Math 324 H Sample Quiz
Use the fundamental theorem of line integrals to compute
F dr
C
where C is the curve starting at (0, 0, 0) travelling along the x-axis to (2, 0, 0)
and then along the straight line path to (0, 0, 3) and F = y cos(xy ), x cos(xy ), 2
Math 324 H Sample Quiz Solution
Use the fundamental theorem of line integrals to compute
F dr
C
where C is the curve starting at (0, 0, 0) travelling along the x-axis to (2, 0, 0)
and then along the straight line path to (0, 0, 3) and F = y cos(xy ), x co
Math 324 H (Ward)
Midterm 1 (50 Minutes)
October 18, 2013
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you
Math 324 H (Ward)
Midterm 1 (50 Minutes)
October 18, 2013
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you
Math 324 H (Ward)
Midterm 2 (50 Minutes)
November 22, 2013
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you
Math 324 H (Ward)
Midterm 2 (50 Minutes)
November 22, 2013
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you
Math 324 H Quiz 1 Solution
Find the volume of the solid formed in the rst octant bounded by the
planes x = 0, z = 0, x y = 0, and 3x + 3y + z = 18 inside the cylinder
x2 + y 2 = 1.
x=0
y=x
2
2
x + y =1
D
=/4
/2
1
(18 3x 3y )dA =
(18 3r cos 3r sin )rdrd
/
Math 324 H Quiz 2
Use a triple integral to nd the volume of the solid inside z 2 = x2 + y 2
between z = 1 and z = 2.
Idea: The surface z =
coordinates: z = r.
x2 + y 2 becomes the familiar cone in cylindrical
z=r
z=2
z=1
E
2
2
1 dV
z
r drddz
=
E
0
2
1
=
1
Math 324 H Quiz 3
Suppose f (r, ) is a function in polar coordinates. Recall that we can
convert this to rectangular coordinates using formulas r(x, y ) and (x, y ).
Part (a) Write down the multivariable chain rule needed to calculuate
f
.
x
f
f r f
=
+
Math 324 H Quiz 4 Solution
1. Compute the directional derivative Du f at (1, 0) where u = 0, 1 and
f (x, y ) = xy + y 2 .
f = y , x + 2y
Du f (1, 0) = 0, 1 0, 1 = 1
2. Is this the maximum value of Dv f for all unit vectors v (You must
justify your answer
Math 324 H Quiz 5
1. Compute the integral
F dr
C
where
r(t) = cos(t), sin(t)
on [0, 2 ]
x2
F(x, y ) = x sin(e ) + 2y, 3x + ey .
2
Apply Greens Theorem: P (x, y ) = x sin(ex ) + 2y and Q(x, y ) = 3x + ey ,
so
Q P
= 3 2 = 1.
x
y
C is just the circle of radi
Math 324 H Quiz 6 Solution
1. Let S be the part of the surface given by the equation 2x + 4y + z = 4
in the rst octant oriented away from the origin (with all components of the
normal positive). Compute the ux of
x
F = 2y, , z
2
through S .
Solution: A pa
Math 324 H Quiz 1
Find the volume of the solid formed in the rst octant bounded by the
planes x = 0, z = 0, x y = 0, and 3x + 3y + z = 18 inside the cylinder
x2 + y 2 = 1.
Math 324 H Quiz 3
Suppose f (r, ) is a function in polar coordinates. Recall that we can
convert this to rectangular coordinates using formulas r(x, y ) and (x, y ).
Part (a) Write down the multivariable chain rule needed to calculuate
Part (b) Calculate
Math 324 H Quiz 4
1. Compute the directional derivative Du f at (1, 0) where u = 0, 1 and
f (x, y ) = xy + y 2 .
2. Is this the maximum value of Dv f for all unit vectors v (You must
justify your answer for full credit)?
Math 324 H Quiz 6
1. Let S be the part of the surface given by the equation 2x + 4y + z = 4
in the rst octant oriented away from the origin (with all components of the
normal positive). Compute the ux of
x
F(x, y, z ) = 2y, , z
2
through S .