Consider the surface S defined by z = f(x,y) =z2 + 4(y- l)2 Graph the level curv
ID: 1945952 • Letter: C
Question
Consider the surface S defined by z = f(x,y) =z2 + 4(y- l)2 Graph the level curves of f in the (x,y) plane. Find delta f and indicate delta f by arrows at some points on the level curves of f.Explanation / Answer
1) Level curves are z = k. First, note that z >= 0, since it is a sum of two squares. When x^2 + 4(y-1)^2 = k, the resulting graph is an ellipse, centered at (0, 1) with major semi axis along the x-axis, length sqrt(k). The minor semi axis is parallel to the y-axis, length sqrt(k)/2. 2) The gradient is grad(f) = (2x, 8(y-1) ). The gradient always points exactly out of the ellipse.