00:01
We're given a contour map for a function, f.
00:04
We're asking you this contour map to estimate partial derivatives of f.
00:10
First, we want to estimate the partial derivative of f with respect to x at the point 2 .1.
00:19
Looking at the contour map, well, the nearest point in the positive x direction to 2 .1 is 2 .7 .1.
00:29
So we have, on the one hand, we can approximate f.
00:37
Of x of 2 .1 as about f of 2 .7 minus f of 2 1 over 2 .7 minus 2 ,000, over 2 .7 minus 2, which is equal to 12 minus 10 over 0 .7, which is equal to 20 -7s.
01:02
On the other hand, the nearest point in the negative x direction, for which we know the coordinate and the function of value, would be 1 .151.
01:14
And so another approximation, e2, of f sub x of 2 1 .15, is approximately f of 1 .15, 1 minus f of 2 1, over 1 .15 minus 2, which is 8 minus 10, over 1 .15 minus 10, over 1 .15 ,000, which is 8 minus 10, over, negative 0 .85 and this is equal to 4017s.
01:55
So the best estimate would be to average these two answers.
02:02
So our final estimate of f sub x of 2 1.
02:06
This is going to be about the average of e1 and e2 which we plug this in is about 2 .6.
02:24
We're also asked to estimate f sub y of 2 1.
02:32
Well once again the nearest point in the negative y direction, for which we know both the coordinates and the functional value is 2 0, so we can approximate this on the one hand as e1, which is about f of 2 1 minus f of 2 0 over 1 minus 0 plugging this in, this is 10 minus 12 over 1, which is about negative 2...