Hi, I need help with problems 5-10 in the attached document
MATH 0240 Final Sample Exam 2
Problem 1. Function f is given by the formula f (x, y) = 2x2 + 3exy .
a) Find the directional derivative of f at the point P = (1, 0) in the
direction of the vector u =< ?1, 2 >.
b) Find the maximal rate of change of f (x, y) at P and the direction in
which it occurs.
Problem 2. The curve is ?
given parametrically by
r(t) =< t + 2 t , 2t ? 1, t + t 5 > .
a) Find its curvature at the point (0, ?1, 0).
b) Set up the integral representing
? the length of the curve from the point
(0, ?1, 0) to the point (10, 3, 4 + 2 5).
DO NOT EVALUATE THE INTEGRAL.
Problem 3. Find an equation of the plane tangent to the surface
x + y 2 z 2 = 8 at the point P = (2, 2, 1).
2 Problem 4. Find all critical points of the function
f (x, y) = x2 + 4xy ? 10x + y 2 ? 8y + 1. For each critical point determine if
it is a local maximum, a local minimum or a saddle point.
Problem 5. Find the work done by the force F(x, y) = 3yi + xj in
moving a particle along the boundary of the trapeziod with the vertices
(0, 0), (1, 1), (2, 1) and (3, 0) in the clockwise direction.
Problem 6. Find the mass of the solid bounded by the surfaces
y + z 2 = 1, x = 0 and x = y 2 + z 2 ? 4, if the density function is given by
the formula ?(x, y, z) = y 2 + z 2 .
2 Problem 7. a) Determine whether the vector field
F(x, y, z) = (2y + 4z)i + (2x + 3z)j + (4x + 3y)k, is conservative or not.
R b) Evaluate (2y + 4z)dx + (2x + 3z)dy + (4x + 3y)dz, where C is the
C curve given by r(t) =< t3 , 2 sin ?t
2 , 3 cos ?t
2 > for 0 ? t ? 1. Problem 8. Find the maximum and minimum values of the function
F (x, y, z) = x ? y on the x2 + y 2 + xy + z 2 = 1
H Problem 9. Evaluate F ? dr, if F(x, y, z) = yi + 2xj + yzk, and C is C the curve of intersection of the part of the paraboliod z = 1 ? x2 ? y 2 in the
first octant (x ? 0, y ? 0, z ? 0) with the coordinate planes x = 0, y = 0
and z = 0, oriented counterclockwise when viewed from above.
Problem 10. Evaluate RR F ? dS, if F(x, y, z) = (yz)i + (x2 y)j + (4zx2 )k S and S is the surface of the solid bounded by the upper hemisphere
x2 + y 2 + z 2 = 1, z ? 0, and the plane z = 0 with the normal pointing
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