Chapter 1: Geometry of R2 and R3
Section 1.3: lines and planes
Math 1229 A
September 22th 2014
Math 1229 A
We know that.
two distinct points in R2 or R3 determine a line.
three non-collinear points in R3 determine a plane.
In Section 1.3.
we will learn ho
Chapter 2: Euclidean m-space and linear equations
Section 2.3: Row Reduction of Linear Systems
Math 1229 A
October 8th 2014
Math 1229 A
In Section 2.3
we will learn how to use special algebraic objects (called matrices) to
simplify and solve linear system
Chapter 1: Geometry of R2 and R3
Section 1.3: lines and planes
Math 1229 A
September 24th 2014
Math 1229 A
Lines
Let P be a point in R2 (or in R3 ) and let v be a nonzero vector in R2
(or in R3 ). Denote by p the vector corresponding to P .
Problem
We wan
Chapter 1: Geometry of R2 and R3
Section 1.2: Dot and cross products
Math 1229 A
September 15th 2014
Math 1229 A
Dot product
Definition
Let u = (u1 , u2 ) and v = (v1 , v2 ) be vectors in R2 . The dot product (or
standard inner product) of u and v is
u v
Chapter 1: Geometry of R2 and R3
Today:
finish Section 1.1+ exercises
begin Section 1.2: Dot and cross products
Math 1229 A
September 12th 2014
Math 1229 A
Difference between two vectors
Difference between two vectors (definition)
The difference of the v
Chapter 3: Matrices
Section 3.2: Matrix Equations and Inverses
Math 1229 A
October 22nd 2014
Math 1229 A
In Section 3.1 we saw the operations that we can do with matrices: sum,
multiplication by a scalar, transpose and multiplication between two
matrices.
Chapter 2: Euclidean m-space and linear equations
Section 2.2: Systems of linear equations
Math 1229 A
October 3rd 2014
Math 1229 A
In Chapter 1 we saw that to find intersection points between lines
and planes we need to solve a set of 2 or 3 equations wh
Chapter 1: Geometry of R2 and R3
Today:
finish section 1.2 and exercises
begin Section 1.3: lines and planes
Math 1229 A
September 17th 2014
Math 1229 A
Cross product
Definition
Let u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ) be vectors in R3 . The cross
Chapter 1: Geometry of R2 and R3
Section 1.1: vectors in R2 and R3
Math 1229 A
September 10th 2014
Math 1229 A
A vector (x, y, z) has the same direction with a given vector
(a, b, c) if there exists a positive number s such that
(x, y, z) = s(a, b, c). Eq
Chapter 2: Euclidean m-space and linear equations
Section 2.3: Row Reduction of Linear Systems
Math 1229 A
October 6th 2014
Math 1229 A
In Section 2.3 we will learn how to use special algebraic objects
(called matrices) to simplify and solve linear system
Welcome to
MATH 1229 A
Methods of Matrix Algebra
Section 003
September 5th 2014
Section 003
Information about the instructor
I am Dr. Myrto Manolaki
email address: [email protected]
(Please write Math 1229A in the subject line of your emails)
Office: MC 120
Chapter 3: Matrices
Section 3.1: Operations on Matrices
Math 1229 A
October 15th 2014
Math 1229 A
In Chapters 1 and 2
1
We saw how to add, subtract and find scalars of vectors in Rm .
2
We introduced matrices to help us solve systems of linear equations.
Chapter 1: Geometry of R2 and R3
Section 1.1: vectors in R2 and R3
Math 1229 A
September 8th 2014
Math 1229 A
In Chapter 1
we will focus on
the set R2 of all ordered pairs (x, y) of real numbers;
the set R3 of all ordered triples (x, y, z) of real numbers
INTRODUCTION
TO REAL ANALYSIS
William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus
Department of Mathematics
Trinity University
San Antonio, Texas, USA
[email protected]
This book has been judged to meet the evaluation criteria set by
the
KING UNIVERSITY COLLEGE
S
at the University of Western Ontario
DEPARTMENT OF ECONOMICS, BUSINESS AND
MATHEMATICS
Mathematics 1600b
Final exam (take home part)
due Thursday, April 23, 2013, 7 p.m.
Instructor S.V. Kuzmin
Name (please print)
Student number
J
from 2011 tests
Page 1
PART A
1
mark
Mathematics 1225B
Test 2 Practice Test
CODE 111
(17 marks)
1. Find the area of the region bounded by the curves y = e3x , y = 0, x = 0 and x = 1.
A: e3 1
B:
e3
3
C:
1
3
D:
e3 1
3
E: e3
Solution: Since e3x > 0 everywher
1
Math 1225B Test 2 Practice Test, based on 2011 tests
Answers
Part A
1.
D
7.
E
13. A
2.
8.
14.
D
B
D
3.
9.
15.
E
A
C
4.
10.
16.
E
B
B
5.
11.
17.
B
E
E
6.
12.
D
B
Part B
18. area =
32
3
19. 3 ln 3 2 ln 2 1
20. 2 ln |x + 2| ln |x 3| + C or ln(x + 2)2 ln |x
based on 2005 abd 2006 tests
Page 1
Mathematics 1225B
Test 2 Practice Test
CODE 111
PART A
(17 marks)
1. Find the area of the region bounded by the curves y = e2x and y = 0 between x =
1
1
A: e e 2
B:
e e2
2
2
1
C: e e 4
1
and x = 1.
2
e2
e
2
2
D:
E: e2 e
from 2011 tests
Page 1
Mathematics 1225B
Test 2 Practice Test
CODE 111
PART A
(17 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET. Only the (scantron) answer sheet will be m
based on 2005 abd 2006 tests
Page 1
Mathematics 1225B
Test 2 Practice Test
CODE 111
PART A
(17 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET. Only the (scantron) answer sh
1
Math 1225B Test 2 Practice Test, based on tests from 2005 and 2006
Answers
Part A
1.
D
7.
E
13. C
2.
8.
14.
E
D
D
3.
9.
15.
Part B
18.
27
4
19. A = 5 and B = 3
20.
ex
(sin x + cos x) + C
2
A
B
D
4.
10.
16.
D
B
B
5.
11.
17.
E
E
B
6.
12.
A
D
from 2011 tests
Page 1
Mathematics 1225B
Practice Final Exam
CODE 111
PART A
(35 marks)
A1. If log3 x log3 x2 = 2, nd x.
A: 6
B: 3
C:
1
3
D: 9
E:
1
9
Solution:
log3 x log3 x2 = 2
A2. If f (x) = 4e
x
x
x2
log3
=2
log3
1
x
=2
1
= 32
x
1
1
=
32
9
x=
, nd f (
based on a 2006 exam
Page 1
Mathematics 1225B
Practice Final Examination
CODE 111
(35 marks)
PART A
A1. Simplify eln 8ln 2 .
A: 4
B: ln 4
C: 6
E: 82
D: ln 6
Solution: We use properties of exponents and logarithms:
eln 8ln 2 =
eln 8
8
= =4
eln 2
2
or
8
eln
1
Math 1225B Practice Final Exam, from 2011 tests
Answers
Part A
1.
E
6.
D
11. B
16. C
21. D
26. A
31. C
2.
7.
12.
17.
22.
27.
32.
B
A
E
A
A
E
B
3.
8.
13.
18.
23.
28.
33.
E
B
D
D
D
D
D
4.
9.
14.
19.
24.
29.
34.
D
E
D
B
A
C
C
5.
10.
15.
20.
25.
30.
35.
C
C