,calc ,a1 ,lab #3

  ,9 ? lab1 ,i'd l 6explore bo?
graphic,y & algebraic,y ! process (
e/imat+ ! slopes ( tang5t l9es 6curves1
& 6tie ? 9 ) limits & derivatives4 ,ag1
pl1se "w 9 pairs if y c4 ,my r1d+ (
previ\s labs c]ta9ly seems 6su7e/ t "o
actu,y le>ns 04cuss+ pro#ms ) o!rs4
  ,z alw1 pl1se expla9 yr r1son+1 &
pl1se write 6be r1d4 ,use ,maple ": x
helps1 & avoid x ": x h9d]s4

  ,let ;f 2 ! func;n giv5 0! rule
f(x) .k >25-x^2"_/4]_4 ,we'll 2 do+ a
lot ) ? func;n1 s x mi<t make s5se
6def9e x 9 ,maple 0do+ "s?+ l
  ;f 3.k ;x -.1 sqrt(25-;x_<2_/4)_2
  ,we'll 2 9t]e/$ 9 ! func;n ;f, & 9 !
slope (! tang5t l9e to y .k f(x) at
x .k -#8, t is1 at ! po9t (-8, 3)_4

  #1_4 ,:at is ! doma9 ( ;f_8

                                      #j
  #2_4 ,make a l>ge & c>e;l plot (!
graph ( ;f, & sket* (,i'd d x by hand) !
tang5t l9e 6! graph at ! po9t x .k -#8_4
,e/imate ! slope ( ? l9e f ! graph4
,pl1se 2 pr9cipl$ & /ick 6yr e/imate
once y h made x4
  #3_4 ,f9d ! ;y coord9ate (! po9t on !
graph ( ;f ) x .k #10_4 ,sket* ! l9e
jo9+ ? po9t 6\r orig9al po9t (-8, 3), &
-pute ! slope ( ? l9e4 ,rep1t ? process
)! l9e 3nect+ (-8, 3) 6! po9t on ! graph
) x .k #6, ! l9e 3nect+ (-8, 3) 6! po9t
on ! graph ) x .k #2, ! l9e 3nect+
(-8, 3) 6! po9t on ! graph ) x .k -#2,
&! l9e 3nect+ (-8, 3) 6! po9t on ! graph
) x .k -#6_4 ,: ( ^! l9es be/
approximates ! tang5t l9e at ! po9t
(-8, 3)_8
  #4_4 ,write an equ,n =! slope (! l9e
jo9+ ! po9t (-8, 3) 6! po9t at
(x, f(x))_4 ,2 z explicit z y c4
  #5_4 ,use yr obs]v,ns 9 (3)1 tgr ) any
o!r calcul,ns y mi<t l 6d1 9 ord] 6get z
a3urate an e/imate z y c =! slope (!
                                      #a
tang5t l9e at (-8, 3)_4 ,h[ well does ?
e/imate agree ) :at y 3jectur$ graphic,y
9 (2)_8
  #6_4 ,once y h ! slope (! tang5t l9e1
write ! equ,n ( ? l9e4 ,plot ! l9e &!
orig9al func;n on ! same axes4 ,d !y
look tang5t8
  #7_4 ,write d[n a limit ^: value is !
slope (! tang5t l9e to y .k f(x) at !
po9t (-8, 3)_4 ,evaluate ? limit
analytic,y4

  #8_4 ,9 class1 we've def9$ ! slope (!
tang5t l9e to y .k f(x) at ! po9t
(x, f(x)) z ! derivative
  "lim%h $o #0] ?f(x+h)-f(x)_/h#_4
,use ? =mula 6-pute ! derivative (
y .k x^3_4 ,plot ;y & y' on ! same
graph4 ,does x make s5se t ! slope ( ;y
%d 2 ! hei<t ( y'_8
  #9_4 ,rep1t ? ) y .k x^4_4 ,c y
3jecture any?+ ab ! derivative ( x^n =
any ;m_8 ,c y >gue = yr 3jecture8

                                      #b
  #10_4 ,rep1t pro#m #8 )! func;n
y .k x^3"-3x_4 ,4cuss :at f1tures (!
graph ( ;y c 2 r1d (f f ! graph ( y' &
vice-v]sa4




















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