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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 65 "Lab # 2: Secant Lines, Ta
ngent Lines, Derivatives(\"D\" and \"diff\")" }}{PARA 256 "" 0 "" 
{TEXT -1 262 "In this lab we investigate the relationship between seca
nt lines and average rates of change, tangent lines and instantaneous \+
rates of change, how Maple can aid in the limiting process that connec
ts these two concepts, and finally the derivative of a function.  " }}
}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 47 "Plotting Secant Lines / Average \+
rates of change" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 685 "As we discusse
d in class, the tangent line to a graph of f(x) at x = a , has slope d
escribing the instantaneous rate of change of y with respect to x, at \+
x = a. We further illustrated that this tangent line can be considered
 the limiting case of  a converging sequence of secant lines. Furtherm
ore,  the slope of a secant line gives the average rate of change of y
 with respect to x over an interval.   Therefore,  the slope of the ta
ngent line (the derivative) at x = a, is the limiting value of a conve
rging sequence of average rates of change, leading to an instantaneous
 rate of change.   In this section we illustrate this concept graphica
lly. First we define the function f(x) = " }{XPPEDIT 18 0 "(x-2)^2;" "
6#*$,&%\"xG\"\"\"\"\"#!\"\"F'" }{TEXT -1 8 " + 1 by:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> (x-2)^2 + 1;" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(,&*$),&9$\"\"\"\"\"#!\"\"F2F1F1F1F1F(F(F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 322 "The goal is to plot a sequence of secant lines that ap
proaches the tangent line of f(x) at x = 2.  Using the point slope def
inition 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 function yields:  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 69 "                                         \+
                     f(x) = " }{XPPEDIT 18 0 "(f(2+h)-f(2))/h;" "6#*&,
&-%\"fG6#,&\"\"#\"\"\"%\"hGF*F*-F&6#F)!\"\"F*F+F." }{TEXT -1 16 " (x -
 2) + f(2)." }}{PARA 0 "" 0 "" {TEXT -1 51 "We now define a function o
f two variables, x and h," }{TEXT -1 131 "  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 259 0 "" }{TEXT -1 1 " " }
{TEXT -1 3 "h. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "secf2h :
= (x,h) -> (f(2 + h) - f(2))/h *(x - 2) + f(2);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'secf2hGf*6$%\"xG%\"hG6\"6$%)operato
rG%&arrowGF),&*(,&-%\"fG6#,&\"\"#\"\"\"9%F5F5-F16#F4!\"\"F5F6F9,&9$F5F
4F9F5F5F7F5F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "This crypti
c notation is for \"sec\"ant line of \"f\" at x = \"2\" for step size \+
\"h\". You may call it anything you want.   " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 79 "plot([secf2h(x,2), secf2h(x,1), secf2h(x,.5), se
cf2h(x,.1), f(x)], x = 0 .. 4);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" 
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45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Notice that this produces a seque
nce of lines that approach the tangent line.  Take a guess at the slop
e of the tangent line. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
# end of section" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 77 "Plotting Ta
ngent Lines/  Instantaneous Rates of change  / The \"limit\" command" 
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 " Now we try to plot the tangent
 line to this curve at x = 2.  Recall that the slope of the tangent li
ne , at x = 2, is defined by the following limit:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 34 "                                  " }{XPPEDIT 18 
0 "m[tan];" "6#&%\"mG6#%$tanG" }{TEXT -1 4 "  = " }{XPPEDIT 18 0 "limi
t((f(2+h)-f(2))/h,h = 0);" "6#-%&limitG6$*&,&-%\"fG6#,&\"\"#\"\"\"%\"h
GF-F--F)6#F,!\"\"F-F.F1/F.\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
107 "The following command in Maple will find this limit. Here we want
 the limit as h goes to zero so we type:  " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "mtanf2 := limit((f(2+h)-f(2))/h, h = 0);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'mtanf2G\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 306 "This notation is for the slope of the tangent line of  t
he function f(x), at x = 2.  Again, you can call it anything you want.
