(1 pt) You are standing above the point (5,5) on the surface z=20-(x^2+y^2).?

Question

You are standing above the point (5,5) on the surface defined by the equation z = 20 - (x² + y²).
(a) In which direction should you walk to descend the fastest? Please provide your answer as a unit 2-vector. 
direction = 
(b) If you begin to move in this direction, what is the slope of your path? 
slope =

Anonymous · Accepted Answer

(a)Grad f = <2x, 2y> = <10, 10> at point (5, 5)

The fastest descent would point the opposite direction,

which is <-10, -10>

You start at the point (5, 5). You should walk toward <-15, -15>

I am not sure what you meant by unit 2 vector.

The unit direction vector would be < – sqrt(2)/2, – sqrt(2)/2>

(b)the slope of the path would be

Sqrt(10^2 + 10^2) = 10sqrt(2)

Malachi Hackett III · Answer

a) To find the fast ascend you find the gradient, so therefore to find the fastest descend is simply the gradients opposite. Start by finding the gradient of z.

The partial with respect to x which is -2x and the partial with respect to -2y which gives us the vector <-2x,-2y>

Then plug in the point getting <2(5),2(5)> which is the fastest ascend making the fastest descend <-10,-10>

Now we want to put this into unit vector form so divide the vector by its length (-10^2+-10^2)^1/2 = square root of 200 so the answer for part a is <-10/sqr(200),-10/sqr(200)>

b) The slope is the negative magnitude of the gradient, aka the negative of the found length so

b) slope =-sqr(200)

Anonymous · Answer

The outside equation has the shape f(x,y,z) = c. Any tangent airplane parallel to -4x+3y+4z = -9 have to obey the equation: grad f = k <-4, 3, four> So <2x, 6y, 2z> = ok<-4, three, four> x= -2k y = okay/2 z = 2k This point is on the surface, so satisfies the equation. 4k2 + three/4 k2 + 4k2 = 1 35/four k2 = 1 k2 = four/35 ok = +- 2/sqr(35) the two elements are (-four/sqr(35), 1/sqr(35), four/sqr(35)) and (four/sqr(35), -1/sqr(35), -four/sqr(35))

Amie Lindgren DDS

(1 pt) You are standing above the point (5,5) on the surface z=20-(x^2+y^2).?

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