00:01
All right, so we are given our demand equation as p equal to 100e to negative 0 .001 x and plus 125.
00:18
So we want to first find the rate of change of the price per bottle, and then that's, we're to find that x equal to 1 ,000, and x equal to 4 ,000.
00:29
So to find d p, dx, we're going to find dp, dx.
00:34
We're going to take the chain rule.
00:37
So we'll have 100, 100e negative 0 .001.
00:47
And times the derivative of the numerator, i mean the exponent to negative 0 .001.
00:56
So simplifying that, we end up with negative.
01:00
So 100 times 0 .001 is 0 .01 and e to negative 0 .001 x.
01:15
So we want to find dp dx at a thousand for x to negative 0 .01 e times a thousand.
01:33
So let's toss out to a calculator negative point zero one e and negative point zero one e and negative point zero one times 1 ,000 so we're going to end up with negative 9 .04 times 10 to negative negative 0 .00904 and then add 4 ,000, pegging in 4 ,000 into this spot...