9147atar notes 3u maths lecture slides final

7/25/2019 9147ATAR Notes 3U Maths Lecture Slides FINAL

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Integration, Acceleration and Induction

HSC Head Start lecture

EXTENSION

M THEM TICS

Presented by:

JACOB SILOVE



TIMETABLE

• Who am I?

• General HSC tips

• Course-specific tips

• Content

Today’s Lecture



HIGH SCHOOL

Year 11 2013):

• Pacific Project

• Mathematics- 96

Year 12 2014):

• International Science Forum

• ATAR- 99.80 –

Physics- 93

–

Chemistry- 93

–

Modern History- 93

–

Extension 2 Maths- 94

–

Advanced English- 95

–

Extension 1 Maths- 98

Who am I?



DON’T PANIC! MESSING UP ONE ASSESSMENT ISN’T THE

END OF THE WORLD

GENERAL HSC TIPS



DON’T COMPETE WITH EACH OTHER. IT ISN’T REALLY WORTH

TRYING TO UNDERSTAND WHY THE HSC WORKS: LEAVE THAT

TO THE MAGICIANS AT THE BOARD OF STUDIES.

GENERAL HSC TIPS



DON’T BE AFRAID TO USE COLOURS

GENERAL HSC TIPS



DON’T JUST WRITE NOTES: WRITE WORKSHEETS

GENERAL HSC TIPS



STUDY IN GROUPS

GENERAL HSC TIPS



PAST PAPERS. PAST PAPERS. PAST PAPERS. PAST PAPERS.PAST PAPERS. PAST PAPERS. PAST PAPERS. PAST PAPERS.

PAST PAPERS. PAST PAPERS. PAST PAPERS. PAST PAPERS.









COURSE SPECIC TIPS













COURSE SPECIC TIPS

IF YOU DON’T UNDERSTAND SOMETHING,

ASK A SMARTER STUDENT. IF YOU ARE

THE SMARTEST STUDENT, ASK YOUR

TEACHER. IF YOU ARE SMARTER THAN

YOUR TEACHER… GET A TUTOR?













COURSE SPECIC TIPS

FORM SMALL STUDY GROUPS













COURSE SPECIC TIPS

FORMULA SHEET













COURSE SPECIC TIPS

UTALISE ONLINE RESOURCES

- KHAN ACADEMY













COURSE SPECIC TIPS

WRITE OUT QUESTIONS

-

QUESTION YOU GOT WRONG

- DIFFICULT QUESTIONS IN PAST PAPERS

- QUESTIONS THAT ALWAYS COME UP

-

etc













COURSE SPECIC TIPS

MATHEMATICS PAPER AS A BREAK



•

Methods of Integration

• Projectile Motion

• Induction

• Binomial Theorem

•

Velocity and Acceleration as a function of x*

• Simple Harmonic Motion**

• Further Probability***

* If we have time

** Highly doubt that we will have time

*** Wow, I’ve seriously misjudged the timing for this presentation

CONTENT



•

Method of finding the area under a curve

• Think of it as reverse-differentiation

• !"#$%&'()*( is the standard form for integration

• Eg. *-*( (# ! "!( , therefore !#% !(*( " (# ! #. (Will

explain +C)

INTEGRATION



•

INTEGRATION



Why plus C?

• Differentiate / "$(# ! #!( #%

• Differentiate / "$(# ! #!( #&

• Differentiate / "$(# ! #!( #'(((((

•

Differentiate / "$(# ! #!( )$*+!,

INTEGRATION



Why plus C?

Integral of /# - "!( could be :

. /" (# !

.

/" (# ! #'((((

. /" (# ! )'((($0$111!'1'0(!10('1!(01'!(!0'(!0

- Etc.

- ALWAYS PLUS C FOR INDEFINITE INTEGRALS!!!!

INTEGRATION



Indefinite vs definite integrals

• Indefinite

– !#%&'()*(

– Finds the primitive graph

–

Eg !#% !(*( "(# ! #.• Definite

– !"#$%&'()*(

– Finds an area beneath a graph

–

Eg ! (# '% !(*( "0(# ! 12 (# ' "' ' )# ! )' ( )# ! "'

INTEGRATION



Some questions

Find the indefinite integral of ) 3 +# 3#

INTEGRATION



Some questions

Find the definite integral of 3 - #43#5 between x = 0

and x = 1

INTEGRATION



Substitution• Always given

Three easy steps

•

Find the derivative

• Cancel terms and substitute

• Convert back into the original variable

METHODS OF INTEGRATION



Find ! 6 4-5 # 4 64+

using the substitution 4" 7 )




Find !#% 35 # 3 638 using the substitution

7" 3 #




•

Find the derivative• Make substitutions and cancel

• Make sure to change variable back into the original one




( "*89:;"<=>=?@ *(-*@ " ( "A=;B<8@/

*# ! (-* @# ! " *( -*@ " ( ""<<=;=C"@8B?

PROJECTILE MOTION



Particle projected verticallyInitially, @ "(8 ( "(8 A "D

PROJECTILE MOTION

Vertical Horizontal



Particle projected at an angle alphaInitially, @ "(8 ( "(8 A "D

PROJECTILE MOTION

!

