specialist maths sample

7/31/2019 Specialist Maths Sample

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VCE Specialist Maths Unit 4 Notes

Written by Abraham Rizkalla

[email protected]

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!Connect!Education!2011! 2!

Contents

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1.1!! ! Set!Notation!! ! ! ! ! ! ! 4!

! 2!!!!!! Calculus:!Differentiation!!

2.1!!! ! The!Fundamental!Theorem!of!Calculus! ! ! ! 5!

2.2!! ! Derivatives!of!x= f(y) 7!

2.3!!! ! Derivatives!of!Inverse!Circular!Functions!! ! ! 8!

2.4.1!! ! Double!Derivatives! ! ! ! ! ! ! 12!

2.4.2!! ! Using!the!Double!Derivative!to!Classify!Stationary!Points! 14!

2.5!! ! Related!Rates! ! ! ! ! ! ! 15!2.6!! ! Implicit!Differentiation! ! ! ! ! ! 18!

! 3!!!!! Calculus:!Integration!

3.1!! ! Integration!of!Polynomials!and!Transcendental!Functions! 22!

3.2!! ! Integration!Involving!Inverse!Circular!Functions! ! 23!

3.3! ! Integration!by!Substitution! ! ! ! ! 25!

3.4! ! Integration!by!Linear!Substitution! ! ! ! 27!

3.5!! ! Integration!Using!Trigonometric!Identities! ! ! 29!

3.6!! ! Integration!with!Partial!Fractions!! ! ! ! 31!

! 4!!!!!! Applications!of!Calculus!

4.1.1! ! Volumes!of!Solids!of!Revolution! ! ! ! ! 33!

4.1.2!! ! Volumes!of!Solids!of!Revolution:!Regions!Bounded!by!!

!!!!!!!!!!!!!!!!!!!!!!!!!Two!Curves! ! ! ! ! ! ! ! 39!

4.2.1!! ! Differential!Equations! ! ! ! ! ! 43!

4.2.2! ! Application!of!Differential!Equations! ! ! ! 46!

4.2.3! ! Differences!of!Rates!(Inflow/Outflow)! ! ! ! 50!4.2.4! ! Other!Methods!of!Solving!Differential!Equations! ! 53!

4.2.5!! ! Eulers!Method!of!Solving!Differential!Equations! ! 54!

4.3! ! Slope!Fields! ! ! ! ! ! ! ! 56!

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Contents (cont.)

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! 5!!!!! Kinematics!

5.1! ! Position,!Velocity,!Acceleration,!Displacement,!Distance!!

!!!!!!!!!!!!!!!!!!!!!!!!!and!Speed!! ! ! ! ! ! ! ! 61!

5.2!! ! Constant!Acceleration!! ! ! ! ! ! 62!

5.3!!! ! Velocity\Time!Graphs! ! ! ! ! ! 64!

5.4!!! ! Velocity!as!a!Function!of!Position!and!Acceleration!as!a!!

! ! Function!of!Velocity!! ! ! ! ! ! 66!

5.5!!!! ! Acceleration!as!a!Function!of!Position!or!Velocity!! ! 68!

! 6!!!!!! Vector!Functions!

6.1! ! Vector!Functions!! ! ! ! ! ! ! 71!

! 7!!!!!!! Dynamics!!

7.1! ! Basic!Terms! ! ! ! ! ! ! ! 74!

7.2! ! Types!of!Forces! ! ! ! ! ! ! 75!

7.3! ! Newtons!Laws!of!Motion! ! ! ! ! ! 76!

7.4! ! Incline!Planes! ! ! ! ! ! ! 79!

7.5! ! Lamis!Theorem! ! ! ! ! ! ! 82!

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Disclaimer

All opinions expressed in these lecture notes are entirely those of the authors and do not

represent those of VCAA or any other body. Information sourced from others has been

appropriately referenced. We are only students, not teachers, doing our best to assist

you in your preparation.

Copyright laws prohibit the distribution or reproduction of these notes.!

