00:01
In this question, we want to find the direction in which the maximum rate of change occurs for the function f of x, y equals 4x times the sine of x, y at the point 4, 2 will give our answer as a unit vector.
00:13
So the maximum rate of change occurs in the direction of the gradient vector.
00:19
So that's what i really need here, the gradient of f.
00:22
So first i'll need my partial of f with respect to x.
00:26
That will require the product rule.
00:28
So first, 4x, derivative of second is cosine of x, y times y by the chain rule plus my second factor, sine of x, y times the derivative of the first, which is 4.
00:50
Then how about my partial with respect to y? that will be, well 4x is now a constant, so that comes along, times the cosine of x, y times x.
01:05
And so now i'm going to go ahead and evaluate this at 4, 2.
01:13
So what am i getting? i'm getting 16 times the cosine of, 4 times 2 is 8, times y is 2, plus 4 sine of 8, and then in my second component, 4x times x, 4 times 4 times 4, 16 times 4 is 64, cosine of 8.
01:40
And now i'm going to head to my calculator.
01:46
I'm getting 32 cosine 8 plus 4 sine of 8, negative .6986, negative .6986, that appears to be.
02:08
And then i'm getting 64 cosine of 8, negative 9 .312, negative 9 .312, 0...