00:01
So if we want to find a tangent line at this intersection between these two curves at this point over here, the first thing we need to do is actually come up with some kind of parametric equation.
00:14
So what i'm going to do, since these are cylinders, i'm going to come over here and first use the fact that if we said x is equal to 5 cosine theta, and then y is equal to 5 sine theta this would parameterize this over here and i guess you could use t if you want but i normally just use this so we have that now we need to solve for z so what we can do is take this come over here plug it in for y and then just solve for z so that is going to give us 25 5 sine squared theta and then plus z squared is equal to 20 so we subtract square root so we get z is equal to first plus or minus the square root of 20 minus 25 sine square of theta and we're going to only use the positive root for this since z is supposed to be 2 if this was negative, then we would want to take the negative root.
01:30
But in this case, we're just going to keep it as the positive root.
01:36
So this is what we have so far.
01:38
So we have some parametric equation for these intersections.
01:44
So the intersection, we can define as r of theta being 5 cosine theta, 5 sine theta, and then the square root of 20 minus 25 sine squared theta.
02:12
So this is going to be our parametric equation.
02:18
Now what we need to do is figure out what is theta going to be at, and then we can take the derivative and kind of go from there.
02:26
So i'm just going to go ahead and use.
02:31
The fact that we know x is supposed to be equal to three at this point so we're going to have three is equal to five cosine theta we can go ahead and take arc or divide by five and then take cosine inverse on each side so we're going to get theta is equal to cosine inverse of three fifths right uh and i guess just kind of simplicity since we'll need to plug this into sign as well i'll go ahead and plug that into there also just so we don't have to work as hard later and think okay what is sign of cosine inverse so this is going to be or for the y is equal to five sine of theta so move things around and we'd get theta is equal to sine inverse of four whips so we'll use both of those later on but in the meantime, we need to go ahead and find our derivative of this.
03:43
So it would be r prime of theta is going to be equal to.
03:50
So the derivative of this is just 5 sine theta.
03:59
The derivative of 5 sine theta is 5 cosine theta.
04:04
And then to take the derivative of this, we're going to have to work a little bit harder.
04:10
So we have to use power rules.
04:12
So it would be one -half, 20 minus 25, sine squared theta.
04:21
Now this would be to the negative one -half power.
04:24
Then we have to take the derivative on the inside.
04:27
So the derivative of 20 is just going to be zero.
04:30
And then it would be minus 25.
04:34
And then we'd use chain rule.
04:36
So it would be 2 sine of theta.
04:39
Then we take the derivative of a sine, which is going to be cosine, and we multiply it...