MATH 23: Calculus III – Final Examination Fall Semester 2006
Instructions. Attempt all questions. Answers must be justified in order
to gain full credit. Calculators are not permitted.
1. Let f(x, y, z)=xey+ln(xz).
(i) (5 points) Find the directional derivative of fat (1,0,1) in the direction
of 2
i+2
j+
k.
(ii) (3 points) What is the direction of maximum rate of change of fat (1,0,1)?
2. (5 points) Show that the following limit does not exist:
lim
(x,y)→(0,0)
x2y
x4+y2
3. (6 points) Use the chain rule to find ∂z/∂u and ∂z/∂v where
z=tan(x+y)with x=ucos vand y=usin v
4. (7 points) Find an equation for the tangent plane to the ellipsoid x2+2y2+4z2=4at the
point (1,1,−1/2).
5. (10 points) Use Lagrange multipliers to find the maximum value of f(x, y )=x2−xy +y2
subject to the constraint x2+y2=1.
6. (10 points) Evaluate the integral 8
02
3
√yx4+1dx dy by reversing the order of integration.
7. (10 points) Find the volume of the region bounded by the plane x+y+z=1and the three
coordinate planes x=0,y=0, and z=0.
8. (7 points) Evaluate the double integral R
sin(x2+y2)dA where Ris the region below.
R
π/2
π
x
y
Please Turn Over
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