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1.1!!Set!Notation!

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Set!notation!is!the!language!used!in!maths!to!convey!information.!Each!symbol!has!

a!unique!meaning,!and!can!be!read!aloud!in!plain!English.!!

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Some!commonly!used!symbols!are:!

"""""""""""""""""is"an"element"of"

"""""""""""""""""is"not"an"element"of"

""""""""""""""intersection""""""""""

U""""""""""""""""union"

\"""""""""""""""""not"including"

|"""""""""""""""""given"that"

""""""""""""""""""complement"(A"is"read"A"complement)"

,""""""""""""""""""where"

:""""""""""""""""""such"that"

!!!!!!!!!!!!!!is!a!subset!of!(but!not!equal!to)!

!!!!!!!!!!!!!!is!a!subset!of!(but!may!be!equal!to)!

!!!!!!!!!!!!is!not!a!subset!of"

{"",""}"""""""""""set"brackets"

["",""]"""""""""""inclusive"interval"

("","")"""""""""""nonGinclusive"interval"

(,)"""""""""""""""ordered"pairs"(coordinates)""

!""""""""""""""""universal"set"

""""""""""""""""the"null"set"

!!!!!!!!!!!!!!!maps!onto!

!!!!!!!!!!!!!implies!that!

"

Common!sets!are:!

R""""""""""""""""""all"real"numbers"

Z"(or"J)"""""""""all"integers"

N""""""""""""""""""all"natural"numbers"

Q""""""""""""""""""all"rational"numbers"

R\Q""""""""""""""all"irrational"numbers"

R+"""""""""""""""""all"positive"reals"

RG""""""""""""""""""all"negative"reals"

C" "all"complex"numbers"

Note!that!0!is!neither!positive!

nor!negative.!

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Since!the!balloon!is!spherical,!we!know!that! 34

3v r= .!

Differentiating!with!respect!to! 24gives! .dv

r rdr

= !Therefore:!!

2 22 4 8dv dr dv r r

dt dt dr = = = .!

Since!dv

dt!is!given!in!terms!of!r,!any!constraint!must!also!be!in!terms!

of!r.!When! 0.036v = litres,!we!can!find!the!radius.!However,!since!

we!have!been!working!in!cms,!we!need!the!volume!in!cm3.!To!

convert!from!litres!to!cm3,!recall!that!1 mL = 1!cm

3,!and!that!there!

are!1000 mL!in!1!litre.!Therefore,!

0.036 0.036 1000 36litres! = = cm3.!

33

36 36 34

When% ,%v r

= = = cms.!

Therefore,!when! 23 8 3 72,"dv

rdt

= = = ,!

and!the!volume!is!increasing!at!a!rate!of!72cm/s.!

ii) The!volume!of!any!prism!is!the!cross!sectional!area! !length.!The!length!is!constant!at!6 m.!The!cross!sectional!area!will!vary!with!h.!

We!need!to!find!this!cross!sectional!area!in!terms!of!h.!

!!

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!

! !

To!find!the!cross!sectional!area!when!the!water!has!depth!h!metres,!

we!need!to!find!x!in!terms!of!h.!Since!~

ABC CED ,!we!can!say:!

0.5,"where" "and" .ED x

AB AX x AXBC AB

= = = !!

( ) ( )3

2 3 3 31 1

2 2 4 2 21

2

!and!h x

hx h x x h hxh x

= = = +

!!

and! 16

(3 2 )x h= .!Now! 1 13 3

2 2 2 (3 2 ) (9 2 )is! .CF x h h+ = + = !

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Solution:!

! i)!1

2 1 22

Let$ .$Then$ and$ .du u

u x xdx

= + = = !

! !

( ) ( )

3

3

2 3

2 3

1 2

2

2

2 2

5 1 5( 1)1(2 1) 2

15 3

2

15 3

4

1 35

4 2

1 110 3 10(2 1) 3

8 8(2 1)20 7 20 7

8(2 1) 8(2 1)

x u

dx u dxx

u u dx

u u du

u u

u u xx

x x

x x

+ " # = +% &

+ ' (

=

=

) *= ++ ,- .

