Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Spring 2013
Math 1225B lesson 21
March 24, 2014
Section 573
Dr. Joseph Turnbull
Email: jturnbu7@uwo.ca
Section 573 (Kings)
Location: W166
M 9:30  11:10 AM
W 10:30  11:20 AM
Check OWL
Lesson notes, announcements,
links, sample tests.
Final exam
Monday, April 28
2:00
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 4:
Antiderivatives and Integration
(text reference: Sections 8.1 and 8.2
custom text pgs. 81  110)
c V. Olds 2012
47
Unit 4
4
Antiderivatives and Integration
Now we start getting into some new calculus, not just review of things youve l
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 10:
Optimization
(text reference: Sections 10.3 and 10.5
custom text pgs. 230  250)
c V. Olds 2012
141
Unit 10
10
Optimization
You will remember that for a function y = f (x), we use the term relative or local extreme values
to refer to
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 8:
Other Integration Methods
(text reference: Sections 9.1, 9.2 and 9.5
custom text pgs. 175  198)
c V. Olds 2012
110
8
Unit 8
Other Integration Methods
In this section we learn two new integration techniques. We know that if an integra
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 9:
Functions of Two or More Variables
(text reference: Sections 10.1 and 10.2
custom text pgs. 207  226)
c V. Olds 2012
128
9
Unit 9
Functions of Two or More Variables
When we talk about a function of a certain number of variables we me
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF MATHEMATICS
Mathematics 1225B
Methods of Calculus
January  April 2013
NOTE: Many science/technically oriented courses and programs, at Western and at other universities, require a fullyear calculus course (
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 2:
Review of Introductory Calculus
Dierentiation Rules and Other Calculus Techniques
(text reference: Sections 5.3 and 5.4 mixed together
custom text pgs. 19  35)
c V. Olds 2012
14
Unit 2
2
Dierentiation of Exponential and Logarithmic F
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 1:
Exponential and Logarithmic Functions
(text reference: Sections 5.1 and 5.2 mixed together
custom text pgs. 3  16)
c V. Olds 2012
1
Unit 1
1
Exponential and Logarithmic Functions
We start the course with a review of exponential and l
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 5:
The Denite Integral
(text reference: Sections 8.4 and 8.5
custom text pgs. 124  142)
c V. Olds 2012
64
5
Unit 5
The Denite Integral
Next, were going to learn about some new notation. Recall that
f (x)dx is the general antideriva
tiv
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 11:
FirstOrder Dierential Equations
(text reference: Section 7.1 and 7.4
custom text pgs. 256  261 and 272  276)
c V. Olds 2012
167
Unit 11
11
FirstOrder Dierential Equations
The last topic of the course is Dierential Equations. This
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 12:
Applications of Dierential Equations
(text reference: Sections 7.1, 5.5 and 7.4
custom text pgs. 261265, 3741 and 276  277)
c V. Olds 2012
175
Unit 12
12
Applications of Dierential Equations
We will nish up the course by looking a
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 7:
Volumes of Revolution
(text reference: Section 5.2
custom text pgs. 160  168)
c V. Olds 2012
92
7
Unit 7
Volumes of Revolution
Next, we want to think about a new kind of problem, which is another application of denite integrals. The
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 3:
Trigonometric Functions
(text reference: Sections 6.2 and 6.4
custom text pgs. 56  76)
c V. Olds 2012
Unit 3
3
31
Trigonometric Functions
In this chapter, our main focus is on doing Calculus with Trigonometric Functions, which is in
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 6:
The Area Between Two Curves
(text reference: Section 5.1
custom text pgs. 148  155)
c V. Olds 2012
76
6
Unit 6
The Area Between Two Curves
In this unit we will learn how to use denite integrals to calculate area. You know how to calc
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Spring 2013
Math 1225B lesson 19
March 17, 2014
Section 573
Dr. Joseph Turnbull
Email: jturnbu7@uwo.ca
Section 573 (Kings)
Location: W166
M 9:30  11:10 AM
W 10:30  11:20 AM
Check OWL
Lesson notes, announcements,
links, sample tests.
Final exam
Monday, April 28
2:00
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Spring 2013
Math 1225B lesson 24
April 2, 2014
Section 573
Dr. Joseph Turnbull
Email: jturnbu7@uwo.ca
Section 573 (Kings)
Location: W166
M 9:30  11:10 AM
W 10:30  11:20 AM
Check OWL
Lesson notes, announcements,
links, sample tests.
Final exam
Monday, April 28
2:00
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Spring 2014
Saturday April 27, 2013
Page 1
Mathematics 1225B
Final Exam
CODE 111
PART A
(35 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
A1. If f (x) = log3 (x 3)8 , find f (4).
8
l
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Spring 2014
Friday April 27, 2012
Page 1
Mathematics 1225B
Final Exam
CODE 111
PART A
(35 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
CODED ON THE SCANTRON SHEET. ALSO CIRCLE YOUR ANSWERS
IN THIS BOOKLET.
ex
, find f (x).
ln x
A1. If f (x) =
A: 0
B:
!
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Spring 2014
from 2011 tests
Page 1
Mathematics 1225B
Practice Final Exam
CODE 111
PART A
(35 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 ma
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Math 1225A/B
Unit 1:
Exponential and Logarithmic Functions
(text reference: Sections 5.1 and 5.2 mixed together
custom text pgs. 3  16)
c V. Olds 2012
1
Unit 1
1
Exponential and Logarithmic Functions
We start the course with a review of exponential and l
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
2
Main Integral Formulas
1.
2.
1
un du = u +1 + C
n
4
3
e.g : 2x x + 1)dx = 2 x )
4
1
u du
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
1
x 2 1
1
2 +1
+x+C
= lnu + C
21
1
1
e.g : x + x2 )dx = lnx + x 2+1 + C = lnx
u
u
e du = e + C
u
u
a du = ln + C
sinu)du = co
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
Friday, March 16, 2012
Page 1
Mathematics 1225B
Test 2
CODE 111
PART A (17 marks)
1. [ 1
1
and the xaxis between x = 2 and x = 5.
x
mark ] Find the area of the region bounded by y =
A: ln 3
B: ln 10
C:
ln 5
ln 2
D: ln
2
5
E: ln
5
2
1
lies above the xaxi
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
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
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
1
Math 1225B Practice Final Exam
(based on a 2006 exam)
Answers
Part A
1.
A
6.
E
11. B
16. B
21. D
26. C
31. C
2.
7.
12.
17.
22.
27.
32.
B
E
D
A
B
C
D
3.
8.
13.
18.
23.
28.
33.
D
A
A
D
E
E
A
4.
9.
14.
19.
24.
29.
34.
B
D
E
D
D
C
E
5.
10.
15.
20.
25.
30.
3
Western University (Ontario)  Also known as University of Western Ontario
MATH 1225

Winter 2012
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