00:02
The situation here is that the crystal is in the shape of a cube, and we're interested in finding the rate of change of the volume with respect to x when x is 3.
00:12
So the volume of a cube is x cubed, and the derivative of that, dvdx, would be 3x squared.
00:21
So then dvdx for x equals 3, we're going to substitute a 3 in there for x, and we get 3 times 3 squared, and that would be 27.
00:31
And the units on volume would be millimeters cubed, and the units on x would be millimeters.
00:41
So what this is telling us is that at this particular size of cube, the volume is growing at a rate of 27 cubic millimeters for every millimeter change in x.
00:58
And in part b of this problem, we're looking at surface area.
01:01
So to get the surface area of a cube, you're going to add the area of every surface.
01:05
And for this cube, every surface has an area of x squared, and there are six of those.
01:10
So the surface area is 6x squared.
01:13
Notice that the derivative of the volume is half of the surface area.
01:18
Now let's take a look at that from a geometric perspective.
01:22
So suppose that the cube grows bigger, and the side length is now delta x.
01:29
Or the side length is now x plus delta x.
01:32
So we added delta x to each side...