,calc ,a1 ,lab #9

    ,9troduc;n4
  ,we're pursu+ two approa*es 6-put+ >1s
"r n[1 "o 9volv+ antid]ivatives & "o
9volv+ sums ( >1s ( n>r[ slices4 ,! f/ (
^! gives rise 6! simple/ =mulas1 b x is
hamp]$ 2c we c't alw take !
antid]ivative (a func;n4 ,z a result1 :5
-put+ >1s 9 r1l life1 "o is (t5 reduc$
6a4+ up ! >1s ( slices4
  ,un=tunately1 we norm,y c't get exact
=mulas =! sum (! >1s (a bun* ( slices1
s9ce we don't h any grt te*nology =
-put+ sums4 ,if we >e reduc$ 6a4+ up !
>1s ( slices1 !n1 we'll 2 j "w+
num]ic,y1 n analytic,y4
  ,! purpose ( ? lab is 6look = ways
6m9imize \r num]ical "w & 6maximize !
a3uracy ( \r approxim,ns if we >e reduc$
6a4+ up slices4

    ,me?ods = approximat+ >1s4

                                      #j
  ,suppose y want 6-pute ! >ea "u !
graph ( y .k f(x) 2t x .k #0 & x .k #3_4
,suppose1 fur!r1 t y c't "w \ ! result
analytic,y1 & t y ne$ 6-pute approximate
num]ical >1s (a bun* ( slices & 6add up
! results4 ,= 3crete;s1 suppose y decide
6cut ! 9t]val @(0, 3@) 9to #4 slices4
,!n ,i c ?9k ( at l1/ #4 me?ods y mi<t
use 6approximate ! >1s4

,me?od #1_3 ,"r-h& rectangles4
  ,9 ? me?od1 : we've us$ 9 class1 y wd
approximate ;f 9 ea* slice 0a 3/ant
func;n ^: hei<t 0 ! value ( ;f at ! "r
h& 5d ( ea* slice4 ,! >ea y're -put+ is
!n %[n 9 ,figure #1_4
  ,=m,y1 ! >ea y're -put+ is
  f(?3_/4#)?3_/4#+f(?6_/4#)?3_/4#
    +f(?9_/4#)?3_/4#+f(?12_/4#)?3_/4#
    .k ".,s%k .k #1<4]f(?3k_/4#)?3_/4#_4

,me?od #2_3 ,left-h& rectangles4
  ,"h y wd use ! same idea1 except y wd
approximate ;f 9 ea* slice 0a 3/ant
                                      #a
func;n ^: hei<t 0 ! value ( ;f at ! left
h& 5d ( ea* slice4 ,! >ea y're -put+ is
!n %[n 9 ,figure #2_4
  ,=m,y1 ! >ea y're -put+ is
  f(?0_/4#)?3_/4#+f(?3_/4#)?3_/4#
    +f(?6_/4#)?3_/4#+f(?9_/4#)?3_/4#
    .k ".,s%k .k #0<3]f(?3k_/4#)?3_/4#_4

,me?od #3_3 ,trapezoids4
  ,9 lab la/ week1 we approximat$ ! >1s
0putt+ trapezoids 9 ea* slice4 ,we cd
write d[n a new =mula = ? >ea1 b "! is a
simpl] way 6?9k ab x4 ,9 ea* slice1 !
>ea (! trapezoid is midway 2t ! >ea (!
left-h& rectangle &! >ea (! "r-h&
rectangle4 ,! same ?+ has 6"w if we add
up all ! trapezoids3 ! >ea has 6be !
av]age (! >1s (! "r-h& & left-h&
rectangles4

,me?od #4_3 ,midpo9ts4
  ,a f9al me?od wd 2 6use rectangles ag1
b ? "t 6let ! hei<t (! rectangle 9 ea*
slice 2 ! hei<t ( ;f at ! midpo9t (!
                                      #b
slice4 ,! >ea1 %[n 9 ,figure #3, wd 2
  f(?0_/4#+?3_/8#)?3_/4#+f(?3_/4#
    +?3_/8#)?3_/4#+f(?6_/4#
    +?3_/8#)?3_/4#+f(?9_/4#
    +?3_/8#)?3_/4#
    .k ".,s%k .k #0<3]f(?3k_/4#
    +?3_/8#)?3_/4#_4

