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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 256 "
" 0 "" {TEXT -1 49 "Lesson 6:  Finding the Equation of a Tangent Line
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart: with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 
"Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 181 "First we'll demonstrate graphically the meanin
g of a tangent line in an animation.  Afterwards, we'll show how to fi
nd the equation of a line tangent to a function at a given point." }}}
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ot(f(x), x=-2..5, y=-8..12, thickness = 3, color = red),\n        anim
ate(T(x,t), x=-2..5, t=-1..5, view=-8..12, color = blue, frames=60));
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[l-%+AXESLABELSG6%Q\"x6\"Q\"yFbj_m%(DEFAULTG-%%VIEWG6$;F,Fh\\m;$Ff[lF.
$\"#7F." 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv
e 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Let's find the \+
equation of the tangent line for the following " }}{PARA 0 "" 0 "" 
{TEXT -1 65 "functions at the indicated points. Plot the function and \+
tangent " }}{PARA 0 "" 0 "" {TEXT -1 23 "line on the same plot. " }}
{PARA 0 "" 0 "" {TEXT -1 21 "a)  x = 2,   f(x)  = " }{XPPEDIT 18 0 "3*
x - x^2 " "6#,&*&\"\"$\"\"\"%\"xGF&F&*$F'\"\"#!\"\"" }}{PARA 0 "" 0 "
" {TEXT -1 19 "b)  x = 1,  g(x) = " }{XPPEDIT 18 0 "x^3 - sqrt(x) " "6
#,&*$%\"xG\"\"$\"\"\"-%%sqrtG6#F%!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 9 "
c)  x =  " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 16 " /6,   h(x)  =
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GF$F$" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 20 "d)  x = 0,  f(x
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&F&F&*$F'\"\"#!\"\"F+" }{TEXT -1 7 "       " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 4 "a)  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := \+
x -> 3*x - x^2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"$*$)F-\"\"#\"\"\"!\"\"F(F(
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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,&
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{MPLTEXT 1 0 18 "df:= x -> 3 - 2*x;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#dfGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"$\"\"\"
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f(2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "df(2);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 66 "The tangent line passes through the point (2,2) and has s
lope -1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "tl:= x -> 2  -
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(\{f(x), tl(x)\}, x = 0..4, col
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F]v$\"3_LL$3_;!o6FE7$Fbv$\"31+++ISX#3\"FE7$Fgv$\"34nm;%RY>+\"FE7$F]w$
\"3W-++vr#z<*F07$Fbw$\"3Qommm6<b$)F07$Fgw$\"3%=++](=#Q\\(F07$F\\x$\"3K
KLLL,CkmF07$Fax$\"3ZMLL$)H(e\"eF07$Ffx$\"3KnmmT2`u\\F07$F[y$\"31,+++YU
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07$F_z$\"3C\"*****\\#f6p)F-7$FdzF(-Fiz6&F[[l$\")#)eqkF^[l$\"))eqk\"F^[
lFedl-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fdz%(DEFAULTG" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}
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" {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "g:=x -> x^3 - sqrt(x);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"$\"
\"\"F1-%%sqrtG6#F/!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "D(g);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6
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G6#F-!\"\"F/F7F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dg:
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"" {XPPMATH 20 "6#>%#dgGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#
\"\"\"\"\"$*&$\"\"&!\"\"F1-%%sqrtG6#F/F6F6F(F(F(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 5 "g(1);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dg(1
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The tangent line passed through th
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1 0 24 "tl:= x -> (2.5)*(x - 1);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#tlGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$$\"#D!\"\"$
F/F0F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot(\{g(x)
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"" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7H7$$\"33++++
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\"34+++I&Q\"GTF3$!3X#H9]Hh:s&F37$$\"3q******>2X;]F3$!3rx=r'*>J?eF37$$
\"30+++Ivv-eF3$!3OJSHv!zOm&F37$$\"3Y+++?2YlmF3$!3.z>Yi>(G?&F37$$\"3')*
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" {TEXT -1 3 "c) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(h);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#R6#%\"xG6\"6$%)operatorG%&arrowGF&,$-%
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "h(Pi/6);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&#\"\"$\"\"#F$-%%sqrtG6#F'F
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0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7i2)f$!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dh(Pi/6);" }{TEXT -1 0 "" }}{PARA 
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"> " 0 "" {MPLTEXT 1 0 49 "tl:= x -> 1 + (3/2)*(sqrt(3)) - (3/2)*(x - \+
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G6\"6$%)operatorG%&arrowGF(,*\"\"\"F-*&#\"\"$\"\"#F--%%sqrtG6#F0F-F-*&
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