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{SECT 0 {EXCHG {PARA 208 "" 0 "" {TEXT 228 56 "Secant Lines, Tangent L
ines, Derivatives(\"D\" and \"diff\")" }}{PARA 209 "" 0 "" {TEXT 229 
262 "In this lab we investigate the relationship between secant lines \+
and average rates of change, tangent lines and instantaneous rates of \+
change, how Maple can aid in the limiting process that connects these \+
two concepts, and finally the derivative of a function.  " }}}{SECT 1 
{PARA 210 "" 0 "" {TEXT 230 47 "Plotting Secant Lines / Average rates \+
of change" }}{EXCHG {PARA 211 "" 0 "" {TEXT 231 685 "As we discussed i
n class, the tangent line to a graph of f(x) at x = a , has slope desc
ribing the instantaneous rate of change of y with respect to x, at x =
 a. We further illustrated that this tangent line can be considered th
e limiting case of  a converging sequence of secant lines. Furthermore
,  the slope of a secant line gives the average rate of change of y wi
th respect to x over an interval.   Therefore,  the slope of the tange
nt line (the derivative) at x = a, is the limiting value of a convergi
ng sequence of average rates of change, leading to an instantaneous ra
te of change.   In this section we illustrate this concept graphically
. First we define the function f(x) = " }{XPPEDIT 18 0 "(x-2)^2;" "6#*
$,&%\"xG\"\"\"\"\"#!\"\"F'" }{TEXT 231 1 " " }{TEXT 231 1 " " }{TEXT 
231 8 " + 1 by:" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 22 "f :=
 x -> (x-2)^2 + 1;" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6#I
\"xGF%F%6$I)operatorGF%I&arrowGF%F%,&*$,&9$\"\"\"!\"#F0\"\"#F0F0F0F%F%
F%" }{TEXT 233 1 " " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 318 "The go
al is to plot a sequence of secant lines that approaches the tangent l
ine of f(x) at x = 2.  Using the point slope definition of a line, the
 secant line through (2, f(2)) and (2+h, f(2+h))  is described by  y -
 y(2) = m (x - 2), where m (slope) is determined from the two points. \+
Writing this as a line we get:  " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 
231 66 "                                                              \+
y = " }{XPPEDIT 18 0 "(f(2+h)-f(2))/h;" "6#*&,&-%\"fG6#,&\"\"#\"\"\"%
\"hGF*F*-F&6#F)!\"\"F*F+F." }{TEXT 231 1 " " }{TEXT 231 1 " " }{TEXT 
231 16 " (x - 2) + f(2)." }}{PARA 211 "" 0 "" {TEXT 231 182 "We now de
fine a function of two variables, x and h,  that describes the secant \+
line.  I make it a function of two variables just so that I don't have
 to re-enter different values of " }{TEXT 231 4 " h. " }}}{EXCHG 
{PARA 211 "> " 0 "" {MPLTEXT 1 232 55 "secf2h := (x,h) -> (f(2 + h) - \+
f(2))/h *(x - 2) + f(2);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#>I'secf2h
G6\"f*6$I\"xGF%I\"hGF%F%6$I)operatorGF%I&arrowGF%F%,&*(,&-I\"fGF%6#,&
\"\"#\"\"\"9%F5F5-F16#F4!\"\"F5F6F9,&9$F5!\"#F5F5F5F7F5F%F%F%" }{TEXT 
233 1 " " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 117 "This cryptic nota
tion is for \"sec\"ant line of \"f\" at x = \"2\" for step size \"h\".