,g5]al =mulas4
  ,9 g5]al1 if we want 6approximate
  !;a^b"f(x)dx
us+ ;n slices1 !n ! "r-h& rectangle
me?od wd tell u 6-pute ! sum
  ".,s%k .k #1<n]f(a+?(b-a)k_/n#)?b
    -a_/n#_4
,! left-h& rectangle me?od says 6-pute
  ".,s%k .k #0<n-1]f(a+?(b-a)k_/n#)?b
    -a_/n#_4
,! trapezoid me?od is ! av]age ( ^! two
sums4 ,! midpo9t me?od says 6-pute
  ".,s%k .k #0<n-1]f(a+,?(b-a)(k
    +?1_/2#),_/n,#)?b-a_/n#_4
  ,all ^! =mulas >e writt5 9 ,maple =m
on ! atta*$ ,maple "w%eet4
                                      #c

    ,example4
  ,suppose y want$ 6-pute ! >ea "u ! s9e
func;n 2t x .k #0 & x .k ?.p_/2#,
  !;0^.p_/2"sin xdx_4
,we've n"e d"o a pro#m l ?1 s x looks l
we'll h 6approximate4
  (,actu,y1 ! situ,n isn't hope.s4 ,if
we 2lieve t ! >ea func;n ,f0(x)
satisfies ,f'0(x) .k sin x & t
,f0(0) .k #0, !n ,f0(x) .k #1-cos x_2 s
! >ea m/ 2
,f0(?.p_/2#) .k #1-cos ?.p_/2# .k #1_4
,? is use;l 6"k z we />t look+ at ! ]ror
9 \r approxim,ns4 ,b let's g back
6assum+ we ne$ to approximate_4)

  #1_4 ,2g9 0fill+ 9 ! foll[+ ta#s1 %[+
! approxim,ns produc$ 0ea* me?od & %[+
_! ]rors f ! exact value ( #1_4 ,foll[ !
advice on ! ,maple "w%eet 6avoid ?r[+
away preci.n unnecess>ily4


                                      #d
  ,approxim,n 6! >ea
  ,',key 6items 9 ! foll[+ ta#
#a ,left rects
#b ,trapezoids
#c ,midpo9ts      ,'
 

--------------------------------
;n         ,"r rects #a  #b  #c
--------------------------------
#1
--------------------------------
#10
--------------------------------
#100
--------------------------------
#1000
--------------------------------
#10,000
--------------------------------
#100,000
--------------------------------
#1,000,000
--------------------------------
                                      #e

  ,]ror 9 ! approxim,n






















                                      #f
  ,',key 6items 9 ! foll[+ ta#
#a ,left rects
#b ,trapezoids
#c ,midpo9ts      ,'
 

--------------------------------
;n         ,"r rects #a  #b  #c
--------------------------------
#1
--------------------------------
#10
--------------------------------
#100
--------------------------------
#1000
--------------------------------
#10,000
--------------------------------
#100,000
--------------------------------
#1,000,000
--------------------------------

                                      #g

  #2_4 ,"s columns 3ta9 approxim,ns t >e
too l>ge (bi7] ?an #1)1 & o!rs 3ta9
approxim,ns t >e too small (less ?an #1
)_4 ,expla9 geometric,y h[ y cd h
det]m9$ 9 adv.e : columns wd 2 :4

  #3_4 (a) ,look at ! ]rors 9 ! "r
rectangle me?od4 ,!y %d =m a r\< patt]n4
,>e !y propor;nal to #1_/n or to
#1_/n^2, or :at8 ,if y ne$$ 6"w \ !
9tegral to #100 digit a3uracy us+ ! "r
rectangle me?od1 r\<ly h[ big wd ;n h
6be8 ,an ord] ( magnitude e/imate is
f9e4 ,don't try actu,y gett+ t level (
a3uracy 9 ! -put,n2 y mi<t n l ! wait4
  (;b) ,rep1t "p (a) =! left rectangle
me?od4
  (;c) ,rep1t "p (a) =! trapezoid me?od4
  (;d) ,rep1t "p (a) =! midpo9t me?od4

  #4_4 ,-p>e ! ]rors (! midpo9t &
trapezoid me?ods4 ,: "w$ bett]8 ,is "!
any rel,n 2t ! sizes & signs (! ]rors 9
                                      #h
! two me?ods8

  #5_4 ,exploit ! 3nec;n 9 ,pro#m #4
6su7e/ a way 6-b9e ! midpo9t &!
trapezoid me?od 96a new me?od m a3urate
?an ei al"o4 ,try \ ! new me?od on \r
sample 9tegral1 & rep1t pro#m #3(a) )!
new me?od4 ,don't quit until y h a me?od
: is way bett] ?an ! midpo9t me?od4 ,j
twice z gd is n 54

  #6_4 ,once y h a re,y gd me?od 9
,pro#m #5, try x \ on a quadratic
func;n4 ,see if y c prove any?+4










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