Math 320 Third Test ______________________
Calculus III Name
This exciting 50-minute test covers sections 12.4-7 of Calculus by Stewart (5ed). Relax
and do well. Each part of each problem is 5 points unless otherwise stated. Check your
work --especially on the easy problems.
1. On the right is a drawing of several level curves of a
function f(x,y) as well as two points. Draw ∇f as a
vector with initial point at each of these points and
clearly indicate their direction and relative lengths.
2. Let f(x,y,z) = x2 + (y + z)2 and P0 = (4,1,2).
a. Find ∇f (at the point P0).
b. Find the equation for the normal line to the surface f(x,y,z) = 25 at P0.
c. Find the equation for the tangent plane to the surface f(x,y,z) = 25 at P0.
d. Find the derivative of f in the direction of the vector i + j + k at P0.
