MATH 226
CALCULUS III
Prof Baxendale
3/25/09
Answers for Sample Second Midterm
BEWARE: The following are just the numerical answers. Since the justication is missing,
they would not receive full credit on the real exam.
1: fxx = 4 + 6xy + 2y 4 , fxy = 3x2
Spring 11 Math 226 Quiz 7 Solution
Name:
(15 minutes, no notes/calculators)
1. Find the local maximum and minimum values, and saddle points, of the following
function. Identify each point of interest as a minimum, a maximum or a saddle point.
f (x, y ) =
1. The dot product is a _ while the cross product is a _
2. Two vectors whose dot product is 0 are _, two vectors whose cross
product is 0 are _
3. Find the volume of the parallelepiped determined by i + 2j 3k, 2i j + 2k, -i j +3k
4. Find parametric equat
Department of Mathematics
University of Southern California
Class Problem; Section 11-8#14
March 4, 2011
Find the minimum and the maximum of the function
f (x, y, z ) = 3x y 3z,
(1)
g (x, y, z ) = x + y z = 0,
(2)
h(x, y, z ) = x2 + 2z 2 = 1.
(3)
subject
Spring 11 Math 226 Quiz 7 Solution
Name:
(15 minutes, no notes/calculators)
1. Find the local maximum and minimum values, and saddle points, of the following
function. Identify each point of interest as a minimum, a maximum or a saddle point.
f (x, y ) =
Department of Mathematics
University of Southern California
February 25, 2011
Tangent Plane/Gradient Notes
For a function F = F (x, y, z ), when it is held constant, say,
F (x, y, z ) = k,
it gives a relationship between x, y , and z . This may be viewed
Department of Mathematics
University of Southern California
April 1, 2011
Homework Problems on 12-4
Problem A.1
(Modication of Problem 14)
For the lamina in the shape of a crescent dened as the region between the two circles,
x2 + y 2 = 2ay and x2 + y 2 =
Exercises 10.9: 2-4, 7, 8, 10, 15, 19, 23, 28, 29, 33, 34
Problems from 11.1 will be due on Feb 17.
Exercises 11.1: 22, 33, 34 (plot contours and graph x-z and the y-z planar sections)
Outside the book: Draw the contours of the double peak below, and gra
Exercises 11.1: 22, 33, 34 (plot contours and graph x-z and the y-z planar sections)
Outside the book: Draw the contours for constant f(x,y) of the double peak below,
and graph the section in the y=0 plane [you may use any plotting software or sketch by
tor
M*t ft zzG
Le
ct
uL6
I Z_3
f
"
,+ t
JL,
*
t4
. Do,^ere
t-
t
NT e 4
c-.-tL-,F,
;"f-L4tr.,tr,-
e-*c,^J1lo-, ;
f
R^rx=t'E'
V y= cfw_E['
J
J l;O *(*ft)
5ro
*,-
. r.L
v, lhJ-
(A-15 | w (o Lt*l- coo ADI rv^ Ts
e ,L
csvrr*<ru\'&
rrtlT
L.-
,gL*lt*
4*
cvrcttkx
>
Math 226: CaJculus 3
0
LuTt o,l
Second Exanination
March 27, 2009
1. (20 points/lO minutes)
In a lab experiment, a charged pa,rticle is placed in vertical and horizontal electric
fields so that
it
experiences accelerations in both directions grven by
a(
Department of Mathematics
University of Southern California
February 25, 2011
Tangent Plane/Gradient Notes
For a function F = F (x, y, z ), when it is held constant, say,
F (x, y, z ) = k,
it gives a relationship between x, y , and z . This may be viewed
Math
Math 226 SI Spring 11 Final Review
w/ Andrew Kupiec and Renee Allison
SGM 124
33-5PM
Find an equation for the plane that passes through the line of intersection of the planes x 2z = 2
and y + z = 1 and is perpendicular to the plane x + y- 2z = 1
Are
Department of Mathematics
University of Southern California
April 15, 2011
Greens Theorem Examples
Sec 13-4, Chapter-End Review Examples, Page 800
Problem 10
The force on a particle is given by
F (x, y, z ) = z + xj + y k.
i
The particle is moved from A(3
tor
M*t ft zzG
Le
ct
uL6
I Z_3
f
"
,+ t
JL,
*
t4
. Do,^ere
t-
t
NT e 4
c-.-tL-,F,
;"f-L4tr.,tr,-
e-*c,^J1lo-, ;
f
R^rx=t'E'
V y= cfw_E['
J
J l;O *(*ft)
5ro
*,-
. r.L
v, lhJ-
(A-15 | w (o Lt*l- coo ADI rv^ Ts
e ,L
csvrr*<ru\'&
rrtlT
L.-
,gL*lt*
4*
cvrcttkx
Department of Mathematics
University of Southern California
Class Problem; Section 11-8#14
March 4, 2011
Find the minimum and the maximum of the function
f (x, y, z ) = 3x y 3z,
(1)
g (x, y, z ) = x + y z = 0,
(2)
h(x, y, z ) = x2 + 2z 2 = 1.
(3)
subject
MATH 226
CALCULUS III
Prof Baxendale
3/25/09
Answers for Sample Second Midterm
BEWARE: The following are just the numerical answers. Since the justication is missing,
they would not receive full credit on the real exam.
1: fxx = 4 + 6xy + 2y 4 , fxy = 3x2
SAMPLE FINAL EXAM
Answer all questions. You must show your working to obtain full credit. Total point
score is 200 points.
1. (a) Find the equation of the line through the point (1, 4, 6) perpendicular to the plane
3x 2y + z = 7. [6 points]
(b) Find the d
MATH 226
CALCULUS III
Prof Baxendale
04/27/09
Answers for Sample Final Exam
BEWARE: The following are just the numerical answers. Since the justication is missing,
they would not receive full credit on the real exam.
1. (a) r(t) = 1, 4, 6 + t3, 2, 1; (b)
Sample First Midterm
Answer all seven questions. You must show your working to obtain full credit. Points may
be deducted if you do not justify your nal answer.
1. Find an equation for the sphere centered at (4, 2, 3) which is tangent to the xz -plane.
2.
MATH 226
CALCULUS III
Prof Baxendale
02/15/09
Answers for Sample First Midterm
BEWARE: The following are just the numerical answers. Since the justication is missing,
they would not receive full credit on the real exam.
1:
2:
3:
4:
x2 + y 2 + z 2 8x + 4y
Sample Second Midterm
Answer all seven questions. You must show your working to obtain full credit. Points may
be deducted if you do not justify your nal answer.
1. Let f (x, y ) = 2x2 + x3 y + x2 y 4 8y .
(i) Find all the second partial derivatives of f