  Using a similar procedure as above, we plot the tangent line (from t
he point slope formula for a line) by:  f(x) = mtanf2 (x - 2) + f(2). \+
 And use this to plot the tangent line. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "tanf2 := x -> mtanf2*(x - 2) + f(2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&tanf2Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&%'
mtanf2G\"\"\",&9$F/\"\"#!\"\"F/F/-%\"fG6#F2F/F(F(F(" }}}{EXCHG {PARA 
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RIu/qN%F27$$\"34++]2%)38RF2$\"3w&)3\"\\D2*fYF27$$\"\"%F)F*-%'COLOURG6&
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[l7$F^rFb[l7$FcrFb[l7$FhrFb[l7$F]sFb[l7$FbsFb[l7$FgsFb[l7$F\\tFb[l7$Fa
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FivFb[l7$F^wFb[l7$FcwFb[l7$FhwFb[l7$F]xFb[l7$FbxFb[l7$FgxFb[l7$F\\yFb[
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LABELSG6$Q\"x6\"Q!Fj^l-%%VIEWG6$;F(Fez%(DEFAULTG" 1 2 0 1 10 0 2 9 1 
4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 165 "Notice above that the tanf2 = 0, and hen
ce the tangent line is horizontal. This means that the instantaneous r
ate of change of y with respect to x at x = 2 is zero. " }}{PARA 0 "" 
0 "" {TEXT -1 114 "Plot the tangent line to f (x) at x = 3 by defining
 a new slope (mtan3) and a new function tanf3. How's it look?  " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "It should be pointed out here tha
t the limit command need not be evaluated at a specific value of x. Fo
r example, try evaluating the derivative of  " }}{PARA 0 "" 0 "" 
{TEXT -1 7 "y = 13 " }{XPPEDIT 18 0 "x^4;" "6#*$%\"xG\"\"%" }{TEXT -1 
5 " - 5 " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 36 " + 2 x \+
- 1 by the limit definition: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "q := x -> 13*x^4 - 5*x^2 + 2*x - 1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$\"\"%\"
\"\"\"#8*&\"\"&F1)F/\"\"#F1!\"\"*&F6F1F/F1F1F1F7F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "limit((q(x+h) - q(x))/h, h = 0);" }
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"
\"#_\"\"#F(*&\"#5F(F&F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 345 "
Recognize this?  This is why we love Maple.  It performs symbolic mani
pulations.  Ie. It doesn'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 a
ble to symbolically determine derivatives. Right?  ... Yes Sir!!!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# end of section" }}}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 42 "Derivatives: The \"D\" and \"diff\" com
mands. " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Maple has two commands \+
to evaluate derivatives of functions.  D(f) and diff(f(x), x). " }}
{PARA 0 "" 0 "" {TEXT -1 42 "D(f)  returns the derivative as  function
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g := x -> x^3 - 4*x;" 
}{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%
)operatorG%&arrowGF(,&*$)9$\"\"$\"\"\"F1*&\"\"%F1F/F1!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(g);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF&,&*
$)9$\"\"#\"\"\"\"\"$\"\"%!\"\"F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 57 "Notice this describes a function and we give it a name by" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "derivg := D(g);" }{TEXT -1 
0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'derivgGf*6#%\"xG6\"6$%)opera
torG%&arrowGF(,&*$)9$\"\"#\"\"\"\"\"$\"\"%!\"\"F(F(F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "or " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "derivg := x -> D(g)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'
derivgGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"\"\"$\"\"%
!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "In either case, yo
u can now evaluate the derivative at a number such as 3 by" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "derivg(3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#B" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "You can al
so plot the derivative with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "plot([g(x), derivg(x)], x = -4 .. 4, color = [red, blue]);" }
{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&
-%'CURVESG6$7S7$$!\"%\"\"!$!#[F*7$$!3ommmmFiDQ!#<$!3w42dS))poS!#;7$$!3
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F_t$!33\"ezl26WF$F07$Fdt$!3)f'\\npU&3o#F07$Fit$!3=#z:&4;xH>F07$F^u$!3u
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A>u(Gr\"GF07$Fbv$\"3#>Dv_6ejI%F07$Fgv$\"3x*))eZG(o-hF07$F\\w$\"3A9IS,?
