Math 21204/22/20

Fromlasth.me⑧

areaof R

= ) ! I ffff DIEmakesvre.TO

OLp-LE2.TLTETE rent

m threat

Ex : Find the area of, one petal in ④-

the 4 - leafed rose givenby

oor=cosYn±#pityf%"

"E-e%I I I"

Y'

FI- I

,

FEI,

/the

÷:÷÷÷÷¥¥÷¥÷

:I -b 'Xt7*16 - vI I✓ \" -Imu-1-• -f y '¥- I l¥* g3h12 J

' '

Itt#2

FINES!!!

'zfoskoDdo=

= Ss!!! I cos 420I do ⑤= S !i

'I LE t E cos ( 40 D

do

p

cosYuI=tztIcos

= Is"

⇐t I, cos(40-1) do

= Got¥¥si¥)" "

S The

= THE I t ITsina.tt/T- L te I I

t 'tsinD

=

'

a fat I - I kit = If =s±aa39

ll.5-linesandc~s.IO

PayA vector -

valued function#

in 3d

is a function ofthe form

Flt ) = L xltbyltbzct) >

where t , Htt, y Ct )

,Z It I

Are realnumbers .

z

Idea is : As youAAAplug in various t÷÷÷÷÷÷f"

"" "

Flt )=

CXl

Ylttzft,,

TBAAs y

x

EgvaHonofalinein3_dpgs@iEPCx.y

#

pfxgyoito)

••

•• #t. ZHI

¥=a, b ,c7-x -Let L be a line in3d . LetPo( xo , yo , to )be a point on the line , Let 8=9,44 bea vector in the

direction of L ,ie let

I be parallel to L .LetPcxy, z ) beanother point on L . Pick t such that→

t = Pop . Then Lxifz )-

- Go, To , Zo ) ttv .= Toa

= Lxo , yo ,z . ) t tea , b ,c ? P③

= C Xo , Yo , Zo ) tSta , Ab , to >

=L Xotta , yottb , Zottc )I T F

E7nofai3#An equation

of the linepassing

through ( Xo , yo ,Zo ) in

the

direction ofthe vector

I - - Caisse )

is givenby

Flt ) = do ,Yo ,

Zo ) ttv

where tis any

real number

⑤×=

Xotta where t

y=y .ttb is any

z =Zottc real

number

Wp pg①Chaptu9stuff@9,2 - a Find the interval

of convergence-

of Ean ( x - a )"

using theratio left

and testing endpoints

• Finding powerseries via integration

and differentiation

9,3 - . Finding Taylor )Maclaurin series

-

either by 9.2methods

norexplicitly calculating f' ) ( al°Kya)

and usingthe formula I t I x . af

k - - ok

91 - . Using the powerseries to solve

limit problems

. differentiate power series (# 35 )

• bien a power seriesturn it

into a function f Hutt 55 , 57,591

⑦ Identifyp⑧

E." " II

⇐ a .ix÷÷ni¥.

= !t¥E¥⇒÷÷'

*

④ p④

⑦Find theMaclaurin series

for f Cx )= 3× and its

radius of convergence .

-

Notes We don'thave a function

to

integrate ordifferentiate like

in the

example In ( It 4x) = ¥4 , DX tc

¥i÷÷÷⇒"""

x

t" '

H=3

×

f' " ( x ) = In (3)• 3

×

f' "( x ) = Ink )

. [ In .3×3=146)) :3

f" '( x ) = [ Ints D

? Inches'T Cusp?5•: kf' "

( x ) =[ In C 3D . 3×

f ' " ( o ) = [ Ints ) ]! 31 In(3) Jk

Where does thisconverge ?

£a!?Ixk ④k = o

L = hi:/ "i¥÷,

I

k !

hi.:i¥iiu¥= tin I I txt

ki x

= txt Liz FLIT1=1×1. 0=0

Since O E La I , theseries

converges

for all x .

EI : Find a power series forP④

ffxl =( ¥72

'

-

¥ ¥2 = - ( ItXY

' 2

. 2x

TEXT- I = - 26¥97

so ,¥⇒z=- I # C ¥ . ]

=

-

t.EC#xzD=-IE.E..c i

:¥÷i..

=- I ¥ §

.

fiykxzk P④-

iTTC - x4k=hs4xyk

=p- I £ , C - ish . 2k x " - I¥÷÷¥÷¥¥So

, g¥gz=- I ⇐ C- 1742k ×

' "

for - I ex- I

.

But it might

also convergeat the endpoints

Since wedifferentiated

to

find The series .

You can check ,it won't

converge

×=

-I

,

at X= Ior