00:01
All right, so we have this function f.
00:02
We're going to evaluate at this point p.
00:07
And we want the directional derivative to have a maximum value of 64 when it's in the x direction.
00:24
All right.
00:26
So what we're going to need to calculate this directional derivative is some arbitrary unit vector that i'm going to write like this, basically using spherical coordinates.
00:49
All right.
00:50
And so its maximum is when theta is pi over two and phi is.
01:13
And it has that value 64 then.
01:17
All right.
01:19
So here's how we're going to have to do this.
01:21
We have to calculate the directional derivative, write it in terms of theta and phi.
01:26
And then we'll get a scalar function of theta and phi.
01:34
And then we have to take the derivatives of that with respect to theta and phi and set them equal to zero when this is true, which will give us a relationship between a, b, and c.
01:47
And then we also have to take the second derivatives and find, make sure we have a maximum and not a minimum.
01:55
All right.
01:58
So the first step is to find the directional derivative, which means calculate grad f.
02:06
So grad f is i times partial f with respect to x, which is 2ax plus j times 2by plus k times 3c, z squared.
02:26
So that's grad f.
02:32
Then we'll take grad f and we'll evaluate it at our point p.
02:48
So that has x equals 1.
02:50
So it's 2a i plus then y is 2b times j plus and z is minus 3c.
03:12
So it's minus 3c times k.
03:18
Actually, it's plus.
03:20
You square that, you get one.
03:24
So it's plus.
03:27
So that's our directional derivative at our point p.
03:32
That's the total derivative.
03:35
Okay...