,calculus ,a1 ,lab #6

  ,! purpose ( ? lab is 6look at an
ele;t>y b nontrivial applic,n (!
derivative1 ) an eye 6%[+ (f "s (! amaz+
p[] (! tools we n[ h4

    ,newton's ,me?od
  ,newton's ,me?od (a /upid "n1 s9ce mo/
( calculus is ,newton's me?od) is a
te*nique = approximat+ roots ( equ,ns y
c't solve exactly4 ,x's us$ 0,maple's
.fsolve -m&1 = 9/.e4 ,newton's ,me?od is
an algori?m = tak+ a f/ approxim,n 6!
root ( an equ,n &= produc+ a bett]
second approxim,n4 ,0us+ ,newton's
,me?od ov] & ov]1 "o c get 9cr1s+ly
a3urate aproxim,ns 6! root y're seek+4
,i'll />t \ illu/rat+ ! process
0approximat+ a root 6! simple equ,n
x^2"-2 .k #0, b we'll !n see t ! same
me?od c 2 us$ 6obta9 roots ( practic,y
any equ,n4 ,i'm only />t+ )a simple
example 6make ! >i?metic easy & 6pick a
                                      #j
case ": we c easily *eck ! a3uracy ( \r
approxim,ns4
  ,"h's ! idea4 ,suppose y want 6f9d a
lot ( digits ( >2], b y don't h a ma*9e
h&y (or y're programm+ ! ma*9e 6d ?
task)_4 ,i le>n$ 9 s*ool h[ 6take squ>e
roots 0h&1 b y probably didn't4 ,:at cd
y d8
  ,y "k t ! numb] y're 9t]e/$ 91 >2], is
a root (! equ,n x^2"-2 .k #0, & t
#1 "k >2] "k #2_4 ,a r1sona# f/
approxim,n to >2] mi<t "!=e 2
>2] ".k<*] x0 .k #1_4 ,"h's h[ ,newton
goes ab gett+ a sequ;e ( bett] & bett]
approxim,ns to >2]_4

  #1_4 (a) ,-pute ! equ,n (! tang5t l9e
6! graph ( y .k x^2"-2 at ! po9t
x .k x0 .k #1_4
  (;b) ,sket*1 ei 0,maple or 0h&1 !
graph ( y .k x^2"-2 al;g ) ? tang5t l9e4
,!y _h bett] 2 tang5t4
  (;c) ,! curve y .k x^2"-2 crosses ! ;x
-axis at x .k +->2]_4 ,is ! po9t at : !
                                      #a
tang5t l9e crosses ! ;x-axis a bett] or
worse approxim,n to >2] ?an ! orig9al
approxim,n x0 .k #1_8
  (;d) ,-pute ! posi;n x1 at : ! tang5t
l9e crosses ! ;x-axis4 ,h[ f> is ? po9t
f >2]_8

  #2_4 ,n[ rep1t ? process />t+ f ! po9t
x1 9/1d () x0_4
  (a) ,-pute ! equ,n (! tang5t l9e 6!
graph ( y .k x^2"-2 at ! po9t x .k x1_4
  (;b) ,sket*1 ei 0,maple or 0h&1 !
graph ( y .k x^2"-2 al;g ) ? tang5t l9e4
,!y _h bett] 2 tang5t4
  (;c) ,is ! po9t at : ! tang5t l9e
crosses ! ;x-axis a bett] or worse
approxim,n to >2] ?an ! approxim,n x1_8
  (;d) ,-pute ! posi;n x2 at : ! tang5t
l9e crosses ! ;x-axis4 ,h[ f> is ? po9t
f >2]_8

  ,newton's ,me?od is j ? process (
improv+ an e/imate (a root 0foll[+ !
tang5t l9e 6! ;x-axis1 rep1t+ ! process
                                      #b
z _m "ts z we l until we h z m* a3uracy
z we ne$4 ,we don't h 6apply x 6a simple
curve l y .k x^2"-2_2 we cd apply x 6any
func;n ;f = : we c -pute values bo? ( ;f
&( f'_4

