Calculus III
Practice Problems 5
1. Let f x y z x ln z 2yz. a) What is ∇ f ? b) Show that
∂2 f
∂ z∂ x
∂2 f
∂ x∂ z
2. Find the direction U of maximal change for the function w
DU w at this point?
x 3 y2 z xyz2 at the point (2,-1,2). What is
3. Suppose that the function w f x y is differentiable at the point P in the plane. Let V I 2J W I J,
and that DV w 2 DW w 3 at P. What is ∇w P ?
4. Suppose z f x y is differentiable at (1,1), and suppose that
d
f 1 t 1 t 2 t dt
0
3
d
f 1 1 t t dt
0
2
What is ∇ f 1 1 ?
5. A particle moves in space according to the equation
X t t 2 I For w xy yz2
xz2 ,
1 t2 J 1 t K
find dw dt along the trajectory. What is dw dt when t
6. Let w xyz and let γ be the helix given by
X t costI sintJ tk
Find dw dt at t
2π 3.
7. Find the equation of the tangent plane to the surface
x1 2
y1 2 z1 2 0
at the point (4,9,25).
8. Find the equation of the tangent plane to the surface
x ln z 2yz 0
at the point
e 2 1 e2 .
2?