Name:
Test #1 Math 2300 Summer 07
1) (15 points) Describe, in detail, the region given by the following inequality:
x 2 + y2 + z 2 - 4 x + 2 y 4
r
r b is orthogonal to a (this is called the
b - proj a
r
a and is denoted orth a b .)
2) (8 points) Show that
e.g.
Find the point(s) of intersection or r (t)= t , 5t , t 2 +4 and
z=x 2 + y 2
1. Substitute parametric linear components into equation of surface.
2
2
t +4=t +25t
2. Solve for t
4
t 2=
25
3. Find r
r
2
t=
2
5
( 52 )
( 52 )=( 25 ,2, 10425 )
( 25 )
4. Fi
Math 2300
Practice Exam
So Lu
Print Name:
Dr. A. Ilarcharras
2
fi'otus
-/
Sign Name:
T.D. ffz
o Exam has 6 problems and 7 pages.
o Exam is closed book;
NO electronic devices are allowed.
o You are allowed to use one side of a 3in by 5in note card.
o Show
Math 2300
Third Exam
Fall 2013
[15] 1. Use cylindrical coordinates to nd the volume of the portion of the sphere
x2 + y 2 + z 2 = 25
between z = 2 and z = 3.
[10] 2. Find the Jacobian of the transformation
x = r cos ,
y = r sin
[10] 3. Set up, but DO NOT
Math 2300
First Exam
Fall 2013
Helfer
Show your work, please.
[20] 1.
For the vectors
a = 2i + j k , b = 3i + 2j , c = 3i + 2j k ,
(a) Find the angle between a and b.
(b) Find the vector projection of a along b.
(c) Find b c.
(d) Find (a b) c.
[5] 2.
You
Math 2300
Final Exam
Fall 2013
Name (print):
Student number:
Signature:
1
Helfer
[10] 1. Find the equation of the plane perpendicular to the line
r(t) = (3 + 2t)i + (2 5t)j + (3 3t)k
and containing the point (1, 1, 2).
[15] 2. Given the vectors
a = 2i j +
Math 2300
Practice for Exam Two
Fall 2013
Helfer
1. If
f (x, y, z) = 3x2 y + 2z sin(xy)
and
x(s, t) = s2 + t3 ,
y(s, t) = se2t ,
z(s, t) = 1/t ,
nd f /t.
2. For the function
f (x, y, z) = x + y 2 z xy ,
(a) nd the gradient; (b) nd the direction of steepes
\
, Math 2300 Exam 1 Dr. A. Harcharras
09/23/16
SO Lennon) 3
Name:
LD. #:
Circle class meeting time: 12:0012:50PM 1:001:50PM
0 Exam has 5 problems and 7 pages (including the cover page).
0 Exam is closed book, no calculator/ computer.
0 You are
Math 2300 Exam 1 Review
12.1/12.2
- The length (aka Norm, Magnitude) of a vector is defined as:
|a|= a 1 +a2 +a3 +.
2
2
2
- The unit vector denoted a^ of
a is defined as:
a^ =
a
|a|
- Vectors are parallel if one is a scalar multiple of the other.
- The d
3. Plug s=3 back into one of the original equations from Step 2:
8 t 15=s 2 8t 15=9 8t =24 t=3
4. Find R 1 (3)
2
2
R 1 (3)=3 , 8(3)15,(3) R 1 (3)=9 , 9,9
13.2 Derivatives/Integrals
- Derivatives/Integrals are done component wise.
- First derivative is kno
2
( )
2
()
b
8
or in this case
=16
2a
2
2
2
2
( x1) +( y2) +( z +8 z+16)=20+16
b) Rewrite z in perfect square notation and simplify
( x1)2 +( y2)2 +( z +4)2=36
5. Now that the equation is in standard form simply note the components (use
2
r =( xx 0 )2 +(
- If
ab=0 then the two vectors are orthogonal.
e.g: Is the triangle formed by the points
P(1,1,1) ,Q(2,4,1) , R(0,3,6) a right triangle?
1. Find PQ
PQ=21,41,11=1,5,2
2. Find PR
PR=01,31,61=1,2,5
3. Find QR
QR=02,3(4)
,6(1)=2,7,7
4. Find PQPR
PR=(11)+(5