In each of the problems, be sure to include clearly drawn sketches of R for double integrals and W for
triple integrals. You might need to change the coordinate system in order to evaluate an integral. Show
all work and use proper notation.
72
1. Change
MATH 223
FINAL EXAM REVIEW PACKET
(Spring 2015)
The following questions can be used as a review for Math 223. These questions are not actual samples
of questions that will appear on the final exam, but they will provide additional practice for the
materia
Math 223 - Section 010
Vector Fields Lab with Solutions In this lab, you will use the applet at http:/www.falstad.com/vector3d/ to develop some intuition about vector fields in space. Some general comments about the applet: The "Particles (Vel.)" and "Str
Math 223 - Section 010
Test 3a with Solutions You have 50 minutes to complete the test. Show all of your work clearly. Unless otherwise stated, each answer must be explained and justified. 1. For each of the following statements, circle TRUE or FALSE. You
Math 223 Vector Calculus Instructor: Arlo Caine
Practice Exam 4 Name:
Directions: Read all questions carefully. Use a pencil and erase all unnecessary marks. Show all of your work in the space provided and display your answer on the line given if requeste
Group Homework for ch.13
Name_
Show all work and include explanations if needed. Proper notation, organization,
and neatness are important. Attach this sheet to the front of your work. Do not
work with someone outside of the class. Due Thursday, September
WHAT NOT TO SAY
Getting a correct result does not always mean the explanation is correct. Choosing a wrong word can
either be incorrect or not what you really mean to say. Here are some examples of what NOT to say
that I have heard over the past few semes
i. [-i(.r.r) and path B using parametcrization {use the given path and orientation].
_. H.
r=-bt+-t _\
._.
3?,
UN;
' : 5t"'+%3
F
56
3-
Uiot L'§k"4~tk') cit .2 ~- : {§>
2. }"][_\'._v] and path B using, the: Irllltdalncntatl Theorem.
{50%} = 511:; + If +k
A
For problems 1 through 5, use the information below to find the value of the line integral using
the indicated technique. Show all work (on separate paper) and use proper notation.
F1 ( x, y) y 2i (2 xy 2 y) j
F2 ( x, y) yi
F3 ( x, y) xyi x 2 j
Path A: Th
Parametric Equations Section 17.1
There are an infinite number of ways to parameterize curves, but we want to be efficient in our
parameterizations and focus on forms that will make our work easier later. The following are
guidelines for best practices fo
In each cafthc problems. he sure to include clearly drawn sketches of}? fur double inlcgrals and H' thr
triple integrals. You might need to chamgc lhc cuordinatc system in urdcr 1L} cwuuic an integral]. ShUW
all work and use pmpcr nuialinn.
i4
1. Change
Chapter 16 Practice
Choose the best coordinate system in each case. If asked to evaluate an integral, make smart
choices for nding antiderivatives or use geometry in cases that allow it.
1. [T44 f(x,y)dydx SketchR.
arctanx
O
"e-a
'2.
2. 3+5: 1 t3
lLijsg\*
Chapter 16 Practice 1
Choose the best coordinate system in each case. If asked to evaluate an integral, make smart
choices for finding antiderivatives or use geometry in cases that allow it.
1
4
0
arctan x
1.
2.
f ( x, y)dydx
8
y 2
4
y 8
f ( x, y)dx
Notation and terminology for plane/wind problems:
i points in the east direction
j points in the north direction
k points upward (related to altitude)
In most problems in section 13.2, the plane is not changing altitude and the wind is blowing
horizontall
WRITING EQUATIONS FOR SURFACES - PRACTICE
The easiest way to create equations for surfaces is to start with a basic surface (see pdf) and modify it
using pre-calculus skills (see video). Although surfaces can be very elaborate, we will focus on the
ones y
The following are basic surfaces that you need to know. You also need to be able to write
equations for transformations (shifting, reflecting).
Surface
Equation
Name
1
x2 y 2 z 2 1
Sphere (ellipsoid)
2
z 2 x2 y 2
Cone (two)
3
z x2 y 2
Paraboloid
4
z y 2 x
Math 223 Vector Calculus Author: Arlo Caine
Practice Exam 4 Name: Solutions
Directions: Read all questions carefully. Use a pencil and erase all unnecessary marks. Show all of your work in the space provided and display your answer on the line given if re
Math 223 Vector Calculus Author: Arlo Caine
Practice Exam 3 Name:
Directions: Read all questions carefully. Use a pencil and erase all unnecessary marks. Show all of your work in the space provided and display your answer on the line given if requested. Y
Math 223 Vector Calculus Instructor: Arlo Caine
Practice Exam 3 Name: Solutions
Directions: Read all questions carefully. Use a pencil and erase all unnecessary marks. Show all of your work in the space provided and display your answer on the line given i
SHORT ANSWERS TO MATH 223 Review Problems
1. (a) Decreasing (b) Increasing 2. A is a cylinder of radius 2 about the y -axis: x2 + z 2 = 4 B is two parallel planes: x = 3 C is a double cone: x + y = z
2 2 2
21. (a) 12x + 4y + 3z = 5 (b) (c)
13 2 13 29
22.
1
MATH 223 REVIEW PROBLEMS
1. You are in a nicely heated cabin in the winter. Deciding that its too warm, you open a small window. Let be the temperature in the room, minutes after the window was opened, feet from the window. (a) Is an increasing or decre
Derivative Review Find the first derivative for each. Use proper notation and simplify. 1. f ( x) " x 2 ! bx ! c a 2. y (t ) " 3 t !2 3. f ( x) " x 2 cos( x) ! sec( x)
4. v " 3 tan(5t )
5. z (t ) " e3
6. w(r ) " # r r #
7. y " ln 5 ! x
2
$ y &5% 8. t ( y