Math 324B
SECOND PRACTICE EXAM
tan z
+ 2 ln y , and v = (2x 3y )3 .
x2
a. Compute w/x, w/y , and w/z in terms of the partial derivatives of f .
b. Considering w as a function of x, y , and z , what is its maximum directional deriva1
1
tive at (x, y, z ) =
Math 324B
FIRST PRACTICE EXAM
(a bit longer than the real exam)
1. Let D be the region between the line y = 3x + 4 and the parabola y = 4 x2 . Express
2x dA as an iterated integral in two ways (dy dx and dx dy ), and evaluate it. (For
D
practice, do it bo
Math 324
Additional review problems for Final Exam
1
1. Consider two vector elds: F = x + z, 1, x and G = y , x, ez .
a) For each of the two elds, determine whether it is conservative. Show your reasoning!
Give a potential function for each conservative e
Math 324B
Midterm 1
April 17, 2009
1
Your Name:
Problem
Points
Possible
1
11
2
10
3
11
4
10
5
8
Total
50
Turn o and put away cell phones, graphing calculators, books, and notebooks.
You may use one 8 1 11 sheet of handwritten notes and a non-graphing ca
M324 CORRECTED Solns. for MT2 Sample Questions Fall, 2011 1
f = (cos y, x sin y + ez , yez ), so
Problem 1. (a)
f (2, 3, 1) = i + ej + 3ek.
(b) Let w be the vector from (2,3,1) to (5,3,5). Then r (0) = 2
so
d
f (r(t) =
dt
(3, 0, 4)
w
= 2
= ( 6 , 0, 8 ),
5
Math 324
Sample Questions for Midterm 2
Autumn, 2011
1
ADDED PROBLEM 4 ON TUESDAY 11/15.
See also: recent quizzes, the actual quiz and the samples, esp. quiz 3 (on parametrizing surfaces);
MT 2 review assignment on WebAssign; and two more, if you need mor
4/23/2017 Assignment Previewer
Change of variables (10556480)
Due: Fri Apr 28 201711:30 AM PDT
Question 123456789101112
1. Question Details SCaIcET8 15.9.001. [3801273]
Find the Jacobian of the transformation.
X=3u+v, y=9uv
A
Solution or Explanation
12
EX
Math 324B
FINAL PRACTICE EXAM
Notes (these pertain to the real exam too): (1) If you use a major theorem (Green,
Stokes, etc.), mention it by name! (2) Its OK to use shortcuts in evaluating integrals (e.g.,
observing that an integral is zero because of so
Math 324
Final Exam
Summer 2011
Print your name:
Problem
Points Score
1
4
2
10
3
10
4
10
5
11
Take home
15
Total
60
Check one:
_ I would like my exam placed in a box outside Nathans office from Saturday morning
until Tuesday morning.
_ Please do not put m
MATH 324A (Autumn 2009)
Practice Final Exam
Student name:
Student number:
Signature:
Do not start working until instructed to do so.
You have 110 minutes.
Please show your work.
Scientic, but not graphing calculators are allowed.
You may use one 8.5 by 11
MATH 324A (Autumn 2009)
Practice Midterm
Student name:
Student number:
Signature:
Do not start working until instructed to do so.
You have 50 minutes.
Please show your work.
Scientic, but not graphing calculators are allowed.
You may use one 8.5 by 11 dou
MATH 324A (Spring 2010)
Midterm
Student name:
Student number:
Signature:
Do not start working until instructed to do so.
You have 50 minutes.
Please show your work.
Scientic, but not graphing calculators are allowed.
You may use one 8.5 by 11 double-sided
MATH 324A (Autumn 2009)
Final Exam
Student name:
Student number:
Signature:
Do not start working until instructed to do so.
You have 110 minutes.
Please show your work.
Scientic, but not graphing calculators are allowed.
You may use one 8.5 by 11 double-s
MATH 324A (Autumn 2009)
Midterm
Student name:
Student number:
Signature:
Do not start working until instructed to do so.
You have 50 minutes.
Please show your work.
Scientic, but not graphing calculators are allowed.
You may use one 8.5 by 11 double-sided
Math 324
Midterm Exam
Summer 2011
Print your name:
This exam has 6 questions on 5 pages, worth a total of 50 points.