 You may call it anything you want.   " }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 232 79 "plot([secf2h(x,2), secf2h(x,1), secf2h(x,.5), secf2
h(x,.1), f(x)], x = 0 .. 4);" }}{PARA 213 "" 1 "" {GLPLOT2D 400 300 
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Fbt$\"3]>/'3\")G*46F27$Fgt$\"3s].aK!>D<\"F27$F\\u$\"3Od+[HmWY7F27$Fau$
\"3p:%e3TMRM\"F27$Ffu$\"3]3\\\"\\Hd`W\"F27$F[v$\"3SAg^kS4o:F27$F`v$\"3
)4F1'4l>#p\"F27$Fev$\"3\"3Cjqg!*=%=F27$Fjv$\"3+=p6+56'*>F27$F_w$\"3EMg
H-E<r@F27$Fdw$\"358jd&Q?gN#F27$Fiw$\"3U%GUH\"\\/kDF27$F^x$\"3#R(R@7\\U
yFF27$Fcx$\"34K$\\Af%*=,$F27$Fhx$\"3W+7LkskdKF27$F]y$\"35^[pY)\\f\\$F2
7$Fby$\"3?XzA\")Gx$y$F27$Fgy$\"3ax32%43X0%F27$F\\z$\"3#3*)RIu/qN%F27$F
az$\"3w&)3\"\\D2*fYF2Fez-%&COLORG62%$RGBG$\"#5F][l$F)F][lFj_nFj_nFh_nF
j_nFh_nFh_nFj_nFj_nFj_nFh_nFh_nFj_nFh_n-%+AXESLABELSG6$Q\"x6\"Q!F_`n-%
%FONTG6$%*HELVETICAGFi_n-%%VIEWG6$;Fj_n$\"#SF][l;$!$;$!\"#$\"$;&F^an" 
1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}}}{EXCHG {PARA 
211 "" 0 "" {TEXT 231 126 "Notice that this produces a sequence of lin
es that approach the tangent line.  Take a guess at the slope of the t
angent line. " }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 16 "# end \+
of section" }}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 230 77 "Plotting Tange
nt Lines/  Instantaneous Rates of change  / The \"limit\" command" }}
{EXCHG {PARA 211 "" 0 "" {TEXT 231 150 " Now we try to plot the tangen
t line to this curve at x = 2.  Recall that the slope of the tangent l
ine , at x = 2, is defined by the following limit:" }}}{EXCHG {PARA 
211 "" 0 "" {TEXT 231 34 "                                  " }
{XPPEDIT 18 0 "m[tan];" "6#&%\"mG6#%$tanG" }{TEXT 231 1 " " }{TEXT 
231 1 " " }{TEXT 231 4 "  = " }{XPPEDIT 18 0 "limit((f(2+h)-f(2))/h,h \+
= 0);" "6#-%&limitG6$*&,&-%\"fG6#,&\"\"#\"\"\"%\"hGF-F--F)6#F,!\"\"F-F
.F1/F.\"\"!" }{TEXT 231 1 " " }{TEXT 231 1 " " }}}{EXCHG {PARA 211 "" 
0 "" {TEXT 231 107 "The following command in Maple will find this limi
t. Here we want the limit as h goes to zero so we type:  " }}}{EXCHG 
{PARA 211 "> " 0 "" {MPLTEXT 1 232 40 "mtanf2 := limit((f(2+h)-f(2))/h
, h = 0);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#>I'mtanf2G6\"\"\"!" }
{TEXT 233 1 " " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 306 "This notati
on is for the slope of the tangent line of  the function f(x), at x = \+
2.  Again, you can call it anything you want.  Using a similar procedu
re as above, we plot the tangent line (from the point slope formula fo
r a line) by:  f(x) = mtanf2 (x - 2) + f(2).  And use this to plot the
 tangent line. " }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 36 "tanf
2 := x -> mtanf2*(x - 2) + f(2);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#>
I&tanf2G6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,&*&I'mtanf2GF%\"\"
\",&9$F/!\"#F/F/F/-I\"fGF%6#\"\"#F/F%F%F%" }{TEXT 233 1 " " }}}{EXCHG 
{PARA 211 "> " 0 "" {MPLTEXT 1 232 35 "plot([f(x), tanf2(x)], x = 0 ..
