Student Name: __ ..__.:L:.__C.--=c=---'---r-:-( _________ _ Period: ----

Taylor Polynomials - Use your calculator only on problem 3.

~ 1. Find a Maclaurin polynomial of degree n for each of the following.

( ) 3x

f x =e , n=4

(1 ....

7 ( Y--) - C "

-f\' x) a ~c:? \' {

3

9

-f "' {_ 'IJ = 2 7e_ 3)(. 2.., 7

t°"J('.x) =- 8lc: 3~ '3I

i (.__x-o)° -t- 3(y._-6)1

+ 'iJ .. J---0" + Z 7 {,:.-oJ + ~r ( y • 6Y

o r. \~ .;i! 5 1 '-l~

f (x) = cosx, n = 6

t C.x):: Ccsx: I

~'(\£, )= -S"t~~ v. D

t ''l_t- ) =- -C.bS .Y -\

{"'(I-)=- "5, i ----X 0

t ('1 '. (i.) ~C osx I

fc s)c,;;- :, ·1"'--x o , ~ ( Ir )/)( 'I :..- (' O'S y__ -

l (y_-o)(; + o(t-·::J'.;. -r (x-::,; ~ ~ l .: t(v.,(t.;. -- 0 '. / '. 2 I 5 ,1 't ~

2. Find a Taylor polynomial of degree n for each of the following.

f(x)=e 5x, n=4, centered at x=2

,- I 0 :7(.

' b zse. ·

(,_,,-;, l Y. - l ) o -I- S, e. io(z-2 ) ~ ?.. s r:. I';( Y- -Z) z...

--~ , ~ - 2,. r./

.j t2.Se..'° l:z.-z ) 5 +- C:,2'>e. 'O c~ ...... -z)

--;-~ ". ~ 1.

?. (y_)~ e '6r S(__'o(x.-(' ) +- L7e. l~(_X_-z) '-'f a ,

-f /_2 5 e.' 0 (. i<.-~) ~ r .~!'i,_d_ Y- - 't., )~.' &, ;t._'-f

f(x)=-1-, n=5, centered at x=2

1- x

--{(_ '/ ) ~ f-:7- -( I - y._y I - l I

f ~y ) -:- -(_/ - )()'l( - t ) ~ (1 - ';lrc. i·-y- • ~ . - ::. ... ~ .. + "c )( ) == - 2 c , -'I-r ? ~ - 'J : -z .... '. . )'C.) - { , .... >

- \.. . _J.:___ t.l

\ -2

Co ., , ( ·' ) --!( (1• I 1 l • ')(.I (_I· / "_, ·- '''/ '- ) : - ~ l - ,.__ - ,,. "-' l. ~ , ~ _ -5 l 'i _ 1,l f c.~·y") =- -z tf(r->'Y -:-, ( - :) -= z-d t-Y-) z-i:-; ; ,.. c' f Cs')C:t-) ;.- I zo ((-'.L-) -<-(-1) _- I ? o (. r-vf ·/ -<.:>- 17.D

3. a. Find a Taylor polynomial of degree n = 4 for f(x ) = e2x centered at c = 3.

-r(~')-=-e.?y e,,,~ \'(.,._) ~ Z e_ ?.v .2. t_.

(..

{"(_'f-.):.L/e_z_x 4,._1,

t ' ''(K')-. Be.2

" 8cc,

s;l~)['l..)=- /{.( Z.-c / t..e. "

b. Find P4 (3 .3 1). What is the value of /(3 .31) and the value of If (3 .3 1 ) - P4 (3.31 )I?

~ (__ ~. ~ , J = 74 er . (,. t) 2-

f-( ~. s i') = 7cf q I Cf '-{ 5"

( 7 <.f9. 7<(,;; -7l(9,"Cl2-/

::: 0,3'-13 (errcr')

4. Suppose that g is a function which has continuous derivatives, and that g{5) = 3, g' {S) =-2, g"(5)=7, g"'(5)=-3.

What is the Taylor polynomial of degree 3 for g centered at x = 5?

