00:01
We need to find this point on the graph such that the tangent line to the function at that point has an x intercept of 6.
00:17
So we need to find this point where the tangent line is touching the graph, but the tangent line will have an x intercept of 6.
00:29
Now, when you have a line that's tangent to a function, the slope of the tangent line is the derivative of the function.
00:45
So the slope of this tangent line is going to be 3x squared.
00:54
So let's keep that in mind.
00:56
The slope of the tangent line of the line will be 3x squared because the slope of the tangent line is equal to the derivative of the function.
01:09
So the slope of this red line is going to be 3x squared, depending on what x is.
01:14
Now, we're looking for this particular point right here.
01:17
It's going to have a particular x coordinate.
01:21
And of course, the derivative of the function at that point will be three times x squared for this particular x.
01:29
The slope of the tangent line will be three times x squared for that very same value of x.
01:37
Now, another way we can look at slope is to change in y over the change in x.
02:14
Okay, so we know the slope of this tangent line is going to be the derivative of the function at this point.
02:22
Three times x squared for whatever this x value is.
02:26
That's the slope of the tangent line.
02:28
But going back to algebra, you could think of the slope of line as the rise.
02:35
Over in the run.
02:36
Well, this rise is basically the difference between this point here and this point here.
02:44
Well, the y coordinate for this point is x -quute, because that's the function value of this point.
02:53
That's the y -cordant, x -cute.
02:55
And of course, down here, the y -coordinate is zero.
02:57
So for this particular x, the rise is x -cute.
03:10
And the run, how much did you run? when you went from this point right here, the x intercept to this point on the graph where the tangent line is touching the curve, you have your run and you have your rise.
03:24
The rise is going to be x cubed depending on this value of x.
03:27
The run from 6 to x, the run is going to be x minus 6, depending on, of course, whatever this x value is.
03:35
So the run is x minus 6.
03:41
So the slope, the slope of the tangent line is the first derivative 3x squared for this particular value of x.
03:52
But the slope of the tangent line using your rise over run definition is x cubed, that's your rise, over your run, x minus 6.
04:10
Your rise over your run, which is another way to find slope, rise over run, x cubed over x cubed over x.
04:17
Minus six.
04:21
Well, these are both telling you the slope of the red tangent line.
04:26
Well, the red tangent line only has, you know, a slope.
04:30
It's not going to, you know, have different slope because you're finding it different ways.
04:34
So if this is the slope of the red tangent line and this is the slope of the red tangent line, these must be equal to each other...