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
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.
FACULTY OF SCIENCE
MATH1020U: Calculus II
Course outline for Winter, 2017
1. Course Details & Important Dates*
Course Type
CRN
Day
Lecture
70154
70155
70167
70851
T/F
M/W
M/R
M/W
Time
Location
6:40 8 pm / 5:10 6:30 pm
9:40 11 am / 8:10 9:30 am
11:10 am 1
MATH1020U: Chapter 14 cont
1
PARTIAL DERIVATIVES cont.
The Chain Rule (Section 14.5 of Stewart)
Recall: In Calc I/Intro Calc, we introduced the chain rule for differentiating composite
functions:
If y f (x) and x g (t ) , then
dy dy dx
dt
dx dt
The Chain
MATH1020: Assignment Worksheet #6
What were working on today: Separable Differential Equations (just some review since we
have time); Parametric Curves
Activity 1 (~15 min): Your TA will run a game to get more practice at some basic concepts
from our work
MATH1020: Assignment Worksheet #4
What we are working on today: Improper Integrals, and Applications of Integration
Activity 1 (~15 min): Your TA will run a game to practice some basic concepts from the past
week!
Activity 2 (~10 min):
Consider the curve
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
1
TECHNIQUES OF INTEGRATION
Integration by Parts (Section 7.1 of Stewart; pg. 464)
Recall: So far, we have learned how to integrate some basic functions.
e.g. x ( x 1)dx
x2
e.g. xe dx
Now we will continue to investigate more advanced
MATH1020U: Chapter 10
1
PARAMETRIC EQUATIONS AND POLAR
COORDINATES
Curves Defined by Parametric Equations (Section 10.1)
Recall: A function is a relationship that passes the vertical line test.
Suppose that x and y are both given as functions of a third v
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
MATH1020U: Chapter 8 cont
1
APPLICATIONS OF INTEGRATION cont
Applications to Physics and Engineering (8.3 of Stewart)
NOTE: Well cover the section on Hydrostatic Pressure and Force only, not Moments
and Centers of Mass.
Here, we are interested in finding
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.
MATH1020: Assignment Worksheet #4
What were working on today: Improper Integrals, and Applications of Integration
Activity 1 (~15 min): Your TA will run a game to practice some basic concepts from the past
week!
Activity 2 (~10 min):
Consider the curve
f
MATH1020U: Chapter 14
1
PARTIAL DERIVATIVES
Functions of Several Variables (Section 14.1 of Stewart)
In the real world, things typically depend on more than 1 independent variable.
Application: Braking distance for a car
Application: Diving Tables. P 1
l
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
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 7 cont
1
TECHNIQUES OF INTEGRATION cont
Integration of Rational Functions by Partial Fractions (7.4)
1
3
dx .
x 1 x 2
Recall: We know how to integrate, for example,
4x 5
dx ?
But what about 2
x x 2
Question: Is there any simila
MATH1020U: Chapter 9 cont
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?
Definition: A separa
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 x2
dx
1
4 x2
dx ?
Note: Technically, ( f ( )
MATH1020: Assignment Worksheet #6
What were working on today: Separable Differential Equations (just some review since we
have time); Parametric Curves
Activity 1 (~15 min): Your TA will run a game to get more practice at some basic concepts
from our work
MATH1010: Chapter 4 cont
1
APPLICATIONS OF DERIVATIVES cont
Antiderivatives (Section 4.9)
Recall: Last day, we studied antiderivatives. Sometimes, it is difficult or impossible to
find the antiderivative of a function, but we can still gather info about
MATH1010: Chapter 3 cont; 4
1
DIFFERENTIATION RULES cont
Linear Approximations and Differentials (Section 3.10)
Recall: We first started studying derivatives because we were interested in finding
tangent lines to curves.
The tangent line is an approximati
MATH1010: Chapter 4 cont
1
APPLICATIONS OF DERIVATIVES cont
Optimization Problems (Section 4.7) cont
Recall: Last day, we reviewed optimization problemslets get some more practice!
Application: Consider a cylindrical tin can which is to be constructed by
MATH1010: Chapter 4 cont
1
APPLICATIONS OF DERIVATIVES cont
How Derivatives Affect the Shape of a Graph (Section 4.3)
Problem: Suppose you are asked to sketch f ( x) 2 x 3 3x 2 12 x . Plotting points is
obviously tedious. Using graphing software may miss
MATH1010: Chapter 5 cont
1
INTEGRALS cont
The Definite Integral (Section 5.2)
Recall: Last class, we were interested in computing the limit of an infinite sum. Just like
f ( x h) f ( x )
earlier in the course when lim
came up frequently and was given the
MATH1010: Chapter 5 cont
1
INTEGRATION cont
The Fundamental Theorem of Calculus (Section 5.3)
Recall: So far, we have been interested in finding derivatives (slope) and integrals (area)
how are these two things related?
x
f (t )dt with a x b
Consider the
MATH1010: Chapter 4 cont
1
APPLICATIONS OF DERIVATIVES cont
Summary of Curve Sketching (Section 4.5)
Recall: Last day, we discussed the importance of Calculus in sketching curvesif you
use software to plot the function, you might miss important features