= + = + ++

+= =

+ +

!

! ii)! 4 1 4Let$ .$Then$ $and$ .du

u x x udx

= + = = !

! ! !

! ! !

!

!

!

!

!

!

!

! !

!

!

!

!

!

12

5 3 12 2 2

7 5 32 2 2

3 32 2

32

2 2

2 2

2

4 ( 4)

8 16

2 16 327 5 3

1 8 16 1 8 162 2( 4) ( 4) ( 4)

7 5 3 7 5 3

16 1282( 4)

7 35 105

x x dx u u du

u u u du

u u u

u u u x x x

x xx

+ =

= +

= +

" # " #= + = + + + +$ % $ %

& ' & '

" #= + +$ %

& '


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Case!2:!There!is!a!repeated!linear!factor!in!the!denominator.!

!2

2 2

2 2

2 2 2 2

2 2

2 4

( 1) ( 2) 1 ( 1) 2

2 4 ( 1)( 2) ( 2) ( 1)

( 1) ( 2) ( 1) ( 2) ( 1) ( 2) ( 1) ( 2)

2 4 ( 1)( 2) ( 2) ( 1)

Let$x x a b c

x x x x x

x x a x x b x c x

x x x x x x x x

x x a x x b x c x

+

= + +

+ +

+ + +

= + +

+ + + +

+ + + + +

!

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49

132 43 9 9

1 1 3

2 4 9

0 4 2 2 2

When ,& &

When ,&

When ,&

x b b

x c c

x a b c a a

= = =

= = =

= = + + = =

!

2

2 2

2 4 13 1 4

( 1) ( 2) 9( 1) 3( 1) 9( 2)

x x

x x x x x

+

=

+ +

!

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Case!3:!There!is!a!non\reducing!quadratic!factor!in!the!denominator.!

2 2

2 3

( 3)( 4) 3 4Let$

x a bx c

x x x x

+

= +

+ +

!and!proceed!as!per!normal.!If!the!denominator!

contains!a!non\reducing!factor!of!degree!n ,!then!the!numerator!term!will!have!

degree! 1n .!

!

Case!4:!If!the!degree!of!the!numerator!is!equal!or!greater!than!the!degree!of!the!

! denominator,!then!the!fraction!is!improper!and!should!be!converted!to!a!

! mixed!number!by!polynomial!division!before!proceeding.!

!

! e.g.,!2

4 1 1 9 1

( 1)(2 3) 2 2( 1)(2 3)

x x x

x x x x

+ +

= +

+ +

,!and!then!proceed!as!per!normal.!

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The!commands!expand!and!propfrac!can!be!used!to!evaluate!these!on!a!

calculator:!

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5.1!!Position,!Velocity,!Acceleration,!Displacement,!Distance!and!Speed!!

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It!is!important!to!have!a!clear!understanding!of!all!these!concepts!before!

proceeding.!

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Position:! ! A!particles!physical!location!at!any!given!time!!vector!!

! ! ! quantity!

Velocity:! ! The!rate!of!change!of!position!with!respect!to!time!!vector!

! ! ! quantity!

Acceleration:! ! The!rate!of!change!of!velocity!with!respect!to!time!!vector!

! ! ! quantity!

Displacement:! The!change!in!position!over!a!given!time!interval!!vector!!

! ! ! quantity!

Distance:! ! The!length!of!the!path!which!a!particle!has!traveled!over!a!

! ! ! given!period!of!time!!scalar!quantity!!

Speed:!! ! The!magnitude!of!velocity!!scalar!quantity!

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!

If!the!position!of!the!particle!is!given!by! ( )x t ,!then:!

!

!

( )

( ) '( )

( ) '( ) ''( )

Position'

Velocity

Acceleration'

x t

v t x t

a t v t x t

=

= =

= = =

!

!

specialist maths sample

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