parabola - jacobs university bremenmath.jacobs-university.de/petrat/teaching/2020_spring...siux...

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1 4 Taylor Series www flax fat parabola in I X b Goal approximate f CH near b by a power series we compute fly dy fix fl b fix f b t ftyldy dd u.ly x integration by parts f 44 Tidy ly x If ly ly Hf hildy recall hyzfjhf.gl g ggtufghgY lx bIf'lbl Slx Hf Holy flxt flh.lt x b f't b tf Ix Hf lyldy linear approximation another integrationbyparts yields check the computation flxt flbltlx h.lt tbh 5 f lb t g f lyldy c Second orderapproximation parabola

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Page 1: parabola - Jacobs University Bremenmath.jacobs-university.de/petrat/teaching/2020_spring...siux EC.it I EoHl x Et ka k 12m11 Kodd A similar cosx 2 f 11 FILH I Et k o k Keren fix e

1 4Taylor Serieswww

flax fatparabola

in

I Xb

Goal approximate fCH near b by a powerseries

we compute flydy fix flb

fix f b t ftyldyddu.ly x

integrationbyparts f 44Tidy ly x If ly ly Hf hildy

recallhyzfjhf.gl g

ggtufghgYlx bIf'lbl Slx Hf Holy

flxt flh.lt x b f'tb tfIx Hf lyldylinearapproximation

anotherintegrationbypartsyields checkthecomputation

flxt flbltlx h.lt tbh 5 f lb t g f lyldycSecondorderapproximation parabola

Page 2: parabola - Jacobs University Bremenmath.jacobs-university.de/petrat/teaching/2020_spring...siux EC.it I EoHl x Et ka k 12m11 Kodd A similar cosx 2 f 11 FILH I Et k o k Keren fix e

Theorem Taylor expansion

Lef f IR SIR be Ntl1 times continuously differentiable on b x Then

f1 1 EN l f tbI t f lyldyko N

Rn H called remainder

Remarks

I other formulas fortheremainder eg the Lagrangeformfint ft

Rn x µ X b for some te Eb X followsfrommeanvaluetheorem

II E f Cbl is calledTaylor seriesof f around b

l f b O it is called Maclaurinseries

If f is arbitrarilyoftendifferentiable andRuhlN

0 forsome x nearb

then f x EI E f lb

Note TheTaylorseries for oftendifferentiable f might

converge to f for some or all

convergebutnot to f if Rdx doesnot convergeto01diverge exceptat b

Page 3: parabola - Jacobs University Bremenmath.jacobs-university.de/petrat/teaching/2020_spring...siux EC.it I EoHl x Et ka k 12m11 Kodd A similar cosx 2 f 11 FILH I Et k o k Keren fix e

Examples

flxl e 3 fYxl e f to e 1

A XkTaylorseries off around b O is EK o k

We alreadyknowthat indeed flxt E.IT for all xelR

fix _six 3 f'lxl cosx

f 1 1 six

f 1 1 cost

f lxl six

letus chooseb O since sinloko cos101 1 we have f 101 4,0

Ntl N 70remainder Lagrangeform 1124 11 H Hye 0 theirNtl IN111

1 sinceeven bounded

siux EC.it I EoHl x Etka k 12m11Kodd

Asimilar cosx 2 f 11 FILH I Etk o k

Keren

Page 4: parabola - Jacobs University Bremenmath.jacobs-university.de/petrat/teaching/2020_spring...siux EC.it I EoHl x Et ka k 12m11 Kodd A similar cosx 2 f 11 FILH I Et k o k Keren fix e

fix e fat EI IE i k Ik

we i f i

ei Eat I 11 IT't ial Il II cos xtis.mx

cos x siux

we have recovered Euler's formula

Application Newton's method

weconsidertheiteration scheme Xµ FW

we are looking for a fixedpoint i e 2 EIR s t 2 Hz EfigXurecall Newton's method forfindingzeros of f

Xm Xu f i.e I 1 1 x FI if flzko Ftz zf xnl f4 1

now we consider the error En Xu z

3 Xµ 2 Em F xn F Z tEn

next time Taylor expansionaround2

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