Question

Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g represents the direction in which g increases the fastest. Notice that this is the direction in the xy plane corresponding to the steepest slope up the surface, with magnitude equal to the slope in that direction.

1. At the point (2, π), find the gradient, and explain what it means.

2. Use it to construct a vector in the tangent plane which corresponds to the steepest direction up. (Hint: the vector will have the same x and y coordinates as the gradient, but you have to find the correct z)

3. Use it to construct a vector in the tangent plane which corresponds to the steepest direction down.

4. Use the gradient to construct a second vector in the tangent plane which will represent a direction with no climb. (Hint: The x and y coordinates will be perpendicular to the gradient. What will the z coordinate be?)

5. Use the gradient to construct a vector normal to the tangent plane. (Hint: It will be perpendicular to the other vectors)

6. Find the equation of the tangent plane and an equation of the normal line.

(b) By rearranging the above equation, write the surface implicitly as a function F(x, y, z) = k, where k is any constant (zero if you like).

(c) The gradient of F represents the direction in which F increases the fastest. Since F is representing the surface implicitly, this instead represents the normal to the tangent plane of the surface.

1. At the point (2, π, g(2, π)), find the gradient of F.

2. Using this gradient, find a vector normal to the tangent plane. (Hint: it should be in the same direction as the previous problem, so it should be a scalar multiple)

3. Use this to find the equation of the tangent plane and an equation of the normal line.

Any help would be greatly appreciated

Answer 1

SOLUTION

b)

wix have CMi anu U