Math 233 Practice Problems for Exam 2 Fall 2006
1. Use Chain Rule to find dz/dt or ∂z/∂u,∂z/∂v.
(1) z=x2y+ 2y3,x= 1 + t2,y= (1 −t)2.
(2) z=x3+xy2+y3,x=uv,y=u+v.
2. If z=f(x, y), where fis differentiable, and x= 1 + t2,y= 3t, compute dz/dt at t= 2
provided that fx(5,6) = fy(5,6) = −1.
3. For the following functions
(1). f(x, y) = x2y+y3−y2, (2) g(x, y) = x/y +xy, (3) h(x, y) = sin(x2y) + xy2.
(a) Find the gradient.
(b) Find the directional derivative at the point (0,1) in the direction of v=<3,4>.
(c) Find the maximum rate of change at the point (0,1).
4. Find an equation of the tangent plane to the surface x2+ 2y2−z2= 5 at the point
(2,1,1).
5. Find parametric equations for the tangent line to the curve of intersection of the surfaces
z2=x2+y2and x2+ 2y2+z2= 66 at the point (3,4,5).
6. Find and classify all critical points (as local maxima, local minima, or saddle points) of
the following functions.
(1) f(x, y) = x2y2+x2−2y3+ 3y2, (2) g(x, y) = x3+y2+ 2xy −4x−3y+ 5.
7. Find the minimum value of f(x, y) = 3 + xy −x−2yon the closed triangular region with
vertices (0,0), (2,0) and (0,3).
8. Use Lagrange multipliers to find the extreme values of the following functions with the
given constraint.
(1) f(x, y) = xy with constraint x2+ 2y2= 3;
(2) g(x, y, z) = x+ 3y−2zwith constraint x2+ 2y2+z2= 5.