  #3_4 ,figure #1 %[s a curve y .k f(x),
an approxim,n x0 6a root ( f(x) .k #0,
&! tang5t l9e 6! curve at ! po9t
(x0, f(x0))_4 ,suppose we "k ! hei<t
f(x0) (! curve at x .k x0 &! slope
f'(x0) (! tang5t l9e4 ,write an equ,n =!
tang5t l9e 9 t]ms (! 83/ants0 f(x0) &
f'(x0), & %[ t ! po9t at : ? tang5t l9e
/rikes ! ;x-axis is
      ,'2g9.(equ,n.),'
x1 .k x0-?f(x0)_/f'(x0)#_4
      ,'5d.(equ,n.),'
  ,equ,n (1) lets u give a qk1 =mal
descrip;n ( ,newton's ,me?od4 ,6f9d a
root 6! equ,n f(x) .k #0, />t ) an
approximate solu;n x0_4 ,use ,equ,n (1)
6get a clos] approxim,n1 x1, 6! root4
,if x1 is / n close 51 !n rep1t !
                                      #c
process us+ x1 z yr new x0_4 ,keep
rep1t+ ! process until y get z close z
6! root z y ne$ 6be4

  #4_4 ,use ,newton's ,me?od
6approximate >2] ag1 />t+ ag at
x0 .k #1_4 ,? "t1 ?\<1 rep1t ! process a
bun* ( "ts1 n j twice4 ,y'll probably
want 6say "s?+ l .,digits 3.k #50 or
.,digits 3.k #100 at ! 2g9n+ 9 ord] 6d !
calcul,ns precisely4 ,al1 2 c>e;l y >e
us+ all ! digits ( ea* approxim,n :5 y
-pute ! next approxim,n4 ,don't j copy !
f/ few digits1 or y won't see :at's go+
on4
  (a) ,>e ! e/imates produc$ 0,newton's
,me?od l>g] or small] ?an >2]_8 ,c y
expla9 :y geometric,y8
  (;b) ,:at c y say ab ! ]ror ( yr
e/imates8 ,try 6be z precise z y c ab h[
fa/ yr ]ror is decr1s+4 ,= 9/.e1 if yr
/>t+ po9t is x0 .k >2]+.@e, s t yr />t+
]ror is .@e, h[ big d y expect ! ]ror (
x1 6be8 ,ano!r "q y mi<t ask is1 if x0
                                      #d
has ;n correct digits1 h[ _m correct
digits d y expect x1 6h8 ,x wd 2 nice
6see an algebraic >gu;t "h1 b x is m
important j 6try 6obs]ve & 6?9k4
  (;c) ,if y want$ 6get #1000 correct
digits ( >2], ab h[ _m "ts wd y h 6apply
,newton's ,me?od />t+ at #1_8 ,:at if y
want$ #1,000,000 digits ( >2]_8 (,do ?
0?9k+1 n 0sett+ .,digits 3.k #1000000 &
try+ x2 ?\< "s?+ l t wd 2 possi# )a bit
( tw1k+ (! s(tw>e_4)

  #5_4 ,! equ,n -x^6"-5x^2"+2x+50 .k #0
has a root "s": 2t x .k -#2 & x .k -#1_4
,use ,newton's ,me?od 6approximate ?
root to #10 digit a3uracy4 ,>e yr
approximants l>g] or small] ?an ! actual
root8 ,c y expla9 :y8

  #6_4 ,:at 3di;ns wd a func;n h
6satisfy 9 ! nei<borhood (a root 9 ord]
=! ,newton's ,me?od approxim,ns 6be
small] ?an ! exact root8 ,l>g]8 ,^ws l 8
9cr1s+10 8decr1s+10 83cave up0 & 83cave
                                      #e
d[n0 may 2 use;l 9 yr answ]s4

  #7_4 ,! equ,n #4x^5"-61x^3"-480 .k #0
has a root 2t x .k -#3 & x .k -#2_4
,expla9 :at happ5s if y try 6approximate
? root us+ ,newton's ,me?od />t+ at
x0 .k -#3_4 ,is "! a way >.d ?8

  #8_4 ,! equ,n <3>x] .k #0 has a root
at x .k #0_4 ,:at wd happ5 if y didn't
"k ? alr & if y tri$ 6approximate ? root
us+ ,newton's ,me?od8 ,is "! a way >.d
?8

  #9_4 ,c y say any?+ 9 g5]al ab :5
,newton's ,me?od does & does n "w8 ,an
answ] )a geometric flavor is f9e4







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