Circular Measure HCI Cheers Ivan (11)| Jeremy (02) 4O1 Copyright © 2010 HCICheers Pte Ltd. All...

Circular Measure

HCI CheersIvan (11)| Jeremy (02)

4O1

Copyright © 2010 HCICheers Pte Ltd. All Rights Reserved. For Educational Purposes only

In conjunction with…

Ω

βα

π

κ

μ

λξ

θ Binomial Theorem

+

-

x

÷

)¯¯¯

Differentiation

Integration

Trigonometry

• Introduction– Pascal’s Triangle

– Thus,

Binomial Theorem

n=0

n=1

n=2

n=3

n=4

654326 615201561)1( bbbbbbb

1 + 5 + 10 + 10 + 5 + 1

1 + 6 + 15 + 20 + 15 + 6 + 1

• Binomial Theorem1.

• Where

• When

2.

Binomial Theorem

nnnnnnnn bbCbCbCb )(...)()1()()1()()1()1( 222

111

00

nbbxx

nnnb

x

nnbn

...123

)2)(1(

12

)1(

11 32

1,10

n

nn

rr

rn

br

nT

r

rnnnn

r

nC

1

!

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

rrnr

nnnnnn

bar

nT

bban

ban

ban

aba

1

332211 ...321

)(

• Binomial Theorem (Advanced)

Binomial Theorem

• Binomial Theorem (Advanced)– Newton's generalized binomial theorem– Around 1665, Isaac Newton generalized the formula to allow real

exponents other than nonnegative integers. In this generalization, the finite sum is replaced by an infinite series.

– In order to do this one needs to factor out (n−k)! from numerator & denominator in that formula, and replacing n by r which now stands for an arbitrary number, one can define:

– Where is the Pochhammer symbol here standing for a falling factorial.

Binomial Theorem

k.

• Binomial Theorem (Advanced)

Binomial Theorem

• Video Presentation (Part1)

Binomial Theorem


Binomial Theorem

• Circular Measure Formula

Circular Measure


Circular Measure

• Trigonometrical Functions

Differentiation

xxdx

d

xxdx

d

xxdx

d

2sectan

sincos

cossin

)(sec)(')(tan

)(sin)(')(cos

)(cos)(')(sin

2 xfxfxfdx

d

xfxfxfdx

d

xfxfxfdx

d

• Diffusing Chain Rule

• Differentiation of Exponential Functions

Differentiation

)(sec)('tan)(tan

)(sin)('cos)(cos

)(cos)('sin)(sin

21

1

1

xfxfnxfdx

d

xfxfnxfdx

d

xfxfnxfdx

d

nn

nn

nn

)()( )('

)(

xfxf

xx

exfedx

d

eedx

d

Differentiation• Laws of Logarithim

a

bb

axbxif

xnx

yxy

x

yxxy

c

ca

ba

an

a

aaa

aaa

log

loglog

,log

loglog

logloglog

logloglog

e

bb

exbxif

xnx

yxy

x

yxxy

c

c

b

n

log

logln

,ln

lnln

lnlnln

lnlnln


Differentiation

• Indefinite Integrals

Integration

dxxgdxxfdxxgxfe

cna

baxdxbaxd

ckxdxkc

dxxkdxkxb

cn

xdxxa

nn

nn

nn

)()()()()

)1(

)()()

)

)

1)

1

1

• Definite Integrals

Integration

dxxgxfdxxgdxxff

dxxfdxxfdxxfe

dxxfdxxfd

dxxfc

ahbhxhdxxfb

cxgdxxfa

a

b

a

b

b

a

c

a

c

b

b

a

a

b

b

a

a

a

ba

b

a

)()()()()

)()()()

)()()

0)()

)()()]([)()

)()()

• Integration of Trigonometric Functions

Integration

cxxdxc

cxxdxb

cxxdxa

tansec)

sincos)

cossin)

