calculus booklet

8/3/2019 Calculus Booklet

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EXPONENTS AND RADICALS

10= x

r

r

r

x x x )

1(

1==

−

r

r x

x=

−

1

sr sr x x x

+=

rssr x x =)(

r

r r

y

x

y

x=)(

sr

s

r

y y

x −=

r r r y x xy =)(

r r

x

y

y

x )()( =−

nn x x =1

nnn y x xy =

nmmn

x x )(=

n

n

n

y

x

y

x=

DERIVATIVES

k is a constant.

)( x f )(' x f k 0

)( x f k )(' x f k

n x 1−

nnx

xe

xe

xln x

1

xsin xcos

xcos xsin−

xtan x2sec

xcot x

2csc−

xsec x x tansec

xcsc x xcotcsc−

x x arcsin,sin1−

21

1

x−

x x arccos,cos 1−

21

1

x−−

x x arctan,tan 1−

21

1

x+

xsinh xcosh

xcosh xsinh

xtanh xh2sec

xcoth xh2

csc−

hxsec xhx tanhsec−

hxcsc xhxcothcsc−

x1sinh

−

1

1

2+ x

x1cosh

−

1

1

2− x

x1tanh −

21

1

x−

x1

coth−

21

1

x−−





INDEFINITE INTEGRALS

A constant of integration should be added inall cases.

)( x f ∫ dx x f )( k kx

)( x f k ∫ dx x f k )(

x

1

||ln x

xe xe

xln x x x −ln

xsin xcos−

xcos xsin

xtan )ln(cos x−

xcot )ln(sin x

xsec ))42

ln(tan(π

+ x

xcsc

))2

ln(tan( x

xsinh xcosh

xcosh xsinh

xtanh ))ln(cosh x

xcoth |))sinhln(| x

hxsec )(tan2 1 xe−

hxcsc

|))2

tanh(ln(|x

22

1

xa −

)(sin1

a

x−

22

1

xa + )(sinh 1

a

x−

22

1

a x − )(cosh 1

a

x−

22

1

a x + )(tan

1 1

a

x

a

−

22

1

a x − )ln(

2

1

a x

a x

a +

−

22

1

xa − )ln(

2

1

xa

xa

a −

+

TAYLOR SERIES (about a x = )

LL+

−

++

−+−+=

)(!

)(

)(!2

)()()()()(

)(

''2

'

a f n

a x

a f a x

a f a xa f x f

nn

MACLAURIN SERIES

LL +++

++=

)0(!

)0(!2

)0()0()(

)(

''2

'

nn

f n

x

f x

xf f x f





REAL FOURIER SERIES (Period T)

∑∑∞

=

∞

=

++=

11

0 )2

sin()2

cos(2

1)(

m

m

m

m t T

mbt

T

maat f

π π

Where

dt t T

mt f

T a

T

m )2

cos()(2

0

∫ =π

dt t T

mt f

T b

T

m )2

sin()(2

0

∫ =π

COMPLEX FOURIER SERIES (Period T)

)2

exp()( t T

mict f

m

m

π ∑∞

−∞=

=

where

dt t T

mit f

T c

T

m )2

exp()(1

0

∫ −=π

Relationship between the above coefficients

of the real and complex forms.

0)(2

1

2

1

0))(2

1

00

<+=

=

>−=

−−mibac

ac

mibac

mmm

mmm

FOURIER TRANSFORM PAIR

dt t it f F ∫ ∞

∞−

−= )exp()()(~ ω ω

dt t iF t f ∫ ∞

∞−

= )exp()(~

2

1)( ω ω

π

POWER SERIES FOR SOME FUNCTIONS

xall for n

x x xe

n x

LL +++++=!!2

12

11

!)1(

32)1ln(

132

≤<−

+−++−=++

x

n x x x x x

nn

LL

L+−+−=−

=

−

!7!5!32sin

753 x x x

xee

x jx jx

L+⋅⋅

⋅⋅

+⋅

⋅++==

−

7642

531

542

31

32

1arcsinsin

7

531

x

x x x x x

L+−+−=+

=

−

!6!4!22cos

642 x x x

xee

x jx jx

L++++=753

315

17

15

2

3

1tan x x x x x

L+−+−==−

753arctantan

7531 x x x

x x x

L++++=−=

−

!7!5!32sinh

753

x x x xee x x x

L+⋅⋅

⋅⋅−

⋅

⋅+−==

−

7642

531

542

31

32

1arcsinsinh

7

531

x

x x x xh x

L++++=+

=

−

!6!4!22cosh

642 x x x

xee

x x x

L+−+−=753

315

17

15

2

3

1tanh x x x x x

L++++==−

753arctantanh

7531 x x x

xhx x





INTEGER SERIES

)1(2

1321

1

+=++++=∑=

N N N i N

i

L

)12)(1(6

1

321 2222

1

2

++

=++++=∑=

N N N

N i N

i

L

23333

1

3 )]1(2

1[321 +=++++=∑

=

N N N i N

i

L

NUMERICAL SOLUTIONS OF f(x) = 0

BISECTION METHOD

1 x and 2 x are estimates of the root that are

on opposite sides of the root (i.e. such that

0)()( 21 <⋅ x f x f ). The next estimate is

given by the arithmetic mean of the previous

two estimates,2

213

x x x

+= . Discard

whichever of 1 x or 2 x is on the same side of

the root as3

x (2

x in the diagram) and repeat

the process. The root of 0)( = x f always

lies between the last two estimates. The

bisection method is slow but always

converges for a 'well behaved' function.

NEWTON METHOD

Start with a single estimate 1 x of the root;then successive estimates are given in

terms of the previous one by

)(

)('1

n

n

nn x f

x f x x −=

+





NUMERICAL INTEGRATION

Let N

abh

−=

Trapezoidal rule

]222

222[2

)(

124

3210

N N N

b

a

f f f f

f f f f h

dx x f

++++

++++≈

−−

∫

L

Simpson rule

The number of intervals N must be even

]42

2424[3

)(

12

43210

N N N

b

a

f f f

f f f f f h

dx x f

+++

+++++≈

−−

∫ L

LAPLACE TRANSFORMS

dt st t f sF t f L ∫ ∞

−

−==

0

)exp()()())((

)(t f )(sF comment

1 1

nt 1

!+n

s

n

at e−

as +

1

t ω sin 22

ω

ω

+s

comment

s

t ω cos 22ω +s

s

t t ω sin 222 )(

2ω

ω

+ss

t t ω cos 222

22

)( ω

ω

+

−

s

s

t ω sinh 22

ω

ω

−s

t ω cosh 22

ω −s

s

t eat

ω sin−

22)( ω ω

++ as

t eat

ω cos−

22)( ω ++

+

as

as

)( τ δ −t τ s

e−

dirac delta

function

)( τ −t H τ se

s

−1

step

function,

)(t f eat −

)( asF + damping in

t becomes

)(kt f )(1

k

sF

k

scale

change

)(t tf ds

dF −

first

Derivative

)(' t f )0()( f ssF − first

derivative

)("' t f )0()0()( '2 f sf sF s −−

second

derivative

∫

t

dx x f 0 )( )(

1sF

s

Integral of

f(t) w.r.t t

∫ t

t f x f 0

21 ()(

)()( 21 sF sF convolution

integral

()( bgt af +

)()( sbGsaF + Linearity

calculus booklet

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