5. Find a fifth-degree Maclaurin polynomial for f (x) = sin(3x) .

f ~) 0 ~ i "'- ( ~- ~ I -) D t ( 5) ( >Z) _- l 'I 3 (. a s c~ ¥ ~ 2-'1.3

t'(.x): ::> C.o'.S,C s)(.',➔.3

\'' (.)t.)-=- -9-si"- C~l(½ O R &)= 3'll -27't.3 + z<J~x..5"

'° "' (~ .. ); -2-7Cos (_~)(.),-2. 7 5 31

• ':> '· (') l <4) 1 .., . - 3 "t -:r:, , 8 I 5 "'~)-=-8ls;..--. C:.'ll.) ~o - y:_ - d-..x ._ 40)(

6. Find a fourth-degree Taylor polynomial for f(x) = ln(x-1) centered at x = 2.

_;l :<-' = \~()(.- /) -:> 0

-\ '(_ 'l_) 7 X -1 - (A.- ,y 1

" x.~ 1 °' I t "( )(.) : - ( _,, - l -: ..::.!.- -'> - I

">(... J ( Y- 1)'

-Py~).- LX-1..) - {y...-z.)2. t- z [x._-2. )3 _ ~(_x.-zJIJ

-z..! ?, ~ 4 ~

~ .,. (_'I,) : ;;_ ( 'f... - I) - 3 -:: (_ :-• 1) > ~ l._

~(4/ 1- ) ~ - ~ (v- 1' · ~ =- ~ .., -)-b - ,, l~- ,)

7. Suppose the function / (x) is approximated near x = o by a third-degree Taylor

polynomial ~(x) =-3+ 7x2 - 2x3. Give the value of:

a. Give the value of: / (0) , / '(O) , /" (O) , and /'"(O).

-\( 6 ~ ~)(-b)

0

-1- \ '(_o J(X.-6)1

-4- ~,,~ o )Cx-6i° + + 111(_0) ()(._-6J~ l 1

, 2 I '? t , v ,

-~ + CJ 'f.._ 4-- 7 v7- - 'l X. 3

\Jc_D_J_-= ___ 3__,l t ·(o) .:: ti 1/-"(o)r- r '~(_c ) ___ ~ I I - =-/ \ -'-..

;.! -~

\ ? 'lo) '-01 -(' '' Cc, J = 7 . 2. ~ { ", ( o) , - .:i: $ 1

8" (p J :=. I '--/ ) t-1'"· (oJ ~ - I "2-j . b. Does f have a local maximum, a local minimum, or neither at x = O? Justify your

answer.

• ~ c1 < i <:__ a. loccJ L re \tz_J ; ve J M; "'- ~ ...,..,.'--- c... V\A-

~ ' Co) =- o ~ 0._& f ,, CD) > D -

8. a. Find a fourth-degree Maclaurin approximation for f(x) = cosx.

-f ( "f-. ') -= msx... I

-\ '(JC.) -=- - "::,\ "'- )(_ 0 ~"C 1')-=- - c.os< - \

0 I -. - ----·--\ u _ 'f--2. ;i,:.... 4

\ \ ~(~): \ -£~ ~

I. 1- cosx

b. Use your answer to (a) to find 1m 2 x - , 0 X

V , y._ --2.. 'i

9. a. Find a third-degree Maclaurin approximation for J(x) = -1

- .

4' (_~) ::. ( \ _ (. ~ · I t 1- 2x

~'(;t.') -: -C t-2 ~-l.c_-i..)-: 2it- 2~y 2 2.

t °')-; - '/( l -li.Y s(-zj ::. 8(1-2-,_ y.s 8

~Ille.~')= -2'-l ( 1-2~r'\_-1...)-· 4i Lt-2)(.r~ L/-3

---------------------

X

Ci ,._., ~ 4 ""l_l< ... +'-i 'f..."L i- 8 X, $ -;(

~"'">6 ',(

- (_~"""'""' ( zx.. -+ t\. ')l._l.. .(.. 3 ~ \ 'l..?0 ~ ~ -:;z:-J

:::_ ~~~ (2 +Lt~+~~~) ~0 10. a. Find a seventh-degree Maclaurin approximation for f(x) = sinx .

.(' c~ ~ ~ $ , "'-.( 0

~1[1--").:: ~SK t

~ ' · C. ~) -=- - .5 , .-...,c 0

~"•C~) ~ - ccs>e -I

\ l <4)CX.) = ~'it'->'- Q

fo")(-,d :: CD¼

~ (.~ ~( ~) -= - 5i; ..... x_ 0

.c o) / y_) =- -c..os~ - ' , L I'- JI sin t . b. Use your answer to (a) to approximate the value of -dt so that the error in

0 t

your approximation is less than ~. Justify your answer. 1

)00

\ (-l- 1! + -ts - {,1) ft - £ j ~I. 51 7! J f 7 \ ' 3, ~ /

1

0 '-- t:

t5 -t -

5, 5 !

I

\ /--l 1. > .f S _ -l 7~ Jf I - _I + I

3<3l, 7;,5 ~

lrt - t·!> 1.

4 t, ·5'. {:7 r,J

\ -'-+ L 13

t:_ - :!-.:- )Jr 5 ! 11_

0

5 \ '- L J_-{ ~ I __ ::

J . .L.. I 17

c 1 8 I~

0

- t. 7 \

7. 7 /) . I

7-1 ~

0

I

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