00:01
We have the cost function c of x equals 44 ,100 plus 700x plus x squared.
00:08
We want to first find the cost production at level one production level 1000.
00:13
So that would simply be c of 1000.
00:17
So we would just plug in 1000 in for x.
00:20
So 44 ,100 plus 700 times 1000 plus 1000 squared.
00:30
That would give us 44 ,100 plus 700 ,000 plus 1 ,000 ,000 and that gives us a total of uh c 1 ,744 ,100.
01:05
So that is our um cost at the production level of 1000.
01:16
Next we want to find the average cost at production level 1000.
01:20
So the average cost the average cost which is usually notated like this would be um that would be found by our c of x over x.
01:40
So that would be um 44 ,100 plus 700x plus x squared over x.
01:55
So that means we would just evaluate that at uh 1000.
02:00
So um if we do that we find our average cost at 1000.
02:10
That would be 44 ,100 plus 700 times 1000 plus 1000 squared over our x of 1000 and that would give us the numerator we'd already plugged in up for part eight.
02:31
So that's going to be 1 ,744 ,100 and we just divide that by 1000 and that gives us an average cost of 1 ,744 .1.
02:51
Next we want to find the marginal cost production at production level 1000.
02:56
So marginal cost would be the derivative of the cost function.
02:59
So we need to find c prime of x.
03:01
So um i'm going to rewrite the c of x first before we find the derivative.
03:08
So we know that c of x is 44 ,100 plus 700x plus x squared.
03:21
So now let's find the derivative of that.
03:25
The derivative of a constant 44 ,100 is just zero plus the derivative of 700x.
03:31
That's a linear function so that would be 700 plus the derivative of x squared.
03:36
We just use the power rule.
03:37
We bring the two down and then we have x.
03:40
We should go down to the first power.
03:42
So our marginal cost function is just 700 plus 2x.
03:47
We want to find that at the production level 1000.
03:49
So that would be c prime of 1000 which would be 700 plus 2 times 1000.
03:58
That would just give us 700 plus 2000 which is 2700.
04:08
And next we want to find the production level that will minimize the average cost.
04:19
So in order to do that we need to first get our average cost function again and that was just the 44 ,100 plus 700x plus x squared over x.
04:42
So we need to find the derivative of this function to find when it will be minimized.
04:47
So let's first simplify this.
04:49
That would be 44 ,100 over x...