,calc ,a1 ,lab #2

  ,i'd l 6d two ?+s 9 ? lab3 f/1 6take a
bit ( "t explor+ limits1 & second1 6play
a bit ) ma!matics 9 ! ,greek /yle1 ask+
\rvs ! "q h[ we wd g ab -put+ if we 7
,>*im$es (or maybe1 ,>*im$es equipp$ )
an e>ly v].n ( ,maple t _h "ey?+ except
.p)_4 ,pl1se "w 9 pairs un.s y're re,y
oppos$ 6! idea4 ,=! m 3jectural "ps ( ?
lab1 especi,y "s (! geometry at ! 5d1 x
may 2 help;l if pairs 3sult ) o!r pairs4
,i'd l ea* pair 6turn 9 a pap]1 b if "w+
9 l>g] gr\ps 9cr1ses yr e6ici5cy1 !n d
s4 ,feel free 6w&] \ 6sit tgr & talk1 -+
back 9 "h :5 & if y ne$ ,maple4

    ,limits4
  ,f/1 ,i'd l u 6look at "s limits us+
bo? ,maple's graph+ capabilities & xs
num]ical p[]4
  #1_4 ,9ve/igate ! limit
  "lim%x $o #2] ?3x^2"-12_/x-2#

                                      #j
  (a) 0plott+ ! func;n ne> x .k #2 &
see+ :at numb]1 if any1 f(x) is gett+
close to4
  (;b) 0-put+ ! value ( f(x) at "s po9ts
close to x .k #2_4 ,a gd 9itial *oice
mi<t 2 #1.9, #1.99, #1.999, #2.001,
#2.01, #2.1_4 ,>e "! patt]ns 9 ! values
y obs]ve : let y 9f] :at ! limit mi<t 28
  (;c) ,d y f9d ! analysis 9 "p (a) or !
analysis 9 "p (;b) m p]suasive8 ,:y8

  #2_4 ,rep1t ? pro#m )! limits
  "lim%x $o #1] ?>x]-1_/x-1#
    "lim%x $o #0] ?1-cos x_/x#
    "lim%x $o #0] ?1-cos x_/x^2"#
  #3_4 ,plot ! pairs ( func;ns
  (a) #3x^2"-12 & #12(x-2) ne> ! po9t
x .k #2_4
  (;b) >x]-1 & ?1_/2#(x-1) ne> ! po9t
x .k #0_4
  (;c) #1-cos x & ?1_/2#x^2 ne> ! po9t
x .k #0_4
  ,h[ d ! limit calcul,ns 9 ,pro#ms #1--
#2 help y "u/& ^! plots8
                                      #a

    ,>*im$es & .p_4
  ,"o (! ?+s ,>*im$es is "kn = is
develop+ a me?od = approximat+ .p z
a3urately z "o likes4 ,he 5ds up giv+ !
approxim,n t is 2t
#3_?1_/7_# ".k<*] #3.142857 &
#3_?10_/71_# ".k<*] #3.140845, ?\< at
l1/ ) ma*9es 6d ! >i?metic1 "o c use 8
me?od 6get bett] approxim,ns /4 ,9 ?
s]ies ( pro#ms1 we won't explore exactly
h[ ,>*im$es 9 fact -put$ .p_4 ,9/1d1
we'll use a geometrical approa* : is 9
keep+ ) :at he mi<t h d"o1 & : is
probably easi] = mod]n folks 6foll[4 ,\r
/>t+ po9t is 6rememb] t .p is ! ratio (!
circumf];e (a circle 6xs diamet]4 ,9
"picul>1 a circle ) radius #1 w h
circumf];e #2.p_4 ,we w try 6e/imate !
circumf];e (a circle ( radius #1 0-put+
! p]imet]s ( polygons 9side & \tside !
circle4 ,09cr1s+ ! numb] ( sides ( ^!
polygons1 we w get bett] & bett]
approxim,ns 6! circumf];e #2.p_4
                                      #b
,>*im$es did "s?+ simil>1 b he us$ >1s
9/1d ( l5g?s1 rely+ on ! fact t ! >ea (a
circle ( radius #1 is .p_4
  #1_4 ,9 a circle ( radius #1, 9scribe
a squ>e l ,figure #1_4
  (a) ,>gue t ! circumf];e (! circle is
l>g] ?an ! p]imet] (! squ>e4
  (;b) ,-pute ! l5g?s (! seg;t a4 jo9+ !
midpo9t (! $ge (! squ>e 6! c5t] (!
circle1 &! seg;t b4 =m+ half ! $ge (!
squ>e4 ,triangles &! ,py?agor1n ,!orem
may 2 use;l "h4
  (;c) ,-pute ! p]imet] (! squ>e4 ,y "k
t ? p]imet] m/ 2 less ?an ! circumf];e
#2.p (! circle4 ,:at e/imate does ? give
y = .p_8
  #2_4 ,n[ use yr squ>e 6build an
octagon l ,figure #2_4
  (a) ,>gue t ! circumf];e (! circle is
l>g] ?an ! p]imet] (! octagon4
  (;b) ,-pute ! l5g?s (! seg;t a8 jo9+ !
midpo9t (! $ge (! octagon 6! c5t] (!
circle1 &! seg;t b8 =m+ half ! $ge (!
octagon4 ,h9t3 ,i wd f/ -pute ! $ge #2b8
                                      #c
(! octagon us+ a "r triangle4 ,i wd !n
f9d ano!r "r triangle & -pute a8_4
  (;c) ,-pute ! p]imet] (! octagon4 ,y
"k t ? p]imet] m/ 2 less ?an !
circumf];e #2.p (! circle4 ,:at e/imate
does ? give y = .p_8
  #3_4 ,d ! same ?+s ) an squ>e & an
octagon circumscrib$ >.d ! \tside (a
circle4 ,^! %d give y e/imates ( t >e
too l>ge1 j z yr calcul,ns 9 ,pro#ms #1
& #2 gave y e/imates t 7 too small4

  #4_4 ,if y >e ambiti\s1 try 6rep1t ^!
calcul,ns )a #16-sid$ polygon or a #32-
sid$ polygon4 ,exactly ! same >gu;ts )
"r triangles %d let y -pute ! p]imet]s (
^! polygons & get q gd e/imates ( .p_4
,if y >e m ambiti\s /1 turn ? 96a g5]al
me?od t lets y -pute ! p]imet] (a #64-
gon1 a #128-gon1 & s on4 ,"o ( my kids &
,i once us$ ? approa* 6-pute >.d #15
digits ( .p_4 ,6me x is ?rill+ t "o c d
? ) re,y r ll "w1 z l;g z "o c -pute
squ>e roots ) fair a3uracy4
                                      #d
  ,bett]1 "o ( my /ud5ts 9 ? class took
? lab & obta9$ a v elegant =mula = .p
us+ ne/$ radicals1 ^: m1n+ is / n cle>
6me1 & : ,i h n f.d else": 9 !
lit]ature1 b : ,i _c 2lieve is new4 ,y
mi<t try ! same ?+ 0"w+ symbolic,y "h r
?an num]ic,y4

















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