 4);" }}{PARA 213 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURV
ESG6%7S7$$\"\"!F)$\"\"&F)7$$\"3Hmmmm;')=()!#>$\"3yz&4#)QZ)eY!#<7$$\"3R
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mm;yYULF6$\"3AdyQ%yLZx$F27$$\"3%HLL$eF>(>%F6$\"3eCF<nrG(\\$F27$$\"3Qmm
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\"3w****\\#G2A3$F2$\"3EMgH-E<r@F27$$\"3;LLL$)G[kJF2$\"358jd&Q?gN#F27$$
\"3#)****\\7yh]KF2$\"3U%GUH\"\\/kDF27$$\"3xmmm')fdLLF2$\"3#R(R@7\\UyFF
27$$\"3bmmm,FT=MF2$\"34K$\\Af%*=,$F27$$\"3FLL$e#pa-NF2$\"3W+7LkskdKF27
$$\"3!*******Rv&)zNF2$\"35^[pY)\\f\\$F27$$\"3ILLLGUYoOF2$\"3?XzA\")Gx$
y$F27$$\"3_mmm1^rZPF2$\"3ax32%43X0%F27$$\"34++]sI@KQF2$\"3#3*)RIu/qN%F
27$$\"34++]2%)38RF2$\"3w&)3\"\\D2*fYF27$$\"\"%F)F*7S7$F($\"\"\"F)7$F-F
iz7$F4Fiz7$F:Fiz7$F?Fiz7$FDFiz7$FIFiz7$FNFiz7$FSFiz7$FXFiz7$FgnFiz7$F
\\oFiz7$FaoFiz7$FfoFiz7$F[pFiz7$F`pFiz7$FepFiz7$FjpFiz7$F_qFiz7$FdqFiz
7$FiqFiz7$F^rFiz7$FcrFiz7$FhrFiz7$F]sFiz7$FbsFiz7$FgsFiz7$F\\tFiz7$Fat
Fiz7$FftFiz7$F[uFiz7$F`uFiz7$FeuFiz7$FjuFiz7$F_vFiz7$FdvFiz7$FivFiz7$F
^wFiz7$FcwFiz7$FhwFiz7$F]xFiz7$FbxFiz7$FgxFiz7$F\\yFiz7$FayFiz7$FfyFiz
7$F[zFiz7$F`zFiz7$FezFiz-%&COLORG6)%$RGBG$\"#5!\"\"$F)Fa^lFb^lFb^lF_^l
Fb^l-%+AXESLABELSG6$Q\"x6\"Q!Fg^l-%%FONTG6$%*HELVETICAGF`^l-%%VIEWG6$;
Fb^l$\"#SFa^l;$\"1-++++++#*!#;$\"$3&!\"#" 1 2 2 0 10 1 2 6 1 4 2 1.0 
45.0 45.0 1 0 "Curve 1" }}}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 165 "No
tice above that the tanf2 = 0, and hence the tangent line is horizonta
l. This means that the instantaneous rate of change of y with respect \+
to x at x = 2 is zero. " }}{PARA 211 "" 0 "" {TEXT 231 114 "Plot the t
angent line to f (x) at x = 3 by defining a new slope (mtan3) and a ne
w function tanf3. How's it look?  " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 
231 148 "It should be pointed out here that the limit command need not
 be evaluated at a specific value of x. For example, try evaluating th
e derivative of  " }}{PARA 211 "" 0 "" {TEXT 231 7 "y = 13 " }
{XPPEDIT 18 0 "x^4;" "6#*$%\"xG\"\"%" }{TEXT 231 1 " " }{TEXT 231 1 " \+
" }{TEXT 231 5 " - 5 " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 
231 1 " " }{TEXT 231 1 " " }{TEXT 231 36 " + 2 x - 1 by the limit defi
nition: " }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 35 "q := x -> 1
3*x^4 - 5*x^2 + 2*x - 1;" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#>I\"qG6\"
f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,**$9$\"\"%\"#8*$F.\"\"#!\"&F.
F2!\"\"\"\"\"F%F%F%" }{TEXT 233 1 " " }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 232 32 "limit((q(x+h) - q(x))/h, h = 0);" }}{PARA 212 "" 1 
"" {XPPMATH 20 "6#,(*$I\"xG6\"\"\"$\"#_\"\"#\"\"\"F%!#5" }{TEXT 233 1 
" " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 345 "Recognize this?  This i
s why we love Maple.  It performs symbolic manipulations.  Ie. It does
n't just evaluate a sequence of numbers, it can do all of the algebra \+
that would be required to determine this limit.  If Maple can perform \+
the above limit calculation then it should be able to symbolically det
ermine derivatives. Right?  ... Yes Sir!!!" }}}{EXCHG {PARA 211 "> " 0
 "" {MPLTEXT 1 232 16 "# end of section" }}}}{SECT 1 {PARA 210 "" 0 ""
 {TEXT 230 42 "Derivatives: The \"D\" and \"diff\" commands. " }}
{EXCHG {PARA 211 "" 0 "" {TEXT 231 86 "Maple has two commands to evalu
ate derivatives of functions.  D(f) and diff(f(x), x). " }}{PARA 211 "
" 0 "" {TEXT 231 42 "D(f)  returns the derivative as  function." }}}
