10. (10 points) Olga Ó. Ølaafsen is the world's toughest...

10. (10 points) Olga Ó. Ølaafsen is the world's toughest mountain climber; she always takes the steepest route up the mountainside. Suppose that she finds herself at the point (1, 3, 67.38) on the surface of Cerro Chirripó, whose elevation above sea level in feet at each point can be modeled (roughly) by $10,000e^{-x^2 - (y-1)^2}$. In what direction should she head to maintain her reputation? N.B.: Chirripó's peak elevation is actually 12,536 feet ? 3,821 m.

Transcript

00:01 For this problem, we are told that the elevation of a mountain above sea level at the point xy is 3 ,000 e to power of negative x squared plus 2y squared over 100 meters.

00:09 So we can say we have f of xy equals 3 ,000 times e to the power of negative x squared plus 2y squared.

00:20 Oops, that's a little bit too sloppy there.

00:23 Let me fix that up.

00:23 X squared plus 2 y squared over 100 meters we have that the positive x -axis points east and the y -axis points north as expected we have that a climber is directly above the point 1010 we are asked then if the climber moves northwest will she ascend or descend and at what slope so if we're moving northwest then we'd be moving in the direction of the unit vector 1 over root 2 or excuse me, it would be negative 1 over root 2, positive 1 over root 2.

01:01 Now, to determine if we are ascending or descending, we want to find the directional derivative of the value of our function, or the, excuse me, the directional derivative of our function, at the point, in this case it's 1010.

01:17 So we need to find the partial derivative, or we want to find, generally speaking, the gradient of our function, which, now let's see here, applying change, rule it would be negative 2x over 100 times 3 ,000 times e to the power of negative x squared plus 2 y squared over 100 so this would then or that would be the x component i should say and the y component would be 4 y squared over 100 which would be 1 over 25 times 3 ,000 times e to the power of or excuse me, that should be negative.

02:03 It would have been negative 4y over 100.

02:06 So then that should be negative y over 25 times 3 ,000.

02:11 E to the power of negative x squared plus 2y squared over 100.

02:18 So evaluating our gradient then at our point 1010, we'll get the gradient here should be equal to negative 600 over ek, cubed minus 1200 over e cubed or we could write this as negative 600 over e cubed times the vector 1 2...

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