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A comprehensive course outline from McMaster Scholars
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McMaster Scholars Math 1ZB3 Summary Booklet
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2 Table of Contents The Purpose of this Booklet .......................................................................................................................................3
Section 6 to 7.8: Review for Integration .....................................................................................................................4
Types of integration................................................................................................................................................4
Formulae/Examples ................................................................................................................................................4
Improper Integrals .................................................................................................................................................5
Convergent/Divergent definitions: .........................................................................................................................6
What about asymptotes? ......................................................................................................................................8
Approximation ........................................................................................................................................................9
Section 11 to 11.10: Sequences/Series ................................................................................................................... 10
Squeeze Theorem: ............................................................................................................................................... 10
Important points about sequences: .................................................................................................................... 11
Series Summary ................................................................................................................................................... 11
Sum of geometric series: ..................................................................................................................................... 13
Integral Test ......................................................................................................................................................... 13
Absolute Convergence ......................................................................................................................................... 13
Power Series ........................................................................................................................................................ 13
Representing functions as power series.............................................................................................................. 15
Taylor/McLaren Series ......................................................................................................................................... 16
Important series .................................................................................................................................................. 18
Section 8 to 8.2: Applications of Integeration ......................................................................................................... 18
Arc Length ............................................................................................................................................................ 18
Area of a Surface of Revolution ........................................................................................................................... 19
Section 9 to 10.5: Differential Equations................................................................................................................. 19
Population Growth .............................................................................................................................................. 19
Separable Equations ............................................................................................................................................ 20
Orthogonal Trajectories ...................................................................................................................................... 20
Linear Equations .................................................................................................................................................. 20
Calculus on Parametric Curves ............................................................................................................................ 21
Arc Length : ...................................................................................................................................................... 21
Surface Area .................................................................................................................................................... 21
Polar Coordinates ................................................................................................................................................ 22
McMaster Scholars Math 1ZB3 Summary Booklet
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3 Conic Sections ...................................................................................................................................................... 22
Ellipses ................................................................................................................................................................. 23
Hyperbolas ........................................................................................................................................................... 23
Section 14: Functions of two variables .................................................................................................................... 24
Level Curves ......................................................................................................................................................... 24
Limits ................................................................................................................................................................... 25
Partial derivatives ................................................................................................................................................ 25
Tangent Planes .................................................................................................................................................... 26
Chain Rule on Multivariable ................................................................................................................................ 27
Directional Derivatives and Gradient Vector....................................................................................................... 28
Integration on Multivariable ............................................................................................................................... 29
Iterated Integrals ................................................................................................................................................ 30
The Purpose of this Booklet
This booklet is a collection of important points in this course. This is also not meant to be heavy reading.
We have tried to condense it into as few pages as possible so that you can jump to chapters or topics, see the
important terms, important formulae, and a quick explanation, and finally important points to keep in mind.
Most of the concepts/equations are from the textbook, the rest is from resources on the MacEng 15
page and web assign assignments.
Good luck on your exams! Here is a link to the MacEng page for further resources! :
https://sites.google.com/site/macengfifteen/home
McMaster Scholars Math 1ZB3 Summary Booklet
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4 Section 6 to 7.8: Review for Integration
Types of integration U subs:
Parts:
Trig Subs:
Partials:
Formulae/Examples Important formulas to memorize:
McMaster Scholars Math 1ZB3 Summary Booklet
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Improper Integrals
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The improper integral could also be flipped:
Convergent/Divergent definitions:
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Therefore:
McMaster Scholars Math 1ZB3 Summary Booklet
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8 What about asymptotes?
( ( )) This wont work because 1/(x-1) is not continuous and does not
obey FTC. Finding the integral of a function with a discontinuity requires finding the integral before and after the
discontinuity.
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9 Approximation
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10 Section 11 to 11.10: Sequences/Series
Definitions:
Convergent/Divergent: is similar to integration basically as n approaches infinity the nth term must approach a
limit to be convergent, etc..
Monotonic Sequence is a bounded sequence and is convergent.
Squeeze Theorem:
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11 Important points about sequences:
Series Summary We will go over each of these types in the review. Here is a summary of that session.
The credit for the summary goes to MacEng page and those who worked on the flowchart.
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12
NO
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13 Sum of geometric series: a / (1-r) if abs of r
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14
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15 Representing functions as power series Recall:
Try and make your function look like the one you know by using algebraic manipulation!
You can
also take the integral and differentiate these sums by the following method:
When taking the derivative the sum index n = 0 goes to n =1 etc
When taking integral the radius of convergence stays the same.
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16 Taylor/McLaren Series Used to approximate functions
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17 The error associated:
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Important series
Section 8 to 8.2: Applications of Integeration
Arc Length
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19 Area of a Surface of Revolution
Section 9 to 10.5: Differential Equations
Population Growth People per time. If M < P the the population is
decreasing. If M>P its increasing. This is known as the logistic
differential equation.
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20 Separable Equations
Then by bringing like terms together:
( ) ( )
Orthogonal Trajectories For example:
Take derivative with respect to y. take negative reciprocal and separate.
( )
Linear Equations
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21
Calculus on Parametric Curves
Arc Length :
Surface Area
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22 Polar Coordinates Converting between polar and Cartesian:
x = r cos , y = r sin if you ever forget, think of SOHCAHTOA.
Also, the equation of a circle: x 2 + y 2 = r 2
Distance between two points:
1. Draw lines from each point to the origin and from one point to the other point.
2. Find angle between lines from origin to each point
3. Cosine law: a = b2 +c 2 2(bc )cosA
Drawing Cartesianpolar form:
1. Assume the vertical axis is r and the horizontal axis is theta.
2. Start at theta = 0 and identify inflection points.
Use these points to identify where the radius is increasing/decreasing as you spin (theta
increases).
Conic Sections
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23 Ellipses
To find ellipse that is rotated 90 degrees in the shape of a standing egg just interchange x and y and foci switches
to (c, 0 ) and vertices switch to (a,0)
Hyperbolas
Verticies/major axis
Minor Axis
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24 Section 14: Functions of two variables
Level Curves
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25 Limits
There are various paths to take for example.
y = x
Y = mx
Y=x^2
Partial derivatives
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26
Tangent Planes
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27 Chain Rule on Multivariable
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28 Directional Derivatives and Gradient Vector
Gradient Vector: ( )
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29
Integration on Multivariable
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30 Iterated Integrals
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