Question

You are standing above the point (1,2) on the surface z=25−(3x2+3y2). (a) In which direction should you walk to descend fastest? (Give your answer as a unit 2-vector.) direction = (b) If you start to move in this direction, what is the slope of your path? slope =

Answer 1

2

Explanation:

To determine the direction in which you should walk to descend fastest, we need to find the gradient vector of the surface z = 25 - (3x^2 + 3y^2) at the point (1, 2).

The gradient vector (∇f) of a function f(x, y) represents the direction of steepest ascent or descent at a given point. It is given by (∂f/∂x, ∂f/∂y).

Let's find the partial derivatives of the surface equation with respect to x and y:

∂f/∂x = -6x

∂f/∂y = -6y

Now, substitute the coordinates of the point (1, 2) into the partial derivatives:

∂f/∂x = -6(1) = -6

∂f/∂y = -6(2) = -12

Therefore, the gradient vector (∇f) at the point (1, 2) is (-6, -12).

To descend fastest, you should move in the opposite direction of the gradient vector.

So, the direction in which you should walk to descend fastest is the negative of the gradient vector: (6, 12).

Next, let's calculate the slope of your path if you start moving in this direction.

The slope of your path is equal to the negative of the ratio of the y-component to the x-component of the gradient vector.

slope = - (-12/6) = 2

Therefore, the slope of your path is 2.