Solutions to the Non-book problems.
(1) Find the perpendicular distance between the parallel planes
2x 3y + 5z = 1 and 2x 3y + 5z = 4
Solution: Pick a point P , such as P = (2, 0, 1) that is in the first
plane and a point Q, such as Q = (2, 0, 0) that is
Worksheet 3
(1) Consider the function z = f (x, y) whose graph contains the point
(2, 3, 7) and which satisfies fx (2, 3) = 2, fy (2, 3) = 5.
(a) Find the gradient of f at (2, 3)
Solution. f (2, 3) =< fx (2, 3), fy (2, 3) >=< 2, 5 >
(b) In what direction
Non-book problems
(Turn in what you can on Monday, September 5th)
(1) Find the perpendicular distance between the parallel planes
2x 3y + 5z = 1 and 2x 3y + 5z = 4
(2) Describe all lines through the origin that make an angle of
3
with
the xy-plane.
(3) Fi
Non-book problems for Section 14.1
The level set of a function w = f (x, y, z) at c = where 0 is given by
(x, y, z) : x2 + y 2 + z 2 =
1
1 + 2
.
Assume the level set for f at c for any c < 0 is the empty set.
(1) Can you tell if f attains a maximum or a m
Non-book problems for Section 14.1
The level set of a function w = f (x, y, z) at c = where 0 is given by
1
(x, y, z) : x + y + z =
1 + 2
2
2
2
.
(1) Can you tell if f attains a maximum or a minimum value?
(2) As , what is happening to the level sets?
(3)
Math 211, Exam I
September 13th, 2016
1
2
3
4
5
6
7
total
14
10
12
18
16
16
14
100
Show all work and write clearly! Do as much as you can and do not assume
the problems are in order of difficulty or importance.
1. Match the descriptions of the geometric o
Department of Mathematics
Bucknell University
MATH 202
Supplementary Handout
Revised Spring 2011
Graphing Polar Functions
The purpose of this handout is to provide some further practice at sketching the graphs of
polar functions. The problems aim to devel
Worksheet 2
Consider the function z = f (x, y) whose graph contains the point (2, 3, 7)
and which satisfies fx (2, 3) = 2, fy (2, 3) = 5.
(1) Find the equation of the tangent line to the trace of f at y = 3.
What is a direction vector for the tangent line
Math 4 Test 2 Study Guide
For the test on Friday, April 11, youre responsible for the sections we covered in chapters 2 and 3 of Lay,
plus section 4.1.
The key to preparing for the test is working lots of problems. Re-working homework problems would be
he
Math 4 Test 2 April 11, 2014
Name _
Put all your answers, together with your work, on separate paper.
You can use a calculator unless the problem says without using a calculator, in which case you need to
show the steps for working the problem manually. H
Math 4 Test 1 March 10, 2014
Name _
Put all your answers, together with your work, on separate paper.
You can use a calculator wherever its possible.
Exact answers only; no decimal approximations.
1.
Consider the following linear system:
2x1 x2 + 3x3 + 4x
Math 4 Test 1 Study Guide
Updated 3/4/2014, 3/8/2014
Youre responsible for all the sections of Chapter 1 in Lay, except section1.6 and 1.10, which are on
applications of linear systems. The applications are important, of course, but since everyone chose w
Math 4 Final Exam Practice Problems
Spring 2014 Jim Riley
3
7t
5
3 8
x2 t
The parametric form is :
5 3
1 11
x3 t
5 3
x4 t
x1
1.
3
7
5
8
3
3
The vector form, which is what the problem was supposed to say, is 5 x4 , where x4 is
11
1
3
Math 4 Final Exam Practice Problems
Spring 2014 Jim Riley
Problems 1 and 4 edited on 5/25/2014.
1.
Solve this linear system by any method and express the solution set in parametric form or vector
form.
3x1 4x2 + 7x3 + 6x4 = 2
x1
+ 3x3 4x4 = 0
2x1 + 7x2 +
Math 4 Syllabus Quiz
Name _
Short Answers
1.
What days does the class meet?
_
2.
What is the name of the textbook? _
3.
What is your instructors name?
4.
Where is your instructors office? _
5.
What day of the week will homework usually be collected? _
6.
Math 4 Final Exam Take-home Portion
Name _
Spring 2014 Jim Riley
This is due at the start of the timed final exam: Wednesday, May 28, 2014, at 11:30 a.m.
Exact answers only no decimal approximations. You can use technology for row-reducing matrices;
there
Math 4 Quiz 1 February 14, 2014
Name _
Due in class on Wednesday, February 19, 2014
This is a take-home quiz. You can use the textbook, your own notes, and any other materials (including
online ones) that are already published. Please dont consult other s
The effect of salt on the
boiling time of water
• Our purpose of the experiment was to see
if the boiling time of water was affected by
the amount of salt added to the water.
• If we found a difference, we wanted to see
if it was significantly different,
Robert Boyle described elements as a substance that could not be broken down any further Law of Mass Conservation Matter cannot be created nor destroyed Law of Definite Proportions Pure substances of the same thing always contain the same proporti
MATH 226 August 27, 2007 Data Collection Numerical Non-numerical (categorical) Thickness of a stack of paper Pages 353-609 = 128 pages Total thickness = 8 cm Thickness of a single piece = 62.5 m Validity make sure that the data expresses the prop
MATH 226 August 29, 2007 Comparative Study testing of a new method against an existing one o Paired data is preferred when collecting Example: Testing a suspension at the beginning and then retest after a certain amount Replication be sure that
MATH 226 September 3, 2007 A simple random sample of size n is a subset of the population containing n elements and chosen in such a way that any subset of the population containing n elements is equally likely to be chosen. Stratified random samp
MATH 226 September 5, 2007 Definition: x1 < x2 < . < xn ordered data For p = (i )/n, I = 1, 2, 3,., n o The quantile p is defined as Q(p) o (i )/10 Q(i )/10) i 1 .05 1.0 2 .15 3.5 3 .25 8.1 4 .35 8.2 5 .45 9.0 6 .55 9.7 7 .65 9.8 8 .75 12.0 9 .
MATH 226 September 10, 2007 Numerical Summaries o Summary of location Sample Median 0.5 quantile; Q(.5); very center of the data range Sample Mean sum of all numbers divided by number of numbers x-bar = sigma(x, 1n)/n Sample Mode most often o
MATH 226 September 12, 2007 Sample o X-bar = 3,000,000 o S = 2,827,948 Population o = 4,632,344 o = 4,500,267 where o s2 = (xi x-bar)2/(n-1) o 2 = (xi )2/N o = sqrt(N-1/N)*s
MATH 226 September 17, 2007 Least Squared Line given n sets of paired (x, y) data points We want to find: y-hat = 0 + 1x we minimize L = (yi yi-hat)2 = (yi 0 + 1xi )2 L/0 = -2(yi 0 + 1xi)(yi 1 1xi) L/1 = -2(yi 0 + 1xi)(yi 0 1xi) SEE BEST FI