Name
Quiz 2
Calculus III 3450:223003 Dr. Norfolk January 21, 2011
Show your work. Unless otherwise specied, give exact answers.
1. Find the exact angle between the vectors a = i j and b =< 1, 2, 2 >.
5 points
2. Find a vector which is orthogonal to c = 2i
Name
Homework #1 Due :
Calculus III 3450:223 Dr. Norfolk
1. Find the centre and radius of the sphere with equation
2x2 + 2z 2 6y = 5x 2y 2 + 7z + 9
2. Consider the intersection of the sphere (x 3)2 + (y + 2)2 + (z 1)2 = 13 with the plane
x + y = 0.
(a) Th
Name
Homework #3 Due :
Calculus III 3450:223 Dr. Norfolk
1. Find the domain of the function r(t) =< sin1
1t
2
, tan(3t2 ), ln(ln t) >.
2. Find the equation of the quadric surface on which the curve r = et cos 10tiet sin 10tj+et k
lies.
(Eliminate t from t
Name
Homework #5 Due:
Calculus III 3450:223 Dr. Norfolk
1. Find and sketch the domain of f (x, y) =
x2 + y 2 1 + ln(4 x2 y 2 )
.
x+y
x2 sin2 y
2. Find the limit, or show that it does not exist :
lim
(x,y)(0,0) x2 + 3y 2
3. Show that u = sin(x at) + ln(x +
Name
Homework #4 Due :
Calculus III 3450:223 Dr. Norfolk
1. Find the length of the curve r(t) = 12ti + 8t3/2 j + 3t2 k, 0 t 1.
2. Let r(t) =< t2 , sin t t cos t, cos t + t sin t >, t 0.
Find the unit tangent vector, T, the unit normal vector, N, and the c
Calculus III
Fall 2005
Due 10/10/2005
Dr. D. P. Story
Assignment #1
1. Find the equations of the planes that are parallel to the plane x + 2y 2z = 1 and two units away from
it. (Hint : Draw a picture and try to use the normal vector of the given plane to
Calculus III
Fall 2005
Name:
Dr. D. P. Story
Quiz #6Section 004
Instructions: (10 points) Solve each of the following problems without error. Show all
details. Box in your answers . Use good notation, you will be marked o for bad notation.
(4pts )
1. Find
(10pts )
Calculus III
Name:
HW #2Section 004
Fall 2005
Dr. D. P. Story
1. (Find Absolute Max/Min.) Consider the function f (x, y ) = 3x + 2y dened over the closed and bounded
region D = cfw_ (x, y ) : x2 y 1 . Find the absolute maximum and absolute minimu
Calculus III
Fall 2005
Name:
Dr. D. P. Story
Quiz #8Section 004
Instructions: (10 points) Solve each of the following problems without error. Show all
details. Box in your answers . Use good notation, you will be marked o for bad notation.
(4pts )
1. Set
Name
Homework #2 Due :
Calculus III 3450:223 Dr. Norfolk
1. Let u = 2i + 3j, v = j + 2k and w = i + j + k.
Evaluate u (v w) and (u v) w
2. Find the value of c so that the following lines intersect
x1
y+1
z2
L1 :
=
=
0
2
3
L2 : x = 2 4t, y = 1 + t, z = c +
Homework #1 (Solutions)
Calculus III 3450:223 Dr. Norfolk
1. Find the centre and radius of the sphere with equation
2x2 + 2z 2 6y = 5x 2y 2 + 7z + 9
Solution: Dividing by 2 and completing the squares gives
3 2
7 2 182
5 2
x
+ y
+ z
=
,
4
2
4
16
5 3 7
182
Test Total
Name
Final Exam Calculus III 3450:223003 Dr. Norfolk Monday, May 2, 2011
Show all of your work. Give exact, reasonably-simplied answers.
Greens Theorem:
P dx + Qdy =
C
D
(Qx Py ) dA
F dr =
Greens Theorem (vector form):
C
F dr =
Stokes Theorem:
Name
Quiz 1
Calculus III 3450:223-003 Dr. Norfolk January 14, 2011
Show your work. Unless otherwise specied, give exact answers.
1. Consider the triangle with vertices A(4, 4, 3), B (3, 2, 1) and C (5, 1, 1).
(a) Show that the triangle is right-angled.
5
Name
Quiz 4
Calculus III 3450:223003 Dr. Norfolk February 4, 2011
Show your work. Unless otherwise specied, give exact answers.
t
1 2
1. Find and simplify r (t), given r(t) =< t tan (t ), sin(cos t),
>.
1+t
5 points
2. Given that r (t) =<
1
1
, 2 sin2 t,
Name
Quiz 3
Calculus III 3450:223003 Dr. Norfolk January 28, 2011
Show your work. Unless otherwise specied, give exact answers.
1. Find the equation of the plane which passes through P (0, 1, 5), and has normal parallel to
the vector P Q, given Q(5, 4, 2)
Name
Quiz 5
Calculus III 3450:223003 Dr. Norfolk February 18, 2011
Show your work. Unless otherwise specied, give exact answers.