{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 20 "g := x -> x^3 - 4*x;" }}
{PARA 212 "" 1 "" {XPPMATH 20 "6#>I\"gG6\"f*6#I\"xGF%F%6$I)operatorGF%
I&arrowGF%F%,&*$9$\"\"$\"\"\"F.!\"%F%F%F%" }{TEXT 233 1 " " }}}{EXCHG 
{PARA 211 "> " 0 "" {MPLTEXT 1 232 5 "D(g);" }}{PARA 212 "" 1 "" 
{XPPMATH 20 "6#f*6#I\"xG6\"F&6$I)operatorGF&I&arrowGF&F&,&*$9$\"\"#\"
\"$!\"%\"\"\"F&F&F&" }{TEXT 233 1 " " }}}{EXCHG {PARA 211 "" 0 "" 
{TEXT 231 57 "Notice this describes a function and we give it a name b
y" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 15 "derivg := D(g);" }
}{PARA 212 "" 1 "" {XPPMATH 20 "6#>I'derivgG6\"f*6#I\"xGF%F%6$I)operat
orGF%I&arrowGF%F%,&*$9$\"\"#\"\"$!\"%\"\"\"F%F%F%" }{TEXT 233 1 " " }}
}{EXCHG {PARA 211 "" 0 "" {TEXT 231 3 "or " }}}{EXCHG {PARA 211 "> " 0
 "" {MPLTEXT 1 232 23 "derivg := x -> D(g)(x);" }}{PARA 212 "" 1 "" 
{XPPMATH 20 "6#>I'derivgG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%--
I\"DG6$I*protectedGF0I(_syslibGF%6#I\"gGF%6#9$F%F%F%" }{TEXT 233 1 " "
 }}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 76 "In either case, you can now
 evaluate the derivative at a number such as 3 by" }}}{EXCHG {PARA 
211 "> " 0 "" {MPLTEXT 1 232 10 "derivg(3);" }}{PARA 212 "" 1 "" 
{XPPMATH 20 "6#\"#B" }{TEXT 233 1 " " }}}{EXCHG {PARA 211 "" 0 "" 
{TEXT 231 37 "You can also plot the derivative with" }}}{EXCHG {PARA 
211 "> " 0 "" {MPLTEXT 1 232 58 "plot([g(x), derivg(x)], x = -4 .. 4, \+
color = [red, blue]);" }}{PARA 213 "" 1 "" {GLPLOT2D 400 300 300 
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ysbg@n\"GF07$Ffp$\"3+XhF%H4OG\"F07$F[q$\"3m(zp/]3XP\"Fbq7$F`q$!3)fjP[R
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7$Fer$!3EzuGW7'4n$F07$Fjr$!3[%*f(e<H%4RF07$F_s$!3%*e+\"HA`***RF07$Fes$
!3X_^7UH!3\"RF07$Fjs$!3)>lbneJ7o$F07$F_t$!33\"ezl26WF$F07$Fdt$!3)f'\\n
pU&3o#F07$Fit$!3=#z:&4;xH>F07$F^u$!3u6$RiWSE/\"F07$Fcu$\"3C,*4IIH@F\"F
bq7$Fhu$\"3'>!*y*QvGW8F07$F]v$\"3\")oA>u(Gr\"GF07$Fbv$\"3#>Dv_6ejI%F07
$Fgv$\"3x*))eZG(o-hF07$F\\w$\"3A9IS,?L`zF07$Faw$\"36T_vArS05F37$Ffw$\"
33w:piWAF7F37$F[x$\"3oT2`&*Q&oZ\"F37$F`x$\"3Nonl%*)4Tt\"F37$Fex$\"3^)>
*p5NF9?F37$Fjx$\"3NSu><n<4BF37$F_y$\"3]@Q.;)R^f#F37$Fdy$\"3UMNZdu_SHF3
7$Fiy$\"3/`])Gr4aE$F37$F^z$\"3**oyk\"p0%GOF37$Fcz$\"3ijI*eq))=*RF37$Fh
zF^[l-%&COLORG6)%$RGBG$\"#5!\"\"$F*FddlFedlFedlFedlFbdl-%+AXESLABELSG6
$Q\"x6\"Q!Fjdl-%%FONTG6$%*HELVETICAGFcdl-%%VIEWG6$;$!#SFddl$\"#SFddl;$
!%#*\\!\"#$\"1-+++++#*\\!#9" 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 
"Curve 1" }}}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 682 "In this graph we
 see the original function in red and the derivative in blue.  From th
is graph you should be able to estimate approximately where the tangen
t line is horizontal.  Better yet, if you click on the graph, you can \+
then use the mouse to have Maple show you the coordinates of any point
 on the graph. Click on the graph and notice that a window opens up on
 the menu bar showing the location of the cursor.  If you click on the
 cursor anywhere on the graph these coordinates will change.  Try this
 and notice that the derivative is equal to zero at about x =   -1.15 \+
and 1.15.  These are the square roots of 4/3 which you can find by set
ting the derivative equal to zero.  " }}}{EXCHG {PARA 211 "" 0 "" 
{TEXT 231 127 "The command diff(f(x), x) returns an expression describ
ing the derivative of f(x) with respect to the independent variable x.