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3]@Q.;)R^f#F37$Fdy$\"3UMNZdu_SHF37$Fiy$\"3/`])Gr4aE$F37$F^z$\"3**oyk\"
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{PARA 0 "" 0 "" {TEXT -1 682 "In this graph we see the original functi
on in red and the derivative in blue.  From this graph you should be a
ble to estimate approximately where the tangent line is horizontal.  B
etter yet, if you click on the graph, you can then use the mouse to ha
ve Maple show you the coordinates of any point on the graph. Click on \+
the graph and notice that a window opens up on the menu bar showing th
e location of the cursor.  If you click on the cursor anywhere on the \+
graph these coordinates will change.  Try this and notice that the der
ivative is equal to zero at about x =   -1.15 and 1.15.  These are the
 square roots of 4/3 which you can find by setting the derivative equa
l to zero.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "The command diff
(f(x), x) returns an expression describing the derivative of f(x) with
 respect to the independent variable x.  " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "h := x -> x^2 + 5*cos(x);" }{TEXT -1 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*
$)9$\"\"#\"\"\"F1*&\"\"&F1-%$cosG6#F/F1F1F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "diff(h(x), x);" }{TEXT -1 0 "" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&%\"xG\"\"#*&\"\"&\"\"\"-%$sinG6#F$F(!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Notice this returns an expression
 for the derivative. In order to enter this as a function we must type
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "derivh := x -> diff(h(
x), x);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'derivhGf*
6#%\"xG6\"6$%)operatorG%&arrowGF(-%%diffG6$-%\"hG6#9$F2F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "The big difference between diff an
d D is that you cannot evaluate the \"diff\" at a number.  " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "derivh(3);" }}{PARA 8 "" 1 "
" {TEXT -1 73 "Error, (in derivh) wrong number (or type) of parameters
 in function diff\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "but you ca
n still plot it. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([
h(x), derivh(x)], x = -2*Pi .. 2*Pi, color = [red, blue]);" }{TEXT -1 
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Again we have the original functi
on in red and the derivative in blue.  Is the slope of h(x) ever negat
ive? " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "For now, the Major diff
erence between D(f) and diff(f(x), x) is  that the D(f) results in a f
unction and diff(f(x),x) results in an expression." }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }{TEXT 258 6 "Aside:" }{TEXT -1 112 "  The benef
it 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#*$%\"tG\"
\"#" }{TEXT -1 5 " + b " }{TEXT 260 1 "t" }{TEXT -1 0 "" }{TEXT -1 65 
" + c and take the derivative with respect to a specific variable:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "quad := (a,b,c,t) -> a*t^2 \+
+ b*t + c;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%quadGf
*6&%\"aG%\"bG%\"cG%\"tG6\"6$%)operatorG%&arrowGF+,(*&9$\"\"\")9'\"\"#F
2F2*&9%F2F4F2F29&F2F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Noti
ce this is just a quadratic in " }{TEXT 261 1 "t" }{TEXT -1 98 ", but \+
also contains the variables a, b, c.  We would expect to differentiate
 this with respect to " }{TEXT 262 1 "t" }{TEXT -1 0 "" }{TEXT -1 3 " \+
by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(quad(a,b,c,t), t
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"aG\"\"\"%\"tGF&\"\"#%\"bG
F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "This gives expected deriva
tive.  Suppose we wanted to differentiate this function with respect t
o a:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(quad(a,b,c,t),
 a);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"tG\"\"#\"
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "This may seem strange bu
t if \"a\" is considered the independent variable then the derivative \+
with respect to \"a\" gives the \"constant\" " }{XPPEDIT 18 0 "t^2;" "
6#*$%\"tG\"\"#" }{TEXT -1 92 ".  We will deal with differentiating wit
h respect to more than one variable in the future.  " }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 9 "Summary: " }{TEXT -1 202 "The \"
D\" command is better when using the derivative as a function because \+
you can evaluate the derivative at a number.  The \"diff\" command wor
ks well if you have more than one variable in the function. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# end of section" }}}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 11 "Assignment " }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 11 "Problem # 1" }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 33 "Consider the function f(x) = x ( " }{XPPEDIT 18 0 "x
^2/3;" "6#*&%\"xG\"\"#\"\"$!\"\"" }{TEXT -1 7 " - 4 x)" }}{PARA 0 "" 
0 "" {TEXT -1 100 "a)  Plot the function and a sequence of  3 secant l
ines that converge to the tangent line at x = 1. " }}{PARA 0 "" 0 "" 
{TEXT -1 53 "b)  Plot the function and the tangent line at x = 1. " }}
{PARA 0 "" 0 "" {TEXT -1 141 "c) For what values of x is the tangent l
ine horizontal?  Support your answer with a graph of the function and \+
it's horizontal tangent lines. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }{TEXT 257 11 "Problem # 2" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 32 "Plot the functions cos(x)  and  " }{XPPEDIT 18 0 "(sin(x+h)-sin
(x))/h;" "6#*&,&-%$sinG6#,&%\"xG\"\"\"%\"hGF*F*-F&6#F)!\"\"F*F+F." }
{TEXT -1 79 "  for incrementally smaller values of  h starting at 1, o
ver the domain  x = -2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 " t
o 2" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "
" {TEXT -1 59 "a) Display one graph for h = .5 and one graph for h = .
1.  " }}{PARA 0 "" 0 "" {TEXT -1 61 "b) At what value of h are the two
 graphs indistinguishable?  " }}{PARA 0 "" 0 "" {TEXT -1 63 "c) What d
o these graphs suggest about the derivative of sin(x)?" }}}}}{MARK "3
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }