University:
Seattle Pacific University
Course:
MAT 1236 | Calculus III
Academic year:

2024
Views:

108

Pages:

2
Author:

Alma Stephenson

Workshop #28 1. Set f (x, y) = y + x2 . (a) On the given graph of z = f (x, y) on the left, roughly sketch some traces and sketch some level curves. Label which are which. (The graph on the right is just for added perspective.) (b) Now, compute the level curve for k = 4 and carefully (to scale) graph it in the appropriate dimension (separate from the graphs above). Sketch several more level curves. (c) We want to carefully graph some vectors in relation to our level curve for k = 4. At the point (x, y) on your level curve, we are going to graph the vector ∇f (x, y). To help you do this, first complete the chart, then graph the vectors on your level curve. What general relationship do the vectors have to the level curve? (x, y) (0,4) (1,3) (2,0) (3,-5) ∇f (x, y) (x, y) (-1,3) (-2,0) (-3,-5) ∇f (x, y) ~ 2. The gradient field of a function f (x, y) is the vector field formed by F(x, y) = ∇f (x, y). Find and sketch the gradient field. (a) f (x, y) = 2x − 4y (b) f (x, y) = 2xy (c) f (x, y) = 2x2 + y 2 (d) f (x, y, z) = y 2 + z 2