2

cbaxa

dxbaxc

cbaxa

dxbaxb

cbaxa

dxbaxa

)tan(1

)(sec)

)sin(1

)cos()

)cos(1

)sin()

2

• Integration of Exponential Functions

• Integration of Logarithmic Functions

Integration

cea

dxec

cedxeb

cedxea

baxbax

xx

xx

1)

)

)

cbaxdxbax

b

cxdxx

a

ln

1)

ln1

)


Integration

Cheers

1

n

n

nxdx

dy

xy

Differentiation -Power Rule

CheersDifferentiation -Chain Rule

)()(

)(

1 abaxndx

dy

baxy

n

n

CheersDifferentiation -Product Rule

)()()( udx

dvv

dx

duuv

dx

d

uvy

CheersDifferentiation -Quotient Rule

2

)()(

v

vdx

duu

dx

dv

v

u

dx

d

v

uy

CheersDifferentiation -Trigonometrical Functions

xxdx

d

xxdx

d

xxdx

d

2sectan

sincos

cossin

CheersDifferentiation -Trigonometrical Functions

)(sec)(')(tan

)(sin)(')(cos

)(cos)(')(sin

2 xfxfxfdx

d

xfxfxfdx

d

xfxfxfdx

d

CheersDifferentiation -Diffusing Chain Rule

)(sec)('tan)(tan

)(sin)('cos)(cos

)(cos)('sin)(sin

21

1

1

xfxfnxfdx

d

xfxfnxfdx

d

xfxfnxfdx

d

nn

nn

nn

CheersDifferentiation -Differentiation of Exponential Functions

)()( )('

)(

xfxf

xx

exfedx

d

eedx

d

)(

)(')(ln

1)(ln

xf

xfxf

dx

dx

xdx

d

Derivatives of Natural Logarithmic Functions

CheersDifferentiation -Indefinite Integrals

dxxgdxxfdxxgxfe

cna

baxdxbaxd

ckxdxkc

dxxkdxkxb

cn

xdxxa

nn

nn

nn

)()()()()

)1(

)()()

)

)

1)

1

1

CheersDifferentiation -Integration of Trigonometric Functions

cxxdxc

cxxdxb

cxxdxa

tansec)

sincos)

cossin)

2

CheersDifferentiation -Integration of Trigonometric Functions

cbaxa

dxbaxc

cbaxa

dxbaxb

cbaxa

dxbaxa

)tan(1

)(sec)

)sin(1

)cos()

)cos(1

)sin()

2

CheersDifferentiation -Integration of Exponential Functions

cea

dxec

cedxeb

cedxea

baxbax

xx

xx

1)

)

)

CheersDifferentiation -Integration of Logarithmic Functions

cbaxdxbax

b

cxdxx

a

ln

1)

ln1

)

CheersDifferentiation -Visit Our Additional Online Website To Find Out More

http://hcicheers.wikispaces.com/




Bibliography

• http://www.wikipedia.org– http://en.wikipedia.org/wiki/Binomial_theorem– http://en.wikipedia.org/wiki/Derivative– http://en.wikipedia.org/wiki/Integral– http://en.wikipedia.org/wiki/Trigonometry

• http://www.youtube.com• Guide Notes• Spark Notes• Past Formulas


http://www.wikipedia.org/

http://www.wikipedia.org/

http://en.wikipedia.org/wiki/Binomial_theorem

http://en.wikipedia.org/wiki/Derivative



http://en.wikipedia.org/wiki/Integral

http://en.wikipedia.org/wiki/Trigonometry

http://www.youtube.com/

Thank you!End of Presentation


Circular Measure HCI Cheers Ivan (11)| Jeremy (02) 4O1 Copyright © 2010 HCICheers Pte Ltd. All...

Documents

Transcript of Circular Measure HCI Cheers Ivan (11)| Jeremy (02) 4O1 Copyright © 2010 HCICheers Pte Ltd. All...