r.

A=2xy

y=Sqrt((r^2)-(x^2))

A(x)=2x(Sqrt((r^2)-(x^2)))

finding A'(x) is my problem

This is what I want to do

A'(x) = (2(Sqrt((r^2)-(x^2))) - ((4x^2)/(Sqrt((r^2)-(x^2))))

I have the worked example and know that A'(x) is:

A'(x) = (2(Sqrt((r^2)-(x^2))) - ((2x^2)/(Sqrt((r^2)-(x^2))))

But why I thought when you differentiate the inside if the bracket (Sqrt((r^2)-(x^2))) you would get 2x, which you would then multiply by the 2x to get 4x^4 and not 2x^2.

Please can someone explain what I am doing worng as I am doing this in every question and therefore geting them wrong, which is infuriating when the rest of my methos is correct.

Thanks Sophie

 

[stapel]

Did you forget that "the square root of" is "to the one-half power", so there should be a "1/2" in front when you differentiate the radical...?

Cancelling the 2 in the denominator with the 4 in the numerator would get you the book's result.

Eliz.

 

[Sophie]

Hello Eliz

Thanks for the reply, but I am still confused

I thought I had to differentiate the inside of the bracket after differentiating the outside and if that was the case I would get my answer. Am I wrong here?

Thanks Sophie

 

[galactus]

Hey Sophie:

Let the area of the rectangle be A=xy

Label the diagram as such:

Therefore, \(\displaystyle \L\\(\frac{x}{2})^{2}+y^{2}=R^{2}\)

Solve for \(\displaystyle \L\\y=\frac{1}{2}\sqrt{4R^{2}-x^{2}}\)

Now, sub into A=xy: \(\displaystyle \L\\\frac{1}{2}x\sqrt{4R^{2}-x^{2}}\). See?.

Now, differentiate:

Use the product rule and get:

\(\displaystyle \L\\\frac{\sqrt{4R^{2}-x^{2}}}{2}-\frac{x^{2}}{2\sqrt{4R^{2}-x^{2}}}\)

Find a common denominator and get:

\(\displaystyle \L\\\frac{2R^{2}-x^{2}}{\sqrt{4R^{2}-x^{2}}}\)

Now, just set the numerator equal to 0 and solve for x.

 

[stapel]

I thought I had to differentiate the inside of the bracket after differentiating the outside....

Yes. But did you remember to account for the 1/2 power when you were doing this "outside" differentiation?

You haven't shown your work, so we cannot know the source of the error. But the book's answer is correct, and experience suggests the 1/2 as being the source of the error.

Please check your work and, if you did account for the 1/2, please reply showing all of your steps, so we can attempt to find the other source of this error.

Thank you.

Eliz.

 

[Sophie]

Thank you soooo much, I was just writing back again with more working but in doing this I finally see it. I can not belive I made such a silly mistake, again and again and again in all my work. That one mistake has given me hours of unessessary greif on so many questions. Such a silly silly mistake...

Thanks Sophie