00:01
So what we have is a semicircle with a radius of 7.
00:06
You might be familiar with the equation of a circle which is centered at the origin.
00:12
It would be x squared plus y squared equals the radius squared, so 7 squared.
00:17
But i'm going to solve for y in this problem, so we just have the semicircle.
00:22
So if 7 squared is 49, i'm going to subtract x squared over.
00:27
And to get y by itself, i have to square root and minus x squared.
00:34
So now what i can do is i can think about creating a rectangle in here, not a rectangle, where this ordered pair would be xy.
00:44
But if we want the largest area, the x only goes from the origin to the right, so it's actually 2x times y.
00:53
Well, now what i can do is replace the y in this problem with what y is equal to.
01:02
Now instead of writing it as the square root, i'm going to write it as 49 minus x squared to the 1 half power.
01:09
So now when i take the derivative, let's write a prime, it's the product rule of the derivative of the left side.
01:15
It's 2, you leave the right side alone.
01:19
Plus now you leave the left side alone, taking the derivative of the right side is the chain rule.
01:26
49 minus x squared is now to the negative 1 half times the derivative of the inside, which is negative 2x.
01:35
So just to clean this up a little bit, like these twos could cancel, i have 2 root 49 minus x squared.
01:45
I'm going to make this a minus, because it's a minus there.
01:49
2x squared over the square root of 49 minus x squared.
01:55
And i can get the same denominator if i multiply this by the square root of 49 minus x squared over itself.
02:01
So that's actually just going to square that piece...