#

D 2 /

D 2

(

Vertical Horizontal



PROJECTILE MOTION



•

Understand initial conditions• Be able to quickly integrate

• Understand what the question is actually asking

• There are only like 3 possible questions!

PROJECTILE MOTION



•

A method used to prove relationships true• We do this by utilising the following steps for set S(n):

– Prove that E 9': is true,

– Assume E 9F : is true,

–

Prove E 9F #': is true, – State true for all n

INDUCTION



• Prove true for n = 1

INDUCTION



• Assume true for n = k

INDUCTION



• Prove true for n = k+1

INDUCTION



• State true for all n

INDUCTION



• Prove true for n = 1

INDUCTION



• Assume true for n = k

INDUCTION



• Prove true for n = k+1

INDUCTION



• State true for all n

INDUCTION



•

Remember the steps required to prove an inductionquestion true

• Make intelligent substitutions

• Make sure to write the conclusions!

INDUCTION



Find the 34th term:

'( #' - $ )# &%

BINOMIAL THEORUM



Find the 34th term:

'( #' - $ )# &% !+1,*((!'*%; '(# '( ; ' ' - $ )# $$ ; (# '!

(+(((((%',$%$; (# '!

BINOMIAL THEORUM



Let’s examine an easier example, to find a general rule.

' '#()#?' '#() ;' '#() ;' '#() ;' '#() ;<;' '#() ? @8>=9

'# G( #H(# ! #.(# $ #<#I(#?nCk '?JF )

BINOMIAL THEORUM




' '#()#?' '#() ;' '#() ;' '#() ;' '#() ;<;' '#() ? @8>=9

'# G( #H(# ! #.(# $ #<#I(#?nCk '?JF )

'?J ( ) #'?J ' )( #'?J ! )(# ! #'?J $ )(# $ #<#'?J? )(#?

BINOMIAL THEORUM




' '#()# ( " ' '#()# ' " # (

' '#()# ! " # ( # (# ! ' '#()# $ " #=( #= (# ! # (# $

' '#()# & " #4( #> (# ! # 4(# $ # (# & ' '#()# % " #?( # 6 (# ! # 6(# $ # ?(# & # (# %

BINOMIAL THEORUM




'

' '

' ! ' ' $ $ '

' & * & '

' % '( '( % '

BINOMIAL THEORUM



KK

KK

'?JF ) "? @ -'? )F) @F @

BINOMIAL THEORUM




'L #()#?

'?J ( )L#? #'?J ' )L#? )' ( #'?J ! )L#? )! (# ! #<#'?J? )' )L(#? )' #'?J? )(#?

BINOMIAL THEORUM



BINOMIAL THEORUM



Greatest term questions' $#%()# !(

BINOMIAL THEORUM




'?JF )"#? )F $#F' !(JF ) $# !()F %#F

BINOMIAL THEORUM




'?JF )"#? )F $#F' !(JF ) $# !()F %#F

If Tk A Tk+1, then Tk will be the greatest term.

M2F #' -M2F " ? @ -'? )F #' ) @'F #' ) @ ; '? )F) @F @ -? @ ;%#F #' ; $# '0)F - %#F ;$# !()F

!()F- 9F #': ;% - $

BINOMIAL THEORUM



!()F- 9F #': ;% - $ This exceeds one between F "' "?* F "'!, but then is less

than one. The greatest term is therefore F "'$, which has

coefficient

' !(J '$ ) $# !()'$ %# '$ " ' !(J '$ ) $# , %# '$

BINOMIAL THEORUM



•

Remember that the term expansion equation is on theformula sheet.

• Try to memorise the rough ways you solve various types of

questions.

•

This is a tough topic; practice practice practice!

BINOMIAL THEORUM



From the Mathematics course:( "*89:;"<=>=?@

*(-*@ " ( "A=;B<8@/ *# ! (-* @# ! " *( -*@ " ( ""<<=;=C"@8B?

VELOCITY AND ACCELERATION




*(-*@ " ( "A=;B<8@/ *# ! (-* @# ! " *( -*@ " ( ""<<=;=C"@8B?

( "" " *A-*@ " *A-*( ; *(-*@





*(-*@ " ( "A=;B<8@/ *# ! (-* @# ! " *( -*@ " ( ""<<=;=C"@8B?

( "" " *A-*@ " *A-*( ; *(-*@( "" " *A-*@ " *A-*( ; *(-*@ " *A-*( ;A





*(-*@ " ( "A=;B<8@/ *# ! (-* @# ! " *( -*@ " ( ""<<=;=C"@8B?

( "" " *A-*@ " *A-*( ; *(-*@( "" " *A-*@ " *A-*( ; *(-*@ " *A-*( ;A

( "" " *A-*@ " *A-*( ; *(-*@ " *A-*( ;A " *-*( ' ' - ! A# ! )




•

Remember that the relevant formulas are on your formulasheet.

• Never forget to + C, because that’s usually half of the

answer!

•

Remember that you can play with derivatives as much asyou want. Flip em, multiply em, divide em up using the

chain rule.


QUESTIONS



QUESTIONS

QUESTIONS



QUESTIONS

Th k !



Jacob Silove

[email protected]

Thanks!

9147atar notes 3u maths lecture slides final

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