2
x + y2 4
1. Sketch the domain of the function f (x, y ) =
.
ln(x + y 2)
5 points
(x + 3)3 (y 2)
.
(x,y )(3,2) (x + 3)4 + 3(y
Homework #3 (solutions)
Calculus III 3450:223 Dr. Norfolk
1. Find the domain of the function r(t) =< sin1
1t
2
, tan(3t2 ), ln(ln t) >.
Solution:
1t
1, or 1 t 3.
2
3
5
tan(3t2 ) is dened for 3t2 = , , , or t =
,
,
, .
2 2
6
2
6
ln(ln t) is dened for ln
Homework #2 (solutions)
Calculus III 3450:223 Dr. Norfolk
1. Let u = 2i + 3j, v = j + 2k and w = i + j + k.
Evaluate u (v w) and (u v) w
Solution:
i
j k
v w = 0 1 2 = 3i + 2j + k
1
1 1
So,
u (v w) =
i j k
2 3 0 = 3i 2j + 13k.
3 2 1
Similarly,
i
j k
3 0 =
Test Total
Name
Final Exam Calculus III 3450:223004 Dr. Norfolk Monday, May 3, 2010
Show all of your work. Give exact, reasonably-simplied answers.
Greens Theorem:
P dx + Qdy =
C
D
(Qx Py ) dA
F dr =
Greens Theorem (vector form):
C
F dr =
Stokes Theorem:
Calculus III
Fall 2005
Date: 12/05/05
Dr. D. P. Story
Assignment #3 - Solution Notes
1. Set up limits of integration. After carefully drawing the region and examining the various projections, we
get:
3/4
3(1x)
(66x2y )/3
(a)
f (x, y, z ) dz dy dx
0
x
3(2z
Calculus III
Fall 2005
Name:
Dr. D. P. Story
Quiz #1Section 004
Instructions: (10 points) Solve each of the following problems without error. Show all details. Box in
your answers . Use good notation, you will be marked o for bad notation.
(5pts )
1. Find
Calculus III
Fall 2005
Name:
Dr. D. P. Story
Quiz #3Section 004
Instructions: (10 points) Solve each of the following problems without error. Show all details. Box in
your answers . Use good notation, you will be marked o for bad notation.
(5pts )
1. Find
Name
Quiz 2
Calculus III 3450:223003 Dr. Norfolk January 21, 2011
Show your work. Unless otherwise specied, give exact answers.
1. Find the exact angle between the vectors a = i j and b =< 1, 2, 2 >.
5 points
2. Find a vector which is orthogonal to c = 2i
Name
Quiz 4
Calculus III 3450:223003 Dr. Norfolk February 4, 2011
Show your work. Unless otherwise specied, give exact answers.
t
1 2
1. Find and simplify r (t), given r(t) =< t tan (t ), sin(cos t),
>.
1+t
5 points
2. Given that r (t) =<
1
1
, 2 sin2 t,
Name
Quiz 5
Calculus III 3450:223003 Dr. Norfolk February 18, 2011
Show your work. Unless otherwise specied, give exact answers.
2
x + y2 4
1. Sketch the domain of the function f (x, y ) =
.
ln(x + y 2)
5 points
(x + 3)3 (y 2)
.
(x,y )(3,2) (x + 3)4 + 3(y
Name
Quiz 10
Calculus III 3450:223003 Dr. Norfolk April 8, 2011
Show your work. Unless otherwise specied, give exact answers.
1. SET UP BUT DO NOT EVALUATE, the triple integral for the volume of the solid E , which
is bounded above by the sphere x2 + y 2
Name
Quiz 11 (take home, due Monday, April 25)
Calculus III 3450:223003 Dr. Norfolk
April 20, 2011
Show your work. Unless otherwise specied, give exact answers.
1. Consider the smooth curve C : r(t) = ti + cos tj + et k , 0 t 1, and the vector eld
1
F = y
Name
Quiz 9
Calculus III 3450:223003 Dr. Norfolk April 1, 2011
Show your work. Unless otherwise specied, give exact answers.
1. Evaluate
R
2 y/x
e dA, where R = [1, 2] [1, 3].
x3
5 points
0
2. Consider
9 y 2
3
2
x dxdy +
3
0
0
9 y 2
9 y 2
x2 dxdy .
(a) Sk
Name
Quiz 8
Calculus III 3450:223003 Dr. Norfolk March 11, 2011
Show your work. Unless otherwise specied, give exact answers.
1. Let f (x, y ) = x4 +
y4
4xy .
256
(a) Show that the only critical points of f (x, y ) are (0,0), (2,8) and (-2,-8).
5 points
Name
Quiz 6
Calculus III 3450:223003 Dr. Norfolk February 25, 2011
Show your work. Unless otherwise specied, give exact answers.
1. Find the rst partial derivatives for the function f (x, y, z ) = xz cos(y 2 /x)
5 points
2. Given g (x, y ) = ln(x + y sin