MATH1020
1
Some Extra Examples from Selected
Harder Topics
(Section )
Chapter 11
Example: Determine whether the sequence
en + en
an = 2
n 1
converges or
diverges. If it converges, find the limit.
Example: What does the divergent test tell us about converg
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Strategy for Integration (Section 7.5 of Stewart)
NOTE: Theres no new math being introduced todaytoday is all about
recognizing which technique to use when (the hardest part!) So well focus on
fig
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Trigonometric Substitution (Section 7.3 of Stewart)
Recall: We know how to integrate, for example,
Question: Now what about something such as, e.g.
x
4 x
2
dx
1
4 x
2
dx ?
Note: Technically, ( f (
Problem 1
A radio antenna broadcasts a 25 kW-power wave. Assume that the radiation is emitted uniformly in all
directions. If Nick is 1.0 km away from the antenna, how far must Anna be from the antenna in order
to receive six times the wave intensity rece
Student ID: -
Electrical, Computer, and Software Engineering
Introduction to Programming for Engineers (ENGR 1200U)
Midterm Exam Spring 2016 (May 31)
Time: 2:30 3:30pm
Student Name
Student ID
Instructions
PRINT your name and student ID on this exam page.
PHY1020U - Physics II
MIDTERM (I) Exam - version A
February 4, 2015
2:10 AM 3:30 PM
First Name
Last Name
Student Number
Signature of Student
INSTRUCTIONS:
Use a pen to fill in the front page. All exam answers should be written in ink in the
provided work
PHY1020U - Physics II
MIDTERM (II) Exam - version B
March 18, 2015
2:10 AM 3:30 PM
First Name
Last Name
Student Number
Signature of Student
INSTRUCTIONS:
Use a pen to fill in the front page. All exam answers should be written in ink in the
provided work
Midterm II Summary
The electric field: The electric field is defined
by the equation
Resistors: The potential difference across a resistor (in the direction of the current) is
~
F~on q = q E.
VR = IR.
Electric potential energy: The electric potential Seri
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Trigonometric Integrals (Section 7.2 of Stewart, pg. 471)
Recall: Weve dealt with integrating trig functions before, e.g. cos x sin8
Question: Now what about, for example,
Example:
sin
5
x
cos
3
x
STUDENTS WILL KNOW:
11.6: ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS
Understand the concept of absolute convergence, and why it is sufficient to check
for absolute convergence to determine if a series is convergent;
How to apply the ratio test (not
STUDENTS WILL KNOW:
15.2: DOUBLE INTEGRALS OVER GENERAL REGIONS
How to evaluate definite integrals over general regions;
How to evaluate definite integrals over type I and type II regions.
11.10: TAYLOR AND MACLAURIN SERIES
The definition of a Taylor and
STUDENTS WILL KNOW:
9.1: MODELLING WITH DIFFERENTIAL EQUATIONS
Basic understanding of the meaning of a differential equation, and how to set up a
differential equation for a simple mathematical modelling scenario;
How to verify if a given function is a so
FACULTY OF SCIENCE
MATH1020U: Calculus II
Course outline for Spring, 2016
1. Course Details & Important Dates*
Course Type CRN
Day
Lecture
10040
11226
M/W
M/W
Time
Location
9:10AM-12:00PM
1:10PM-4:00PM
UA1350
UA1350
Classes Start
Classes End
Final Exam Pe
ASSIGNMENT #2
DUE DATE: This assignment is to be submitted entirely on
paper on Tuesday May 23rd by 12pm in your TAs drop box
located on the 4th floor of the UA building (near the science
main office). Hand in one copy per pair.
ASSIGNMENT (27 marks total
Assignment #1
DUE DATE: This assignment is to be submitted entirely on
paper on Tuesday May 16th by 12pm in your TAs drop box
located on the 4th floor of the UA building (near the science
main office). Hand in one copy per pair.
ASSIGNMENT (25 marks total
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Trigonometric Integrals (Section 7.2 of Stewart, pg. 471)
Recall: Weve dealt with integrating trig functions before, e.g.
Question: Now what about, for example,
Example:
sin
5
x cos x dx
cos
3
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Approximate Integration (Section 7.7 of Stewart)
We have now learned several techniques of integration, but do they always work?
Recall: Back in Calc I/Intro Calc, we studied several numerical app
MATH1020U: Chapter 9 cont & Chapter 10
1
DIFFERENTIAL EQUATIONS cont
Separable Equations (Section 9.3 of Stewart)
Recall: Weve just spent time solving differential equations graphically and numerically,
but is it possible to get an exact solution?
Definit
MATH1020: Assignment Worksheet #1
What were working on today: Integration by Parts and Trig Integration
Activity 1 (~20 min total): [This is pg. 514 #3]
a) (5 min max) Use Maple to evaluate the following integrals.
i)
ln xdx
ii)
x ln xdx
iii)
x
2
ln xdx
MATH1020U: Chapter 10 cont & Chapter 14
1
PARAMETRIC EQUATIONS AND POLAR
COORDINATES cont
Polar Coordinates (Section 10.3 of Stewart)
Recall: In Calculus I, we studied the unit circle.
x r cos( )
y r sin( )
y
tan( )
x
r 2 x 2 y 2
Example: The Polar coord
MATH1020U: Chapter 7 cont
1
TECHNIQUES OF INTEGRATION cont
Trigonometric Substitution (Section 7.3 of Stewart)
Recall: We know how to integrate, for example,
Question: Now what about something such as, e.g.
Question: What have we done?
x
4 x2
dx
1
4 x2
dx
MATH1020U: Chapter 7 cont and 8
1
TECHNIQUES OF INTEGRATION cont
Improper Integrals (Section 7.8 of Stewart) cont
Recall: Last day, we evaluated Type I improper integralslets get some more practice.
Application: The rate at which a pollutant is being dump
MATH1020U: Chapter 9
1
DIFFERENTIAL EQUATIONS
Modelling with Differential Equations (9.1 of Stewart)
Recall: Throughout your mathematics education, youve learned how to solve various
types of equations. However, other types of equations are also possible.
MATH1020U: Chapter 15 cont & Chapter 11
1
MULTIPLE INTEGRALS cont.
Double Integrals over General Regions (Section 15.2 of Stewart)
Recall: Last lecture, we looked at double integrals over rectangular regions. The
problem with this is that most of the regi