00:01
Okay, so we have here the total cost function, which is a function of x, and x represents the number of hundreds of units produced of whatever product.
00:11
Okay, and what we want to know is how many units will minimise average cost.
00:18
Okay, so first of all let's find out what average cost is.
00:22
So our average cost, c of x, is going to be the cost divided by x itself.
00:28
So it's going to be 300 times 0 .02 x plus 2 all cubed divided by x.
00:42
Now we're going to differentiate this, and there's a few different ways to do it.
00:46
We could use the quotient rule, and we would have to use the chain rule, or we could expand out the numerator, and then the differentiation is a little bit simpler.
00:58
So let's go ahead and do that.
01:00
Okay, so if we expand out the cubed term on top, we end up with a 1 over 1, 2, 5, 0, 0, 0, x cubed plus 0 .0024 x squared plus 0 .24 x plus 8.
01:28
Okay, and that's all divided by x.
01:33
And now we can multiply through by 300, and we're going to get 0 .0024 x cubed plus 0 .72 x squared plus 72 x plus 2400, and that is all divided by x.
02:03
And so finally we can write this as 0 .0024 x squared plus 0 .72 x plus 72 plus 2400 x to the minus 1.
02:21
Okay, and now we want to differentiate our average cost function with respect to x.
02:29
So 0 .0024, we need to multiply that by 2, so we're going to get a 0 .0048 x and then plus 0 .72, the x just differentiates to 1...