Question

Please help with the explanation questions as well, Thank you! *NOTE: we are discussing functions of 2 variables f(x,y) in this project - NOT functions of 3 variables f(x,y,z). Functions of 3 variables have 4 total variables. In other words, you should not be writing f(x,y,z) anywhere in this project.* EQUATIONS that we are comparing: z = x^2 + y^2 z = (x^2 + y^2)^(1/2) The point (1,0,1) lies on both of these surfaces. Use directional derivatives to find the direction of greatest increase at this point on each surface, and also find the magnitude of the directional derivative in that direction**. In each of your contour maps from Part 1, include a vector with tail located at (x,y) = (1,0) and pointing in the direction of greatest increase. Does the direction of greatest increase for each surface make sense in terms of the contour map? Why or why not? What do the values of the directional derivatives in those directions tell you about the the shape of each surface (if you were to continue in those same directions)?

Answer 1

To find the direction of greatest increase at the point (1,0,1) on the surfaces z = x^2 + y^2 and z = (x^2 + y^2)^(1/2), we need to calculate the directional derivatives in the direction of the unit vector u = <a,b>, where a and b are scalars.

Let's start with the surface z = x^2 + y^2. The gradient of the surface is:

grad(z) = <2x, 2y>

To find the directional derivative of z at (1,0,1) in the direction of u = <a,b>, we take the dot product of the gradient and the unit vector:

D_u(z) = grad(z) · u/|u| = <2a, 2b> · <a^2 + b^2>^(1/2) = 2(a^2 + b^2)^(1/2)

To find the direction of greatest increase, we need to choose the unit vector u that maximizes D_u(z). Since a^2 + b^2 = 1 (since u is a unit vector), we want to maximize 2(1)^(1/2) = 2.

Therefore, the direction of greatest increase at point (1,0,1) for the surface z = x^2 + y^2 is in the direction of the unit vector u = <1,0>. The magnitude of the directional derivative in this direction is 2.

For the surface z = (x^2 + y^2)^(1/2), the gradient of the surface is:

grad(z) = <x/(x^2 + y^2)^(1/2), y/(x^2 + y^2)^(1/2)>

To find the directional derivative of z at (1,0,1) in the direction of u = <a,b>, we take the dot product of the gradient and the unit vector:

D_u(z) = grad(z) · u/|u| = <a/(a^2 + b^2)^(1/2), b/(a^2 + b^2)^(1/2)> · <a^2 + b^2>^(1/2) = a + b(0)

To find the direction of greatest increase, we need to choose the unit vector u that maximizes D_u(z). Since a^2 + b^2 = 1 (since u is a unit vector), we want to maximize a.

Therefore, the direction of greatest increase at point (1,0,1) for the surface z = (x^2 + y^2)^(1/2) is in the direction of the unit vector u = <1,0>. The magnitude of the directional derivative in this direction is 1.

In the contour maps from Part 1, the directions of greatest increase for each surface make sense in terms of the contour lines. The directional derivatives indicate the rate of change of the surface in a particular direction. In the direction of greatest increase, the surface is changing at the fastest rate. In both cases, the direction of greatest increase is perpendicular to the contour lines at the point (1,0,1). This makes sense because the contour lines represent points where the surface has the same value, so the direction of greatest increase would be perpendicular to those lines.

The values of the directional derivatives in those directions tell us about the shape of each surface. For the surface z = x^2 + y^2, the directional derivative in the direction of greatest increase is 2, which means that the surface is changing at a fast rate in that direction. This indicates that the surface is steeply sloped in the direction of greatest increase. For the surface z = (x^2 + y^2)^(1/2), the directional derivative in the direction of greatest increase is 1, which means that the surface is changing at a slower rate in that direction. This indicates that the surface is less steeply sloped, and closer to being flat in the direction of greatest increase.