  " }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 25 "h := x -> x^2 + 5
*cos(x);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#>I\"hG6\"f*6#I\"xGF%F%6$I
)operatorGF%I&arrowGF%F%,&*$9$\"\"#\"\"\"-I$cosG6$I*protectedGF4I(_sys
libGF%6#F.\"\"&F%F%F%" }{TEXT 233 1 " " }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 232 14 "diff(h(x), x);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#
,&I\"xG6\"\"\"#-I$sinG6$I*protectedGF*I(_syslibGF%6#F$!\"&" }{TEXT 
233 1 " " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 104 "Notice this retur
ns an expression for the derivative. In order to enter this as a funct
ion we must type:" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 29 "de
rivh := x -> diff(h(x), x);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#>I'der
ivhG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%-I%diffGI*protectedGF.6
$-I\"hGF%6#9$F3F%F%F%" }{TEXT 233 1 " " }}}{EXCHG {PARA 211 "" 0 "" 
{TEXT 231 91 "The big difference between diff and D is that you cannot
 evaluate the \"diff\" at a number.  " }}}{EXCHG {PARA 211 "> " 0 "" 
{MPLTEXT 1 232 10 "derivh(3);" }}{PARA 214 "" 1 "" {TEXT 234 72 "Error
, (in derivh) wrong number (or type) of parameters in function diff" }
{TEXT 234 1 "\n" }}}{EXCHG {PARA 211 "" 0 "" {TEXT 231 27 "but you can
 still plot it. " }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 64 "plo
t([h(x), derivh(x)], x = -2*Pi .. 2*Pi, color = [red, blue]);" }}
{PARA 213 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7Y7
$$!3)****>YH&=$G'!#<$\"3],HlW<%yW%!#;7$$!3dp3dxTF4gF*$\"3KQXMnx\\#4%F-
7$$!3W07Azb%4x&F*$\"3-BY\\bd?mPF-7$$!3[DqOW,#H]&F*$\"3]\\XEEld$Q$F-7$$
!3()o*[H1=JB&F*$\"3]sf&[#)3t)HF-7$$!3gAxs.$)fk\\F*$\"3)\\83!Hc\\*e#F-7
$$!3/sy+9hk:ZF*$\"3P\"z(=\"Rg`A#F-7$$!3ellUT0(yX%F*$\"3#\\5#=\"Gr8'=F-
7$$!3!>@6bny7>%F*$\"3fZ*ff+gx]\"F-7$$!3/t34t<aDRF*$\"3uEWh([Ap=\"F-7$$
!3wfUt0()>_OF*$\"3AJ.e`HJw*)F*7$$!35AGA$)zV6MF*$\"3'*H8%>T[)=oF*7$$!3p
xppL%)RSJF*$\"3I)))ox)e5i[F*7$$!3OO)R(pfCoGF*$\"3M$=!*zwqCT$F*7$$!3In=
4Um(fg#F*$\"3W%*H[oAM\"\\#F*7$$!3C!Q9W!Q*o[#F*$\"3KEaq*3%[=AF*7$$!3=$*
otm4\"yO#F*$\"3&RG>&*)Q;I?F*7$$!3'\\q*za*4iA#F*$\"3V)oPEnu&3>F*7$$!3I;
D'G%*3Y3#F*$\"3mn/G#z)3))=F*7$$!3s.k\">JaY'>F*$\"3j9k&QD(4T>F*7$$!38\"
Hq4o*pW=F*$\"3O(\\L9Cg/0#F*7$$!3%zq%)f<Gcc\"F*$\"3'G\"[XCA.xCF*7$$!3kG
dA2Eg=8F*$\"3/bLrdsN')HF*7$$!3YT1&oSwv/\"F*$\"3y#*4rbcx&f$F*7$$!3q(\\-
)>P&\\*y!#=$\"3EMZI$R@V9%F*7$$!3WCVrl)[@?&F^s$\"3X_^2ia=4YF*7$$!3S5B'p
S8$HFF^s$\"33eZ(p*eT*)[F*7$$!3.VX:B'HE?'!#?$\"3u**pQ\"HU***\\F*7$$\"3b
6>V%QN&3FF^s$\"3'3'RCQg2\"*[F*7$$\"3H.`,suJ?^F^s$\"3j_RH\"yI4i%F*7$$\"
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$\"3tD!)H-M([I\"F*$\"3z%Hlf\"Rp;IF*7$$\"3oO`QHJff:F*$\"3_tZ]WbM)[#F*7$
$\"3g&>\"zdfTU=F*$\"3uRPjkj.`?F*7$$\"3)pDzHS![p>F*$\"33_6wgK#y$>F*7$$
\"3M=t;[[a'4#F*$\"33JG/7u>')=F*7$$\"3ea$oZE8AB#F*$\"3u?A')G!)f6>F*7$$
\"3!3Rp8o\")yO#F*$\"3y)eI$)o^-.#F*7$$\"3w\"R,6V<3\\#F*$\"3Jj7r[%*4EAF*
7$$\"3q#RL3=`Ph#F*$\"3yh<!zJ>A^#F*7$$\"3SB4#HnbD)GF*$\"3!Rw\"*)*oTfZ$F
*7$$\"3>LsqpyZNJF*$\"3ytteZ\\JJ[F*7$$\"3sPtcTW&)*R$F*$\"3E_lMJL$[s'F*7
$$\"3u[$)>k3LeOF*$\"3)=>,&=$3i.*F*7$$\"3+;\\MC<$*GRF*$\"3em%[(QRy!>\"F
-7$$\"3Oz6M9Db*=%F*$\"3Vb(p$*=lb]\"F-7$$\"3o8rS9\\2cWF*$\"379<L#)>!*e=
F-7$$\"3wWiVu.R?ZF*$\"3(Q(R/$=4AB#F-7$$\"3Im7YJ)oK'\\F*$\"3yN_MI>`(e#F
-7$$\"3M`<._\\jT_F*$\"3a]eJ0]\"***HF-7$$\"3w8mpG*31\\&F*$\"3%\\5,A1&ol
LF-7$$\"3Ql7(o6ngv&F*$\"3eP#=P.h`u$F-7$$\"3b(*fluW95gF*$\"31SNqQ8m$4%F
-7$$\"3)****>YH&=$G'F*F+7]o7$F($!3A$z._1PmD\"F-7$F/$!3RxeXXU5P8F-7$F4$
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-7$F>$!3]gtz`>N![\"F-7$$!3GX$QL=e))4&F*$!3EG-nmW*G[\"F-7$FC$!3gr0\"=R,
rZ\"F-7$$!3(ozn)3A7S[F*$!3-LFFS5&RY\"F-7$FH$!3HfSdql7V9F-7$$!3O=srF$en
e%F*$!3=C#zs36MT\"F-7$FM$!3G>w!eVmaP\"F-7$FR$!3E<;K8#*)=F\"F-7$FW$!3)e
5D)f!\\\"Q6F-7$Ffn$!3KUb&3%y#zu*F*7$F[o$!3c)>\\**z(yb\")F*7$F`o$!39p\"
\\$pd#[F'F*7$Feo$!3_G#RB2:nQ%F*7$Fjo$!3M(yYS*))4gEF*7$Fdp$!3+)=/[4'RT7
F*7$F^q$\"3S])eQ(zm^=F^s7$Fhq$\"3&R$pO'o7U7\"F*7$$!3a*\\x%GR;0<F*$\"3p
X%[<Q.Ya\"F*7$F]r$\"3!)=Astono=F*7$$!3H=_g\"R:@W\"F*$\"3/<$p6NHW2#F*7$
Fbr$\"3[v'\\ZcJY?#F*7$$!3/&=Qq]*3$=\"F*$\"3!)GXrCLriAF*7$Fgr$\"3t@$yN.
@fB#F*7$$!3;cW:%*)e`=*F^s$\"3k3$\\27(\\O@F*7$F\\s$\"3YEgjl#**4(>F*7$Fb
s$\"3)\\A\\@F.\\W\"F*7$Fgs$\"3;E\"4=cW\">!)F^s7$F\\t$\"3O^l.,!p2'=!#>7
$Fbt$!3umb2ypigzF^s7$Fgt$!3,tN)pn$oD9F*7$F\\u$!3jHvY%HcY%>F*7$$\"3%Hbw
(\\+hq!*F^s$!3?s?Du1OC@F*7$Fau$!3\"pEwECrGB#F*7$$\"3!=$*\\hHUK<\"F*$!3
?n#*RB\"pNE#F*7$Ffu$!3-5_WK]]9AF*7$$\"3?\"oTeELAV\"F*$!3q6nI74h(3#F*7$
F[v$!3iG2Vl**\\!)=F*7$$\"39m#)eVX+,<F*$!3-L***QJlcb\"F*7$F`v$!3(=`fU@c
=8\"F*7$Fjv$!3=f>cm&HlJ\"F^s7$Fdw$\"3!=,MTV!zT7F*7$F^x$\"3%3)zrQh;4FF*
7$Fcx$\"3C;)eGWiV[%F*7$Fhx$\"3$*\\2`5?QSiF*7$F]y$\"3y;xe^2rw!)F*7$Fby$
\"3=4!\\'4g*oy*F*7$Fgy$\"3N!\\pMGD+9\"F-7$F\\z$\"3iEaJcO6r7F-7$Faz$\"3
=%)e[z/)[P\"F-7$$\"3Az;UWEB)e%F*$\"32CLtxxz89F-7$Ffz$\"3_\\fRp?1W9F-7$
$\"3`b([HgH=%[F*$\"3#)3n3ZI=k9F-7$F[[l$\"3y>PK]4+x9F-7$$\"3#)4lu\"*=X-
^F*$\"3EMtF[H$H[\"F-7$F`[l$\"39fGC'*4#*z9F-7$$\"3a$=k.%>7m`F*$\"3CYEI[
R8q9F-7$Fe[l$\"3I0k7GR?a9F-7$Fj[l$\"3$p^<K#et-9F-7$F_\\l$\"3=TuOF$foL
\"F-7$Fd\\l$\"3A$z._1PmD\"F--%&COLORG6)%$RGBG$\"#5!\"\"$\"\"!Fa[mFb[mF
b[mFb[mF_[m-%+AXESLABELSG6$Q\"x6\"Q!Fh[m-%%FONTG6$%*HELVETICAGF`[m-%%V
IEWG6$;$!+3`=$G'!\"*$\"+3`=$G'Fd\\m;$!21p'*3>4:g\"!#:$\"1k$z)okXmX!#9"
 1 2 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}}}{EXCHG {PARA 
211 "" 0 "" {TEXT 231 108 "Again we have the original function in red \+
and the derivative in blue.  Is the slope of h(x) ever negative? " }}}
{EXCHG {PARA 211 "" 0 "" {TEXT 231 143 "For now, the Major difference \+
between D(f) and diff(f(x), x) is  that the D(f) results in a function
 and diff(f(x),x) results in an expression." }}}{EXCHG {PARA 211 "" 0 
"" {TEXT 235 6 "Aside:" }{TEXT 231 112 "  The benefit of the \"diff\" \+
command is that we may describe functions with more than one variable \+
such as y = a " }{XPPEDIT 18 0 "t^2;" "6#*$%\"tG\"\"#" }{TEXT 231 1 " \+
" }{TEXT 231 1 " " }{TEXT 231 5 " + b " }{TEXT 236 1 "t" }{TEXT 231 
65 " + c and take the derivative with respect to a specific variable:"
 }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 37 "quad := (a,b,c,t) ->
 a*t^2 + b*t + c;" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#>I%quadG6\"f*6&I
\"aGF%I\"bGF%I\"cGF%I\"tGF%F%6$I)operatorGF%I&arrowGF%F%,(*&9$\"\"\"9'
\"\"#F2*&9%F2F3F2F29&F2F%F%F%" }{TEXT 233 1 " " }}}{EXCHG {PARA 211 ""
 0 "" {TEXT 231 35 "Notice this is just a quadratic in " }{TEXT 236 1 
"t" }{TEXT 231 98 ", but also contains the variables a, b, c.  We woul
d expect to differentiate this with respect to " }{TEXT 236 1 "t" }
{TEXT 231 3 " by" }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 23 "dif
f(quad(a,b,c,t), t);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#,&*&I\"aG6\"
\"\"\"I\"tGF&F'\"\"#I\"bGF&F'" }{TEXT 233 1 " " }}}{EXCHG {PARA 211 ""
 0 "" {TEXT 231 100 "This gives expected derivative.  Suppose we wante
d to differentiate this function with respect to a:" }}}{EXCHG {PARA 
211 "> " 0 "" {MPLTEXT 1 232 23 "diff(quad(a,b,c,t), a);" }}{PARA 212 
"" 1 "" {XPPMATH 20 "6#*$I\"tG6\"\"\"#" }{TEXT 233 1 " " }}}{EXCHG 
{PARA 211 "" 0 "" {TEXT 231 133 "This may seem strange but if \"a\" is
 considered the independent variable then the derivative with respect \+
to \"a\" gives the \"constant\" " }{XPPEDIT 18 0 "t^2;" "6#*$%\"tG\"\"
#" }{TEXT 231 1 " " }{TEXT 231 1 " " }{TEXT 231 92 ".  We will deal wi
th differentiating with respect to more than one variable in the futur
e.  " }}}{EXCHG {PARA 211 "" 0 "" {TEXT 235 9 "Summary: " }{TEXT 231 
202 "The \"D\" command is better when using the derivative as a functi
on because you can evaluate the derivative at a number.  The \"diff\" \+
command works well if you have more than one variable in the function.
