Calculus 3 (MA 113), Winter, 2003—2004
TakeHome Quiz 12 (due Monday, Feb 9, 2004)
Instructions: Must be done by yourself. If you have questions, please ask during the class
period.
1. Let z = 2x2 y − 2x + y
∂z
=
(a)
∂x
(c)
∂2z
=
∂x2
(b)
(d)
∂z
=
∂y
∂2z
=
∂x∂y
(e) ∇z(x, y) =
→
−
→
−
(f) Find the directional derivative of z at the point (1,1) in the direction i + 3 j .
(g) Find the direction (give as a unit vector) in which the directional derivative at the point
(1,1) is the largest.
(h) What is the value of the directional derivative in the direction found in (g)?
(i) In what directions (there will be two and give as unit vectors) will the directional derivative
at the point (1,1) be 0?
(j) Give the parametric form of the tangent plane to the graph of z = 2x2 y − 2x + y at the
point (1, 1, 1).
(k) Give the Cartesian form of the tangent plane to the graph of z = 2x2 y − 2x + y at the
point (1, 1, 1).
2. Given the following information (some of it may not be needed),
∂f
∂2f
∂f
(1, 2) = −1,
(1, 2) = 4,
(1, 2) = 5, f (3, −1) = −6,
f (1, 2) = 3,
∂x
∂y
∂x∂y
(a) ∇f (1, 2) =
→
−
→
−
(b) The directional derivative of f at the point (1, 2) in the direction i + 3 j is
(c) Find the directions (give as unit vectors) in which the directional derivative of f at the
point (1, 2) is 0.
(d) Find the direction (give as a unit vector) in which the directional derivative at the point
(1,2) is the largest.
(e) What is the value of the directional derivative in the direction found in (d)?
(f) Find the equation of the tangent plane (give in Cartesian form) to the graph of z = f (x, y)
at the point (1, 2, 3).
3. Find the equation of the tangent plane at the point (1, 0, 1) to the surface defined by the
equation x2 y + eyz − y 3 − x + z = 1.
4. Determine the critical point(s) of the function f (x, y) = 2x2 + 2xy + y 2 + 2x − 3 = 0. Use
your common sense to determine whether the critical point(s) is a max or min.
5. Use Lagrange multipliers to solve: minimize x2 − y 2 subject to x − 2y + 6 = 0. Clearly
show your work.
6. Find the absolute minimum value (and where it occurs) of f (x, y) = 2xy − xy 2 + 3x2 on
the unit disk ({(x, y) | x2 + y 2 ≤ 1}). Show your work.
7. Set up a double integral to find the volume of the solid bounded by the graphs of the
equations: z = xy, z = 0, y = x, x = 1. first octant. Even if you use Maple, be sure to give
a rough sketch of the solid and the iterated integrals you are using.
8. No maple on this problem. Rewrite the iterated integral
evaluate one of the iterated integrals.
R2R3
0
x
y dy dx using dx dy, and