1zb3 mcmaster scholars - booklet content

McMaster Scholars Math 1ZB3 Summary Booklet

2

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


3

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


4

4 Section 6 to 7.8: Review for Integration

Types of integration U subs:

Parts:

Trig Subs:

Partials:

Formulae/Examples Important formulas to memorize:


5

5

Improper Integrals


6

6

The improper integral could also be flipped:

Convergent/Divergent definitions:


7

7

Therefore:


8

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.


9

9 Approximation


10

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:


11

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.


12

12

NO


13

13 Sum of geometric series: a / (1-r) if abs of r


14

14


15

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.


16

16 Taylor/McLaren Series Used to approximate functions


17

17 The error associated:


18

18

Important series

Section 8 to 8.2: Applications of Integeration

Arc Length


19

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.


20

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


21

21

Calculus on Parametric Curves

Arc Length :

Surface Area


22

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


23

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


24

24 Section 14: Functions of two variables

Level Curves


25

25 Limits

There are various paths to take for example.

y = x

Y = mx

Y=x^2

Partial derivatives


26

26

Tangent Planes


27

27 Chain Rule on Multivariable


28

28 Directional Derivatives and Gradient Vector

Gradient Vector: ( )


29

29

Integration on Multivariable


30

30 Iterated Integrals

1zb3 mcmaster scholars - booklet content

Documents