Worksheet on gradients and level sets PDF

Preview

Summary

Questions ...

Description

MATH 228: Calculus III Worksheet on Gradients and Level sets Do the following exercises, starting with f (x, y) = x2 + y 2 . Then repeat these exercises for the p 2 function f (x, y) = 4 − x − y 2 . And then for f (x, y) = x2 − y 2 . 1. Sketch the graph of this function and recognize it as a surface of revolution by identifying a curve in the first quadrant of the xz-plane that sweeps out the graph when rotated about the z-axis. 2. Make a contour map by sketching a few level sets in the domain of the function. ∂f ∂f i+ j at several points ∂x ∂y for each level set. (Hint: the gradient vector is perpendicular to the line tangent to the level set.)

3. Sketch the gradient vector field by plotting the vector ∇f =

4. Find a vector w tangent to the graph of f (x, y) at the point (x0 , y0 , z0 ) where (x0 , y0 ) = (1, 0) and z0 = f (x0 , y0 ) by forming a parametrized line r(t) satisfying r(0) = (x0 , y0 ), d  dr then setting w = (0) +  f (r(t))k. (Remark: Among the infinitely many possible dt dt t=0 dr choices for (0), the simplest are i and j.) dt 5. Find the tangent plane to the graph of f (x, y) at the point (x0 , y0 , z0 ) by finding a second vector tangent to the graph of f (x, y) and computing the cross product with the vector w you found in the previous exercise. 6. Compute the directional derivative of f (x, y) at (x0 , y0 ) in the direction of

dr (0). dt...