Problem Points Score
1
4
2
6
3
10
4
10
5
10
6
10
Total
50
You should:
write complete solutions or you may not receive credit.
box your f
Math 324
Final Exam, page 1 of 4
Summer 2011
1. (4 points) Let f = x + xy + y , and let C be the curve below, with endpoints (4, 0) and
(5, 3), and oriented in the clockwise-ish direction.
(5,3)
3
2
1
1
2
(4,0)
4
3
5
-1
f dr.
Determine
C
Solution: Using t
4/22/2017 Assignment Previewer
Integration with polar coordinates (10491861)
Due: Fri Jun 9 2017 11 :30 AM PDT
Question 123456789101112131415
1. Question Details SCaIcET8 15.3.008. [3800990]
Evaluate the given integral by changing to polar coordinates.
ff
4/22/2017 Assignment Previewer
Center of mass and probability (15.5) (10511170)
Due: Fri Jun 9 2017 11 :30 AM PDT
Question 12345678910
1. Question Details SCaIcET8 15.4.005. [3801393]
Find the mass and center of mass of the lamina that occupies the region
5/16/2017 Assignment Previewer
Surface integral of functions (10636366)
Due: Fri May 26 2017 11:30 AM PDT
Question 1234567891011
1. Question Details SCaIcET8 16.7.005. [3802079]
Evaluate the surface integral.
f] (X + y + 2) d5, Sis the parallelogram with
Quiz 5. MATH 324 C
Name:
Student ID:
For full credit, please show your work.
1. (6 points) Consider the vector field
F = sin(y)i + (x cos(y) + cos(z)j y sin(z)k.
It is known that F is conservative.
(a) (3 points) Find a function f such that f = F .
R
R
R
Quiz 3. MATH 324 C
Name:
Student ID:
For both problems, E is p
a solid that lies within the sphere x2 + y 2 + z 2 = 16, above the xyplane
and below the cone z = x2 + y 2 . The solid is the same in Question 25, HW4.
1. (4 points) Write an integral in cylin
Graphs of quadric surfaces
FI'
Ellipsoid
Elliptic Paraboloid
2
+=1
a1 c1
All traces are ellipses.
If a = b = c, the ellipsoid is
a sphere.
z I: F:
_=_+_
c a: b3
Horizontal traces are ellipses.
Vertical traces are parabolas.
The variable raised to the
rst
Conservation of Energy Discussion (from 16.3)
Here is a brief discussion of the origin of the term conservative for a vector eld, F, that is the gradient
of some potential function, f. Mathematically, this relationship is F = V f , but lets see where the
FINDING POTENTIAL FUNCTIONS FOR CONSERVATIVE VECTOR
FIELDS.
Say that we are given a vector fields F = P i + Qj + Rk and we know that F is conservative.
Our goal is to find a potential function, this is: a function f such that
F = f.
We then need that fx =
Quiz 2. MATH 324 C
Name:
Student ID:
1. Consider the plate bounded by the curve r() = 4 3 cos(). The density of this plate equals
1 if y 0 and equals a if y < 0.
(a) (2 points) Sketch the region described by the curve r() = 4 3 cos(). Make sure to label
t
Computing the tangent plane to a surface.
Let S = cfw_r(u, v), (u, v) D be a surface. We want to compute the equation of the tangent plane
at a given point.
Say that we are given with the value of the parameters u = u0 and v = v0 . Plug in into the
parame
Assignment Previewer
4/28/2017
Vector elds and Line Integrals (10519339)
SCalcETS 16.1.001. [3801859] _
Due: Fri May 5 2017 11:30 AM PDT
Question Details
Question |1|2|3I4|5|6|7|8|9|10|11|12|13|14|15|16|
1.
Y
Sketch the vector eld F by drawing a dia
5/9/2017 Assignment Previewer
Green's Theorem (10612841)
Due: Fri May 19 2017 11:30 AM PDT
Question 123456789101112131415
1. Question Details SCaIcET8 16.4.005. [3801018]
Use Green's Theorem to evaluate the line integral along the given positively oriente
5/3/2017 Assignment Previewer
Conservative vector elds. (10598335)
Due: Fri May 12 2017 11:30 AM PDT
Question 123456789101112131415
1. Question Details SCaIcET8 16.3.003. [3801806] _
Determine whether or not F is a conservative vector field. If it is, fin