 " }}}{EXCHG {PARA 211 "> " 0 "" {MPLTEXT 1 232 16 "# end of section" 
}}}}{SECT 1 {PARA 210 "" 0 "" {TEXT 230 10 "Assignment" }}{PARA 211 ""
 0 "" {TEXT 231 204 "Remember: Your Name, Section, and lab number in u
pper left.  You may put all of this in one line if you want to save ve
rtical space.  These two graphs and corresponding written portions mus
t be put on to " }{TEXT 235 3 "one" }{TEXT 231 15 " printed page. " }}
{EXCHG {PARA 211 "" 0 "" {TEXT 235 11 "Problem # 1" }{TEXT 231 2 ": " 
}}{PARA 211 "" 0 "" {TEXT 231 29 "Consider the function f(x) = " }
{XPPEDIT 18 0 "x/10;" "6#*&%\"xG\"\"\"\"#5!\"\"" }{TEXT 231 1 " " }
{TEXT 231 1 " " }{TEXT 231 3 " ( " }{XPPEDIT 18 0 "x^2/3;" "6#*&%\"xG
\"\"#\"\"$!\"\"" }{TEXT 231 1 " " }{TEXT 231 1 " " }{TEXT 231 10 " - 4
.1  x)" }}{PARA 211 "" 0 "" {TEXT 231 4 "a)  " }{TEXT 235 6 "Graph:" }
{TEXT 231 401 " Plot the function and the tangent to the curve at x=6.
 Plot the secant lines through (6,f(6)) and (6+h,f(6+h)) for h=3 and h
 = 2.  Put all this (function, tangent, 2 secants) on one graph plotte
d over the domain [-2,12]. The secant lines should appear to be conver
ging to the tangent line. Label the curve and label the lines as secan
t or tangent. This labelling is probably best done by hand.       " }}
{PARA 211 "" 0 "" {TEXT 231 4 "b)  " }{TEXT 235 10 "Questions:" }
{TEXT 231 107 " For what value(s) of x is the tangent to the curve y =
 f(x) a horizontal line. Give these values exactly. " }}}{EXCHG {PARA 
211 "" 0 "" {TEXT 235 11 "Problem # 2" }}{PARA 211 "" 0 "" {TEXT 231 
3 "a) " }{TEXT 235 6 "Graph:" }{TEXT 231 33 " Plot the functions cos(x
)  and  " }{XPPEDIT 18 0 "(sin(x+h)-sin(x))/h;" "6#*&,&-%$sinG6#,&%\"x
G\"\"\"%\"hGF*F*-F&6#F)!\"\"F*F+F." }{TEXT 231 1 " " }{TEXT 231 1 " " 
}{TEXT 231 139 "  for values of  h = 1, 0.5, and 0.2. Put these on one
 graph plotted over one period. Label the curves by hand indicating th
e value of h.  " }}{PARA 211 "" 0 "" {TEXT 231 3 "b) " }{TEXT 235 10 "
Question: " }{TEXT 231 60 "What do these graphs suggest about the deri
vative of sin(x)?" }}}}{PARA 215 "" 0 "" }{PARA 216 "" 0 "" }}
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }