Midterm 2: Practice Exam
Math 2210-003
Fall 2015
Warnings:
(1) Please bring your student ID with you during midterm exams.
(2) No references or calculators can be used for midterm exams.
(3) Chapter 12 & 13 &14.1 (13.5 excluded)
Practice Exam
(1) Find the slope of the tangent to the curve of intersection of the surface
36z = 4x2 + 9y 2 and the plane x = 3 at the point (3, 2, 2).
p
(2) The point P (1, −1, −10) is on the surface z = −10 |xy|. Starting at
P , in what direction u = u1 i + u2 j should one move in each case?
(a) To climb most rapidly.
(b) To stay at the same level.
(3) Consider the equation xz 2 + 2yz − 6 ln z = 2 as defining z implicitly as
a function of x and y. Find ∂z/∂x and ∂z/∂y at (0, 1, 1).
(4) Find the equation of the tangent plane and the normal line to the surface
x2 + y 2 + 2z 2 = 23 at (1, 2, 3).
(5) Find the maximum and minimum values of f (x, y) = 2 + x2 + y 2 on the
closed and bounded set S = {(x, y) : x2 + 14 y 2 ≤ 1}.
(6) What are the dimensions of the rectangular box, open at the top, that
has maximum volume when the surface area is 48?
(7) Use double integration to find the volume of the tetrahedron bounded
by the coordinate planes and the plane 3x + 6y + 4z − 12 = 0.
(8) Evaluate:
ZZ
sin y 3 dA
D
where D is the region bounded by y =
√
x, y = 2 and x = 0.
(9) Find the area of the surface z = x2 + y 2 below the plane z = 9.
(10) If S the rectangular region in the xy−plane that is bounded by the lines
x = 0, x = 1,√y = 0 and y = 2, find the area of the part of the cylindrical
surface z = 4 − x2 that projects onto S.
(11) Find the area of the region in the first quadrant bounded by the curves
y 2 = x, y 2 = 2x, x2 = 8y and x2 = 9y.
(12) Find
√ the volume of the solid sphere inside both of the spheres ρ =
2 2 cos φ and ρ = 2.
(13) Find the mass of a solid sphere S if its density δ is inversely proportional
to the distance from the center.
(14) Compute the volume of the solid bounded by z = x2 + y 2 , z = 0 and
x2 + (y − 1)2 = 1.
(15) A vector field F is given:
F (x, y, z) =< x2 y, sin z + x, exyz >
(a) Find the divergence of F .
(b) Find the curl of F .