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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "Module  8 : Differential Calculus
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 3 "" 0 "" {TEXT -1 26 "803 : Fun With Derivatives" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 17 "O B J E C T I V E
" }}{PARA 0 "" 0 "" {TEXT -1 255 "In the project, we will learn variou
s ways to take the derivative of functions, how to create derivative t
ables to see trends, how to take higher order derivatives, how to cons
truct tangent lines, and how to graph a function together with its der
ivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" 
{TEXT -1 254 "In this project we will use the following command packag
es. Type and execute this line before beginning the project below. If \+
you re-enter the worksheet for this project, be sure to re-execute thi
s statement before jumping to any point iin the worksheet." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rest
art;  with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the prote
cted names norm and trace have been redefined and unprotected\n" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________
____________________________________________________________________" 
}}{PARA 4 "" 0 "" {TEXT -1 17 "A. The Derivative" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "_________________________________________________________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 200 "In the last proj
ect, we learned how to find derivatives the \"long way\". We will now \+
take them directly in a single command. there are two different ways t
o compute derivatives using Maple - using the " }{TEXT 256 4 "diff" }
{TEXT -1 9 " and the " }{TEXT 257 1 "D" }{TEXT -1 10 " commands." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "f := x -> sqrt( 1 + x^2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\"\"\"F0*$)9$\"\"#F0F0
F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 
"The " }{TEXT 258 4 "diff" }{TEXT -1 120 " command requires two parame
ters - the function and the variable with to respect to which the deri
vative is being taken." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff( f(x), x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#*&%\"xG\"\"\"*$-%%sqrtG6#,&F%F%*$)F$\"\"#F%F%F%!\"\"
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Alth
ough you may not be as familiar with the differential operator, " }
{TEXT 259 1 "D" }{TEXT -1 109 " , which acts on functions by taking th
eir derivative, it is a very convenient way to compute the derivative.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "D(f)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\"*$-%%s
qrtG6#,&F%F%*$)F$\"\"#F%F%F%!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 54 "Here is another way to compute derivative
s. Using the " }{TEXT 260 4 "diff" }{TEXT -1 96 " command, Maple will \+
create an expression showing a derivative but not actually compute it.
 the " }{TEXT 261 6 "value " }{TEXT -1 25 "command forces an answer." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "f := x -> x^100:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "D
iff( f(x), x);   % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%D
iffG6$*$)%\"xG\"$+\"\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Di
ffG6$*$)%\"xG\"$+\"\"\"\"F),$*$)F)\"#**F+F*" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 83 "_____________________________________
______________________________________________" }}{PARA 4 "" 0 "" 
{TEXT -1 20 "B. Derivative Tables" }}{PARA 0 "" 0 "" {TEXT -1 83 "____
______________________________________________________________________
_________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "Instead of taking the derivative \+
of a single function, we can create a command which will create a tabu
lar array of derivatives of similar functions. This makes it easy to s
ee patterns and trends." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 43 "Here are powers of x and their derivatives." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "transpose(array( [seq( [ x^k, diff(x^k,x) ],k = 1..11)]));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7-%\"xG*$)F(\"\"#\"\"\"*
$)F(\"\"$F,*$)F(\"\"%F,*$)F(\"\"&F,*$)F(\"\"'F,*$)F(\"\"(F,*$)F(\"\")F
,*$)F(\"\"*F,*$)F(\"#5F,*$)F(\"#6F,7-F,,$F(F+,$F)F/,$F-F2,$F0F5,$F3F8,
$F6F;,$F9F>,$F<FA,$F?FD,$FBFG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 104 "Here is a table showing the derivatives \+
of both positive and negative powers of x. Do you see a pattern?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "traspose(array( [ seq( [ x^k, diff(x^k,x) ], k= -5..5)]));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)trasposeG6#-%'matrixG6#7-7$*&\"\"\"
F,*$)%\"xG\"\"&F,!\"\",$*&F,F,*$)F/\"\"'F,F1!\"&7$*&F,F,*$)F/\"\"%F,F1
,$F+!\"%7$*&F,F,*$)F/\"\"$F,F1,$F9!\"$7$*&F,F,*$)F/\"\"#F,F1,$F@!\"#7$
*&F,F,F/F1,$FGF17$F,\"\"!7$F/F,7$*$FIF,,$F/FJ7$*$FBF,,$FTFC7$*$F;F,,$F
WF<7$*$F.F,,$FZF0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 101 "Here is table of derivatives of sin(x), sin(2x), sin(3x)
, sin(4x), and sin(5x). Do you see a pattern?" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "transpose(ar
ray( [seq( [sin(k*x), diff( sin(k*x),x) ], k =1..5)]));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7'-%$sinG6#%\"xG-F)6#,$F+\"\"#-F)6
#,$F+\"\"$-F)6#,$F+\"\"%-F)6#,$F+\"\"&7'-%$cosGF*,$-F>F-F/,$-F>F1F3,$-
F>F5F7,$-F>F9F;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 83 "_______________________________________________________________
____________________" }}{PARA 4 "" 0 "" {TEXT -1 26 "C. Evaluating A D
erivative" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________
_______________________________________________________" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 312 "Since the derivative of a function is a function itself,
 it can be evaluated at various values of x. In order words, we can pl
ug in a value like x = a into f'(x) to find the value f'(a). what we a
re doing geometrically is finding the slope of the tangent line at (a,
f(a)). There are two ways to do this in Maple." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 262 4 "diff" 
}{TEXT -1 170 " command requires one statement to compute the derivati
ve and a second statement to evaluate it. On the other hand, using the
 D command you can do this in one quick step." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := x -> x^
4 + x^3 - 5*x^2 + 60;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"x
G6\"6$%)operatorG%&arrowGF(,**$)9$\"\"%\"\"\"F1*$)F/\"\"$F1F1*&\"\"&F1
)F/\"\"#F1!\"\"\"#gF1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "diff( f(x), x); eval(%, x=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
(*$)%\"xG\"\"$\"\"\"\"\"%*&F'F()F&\"\"#F(F(*&\"#5F(F&F(!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"$0\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "(D)(f)(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$0\"
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "As y
ou can see, the D command is more convenient in this case." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________
____________________________________________________________" }}{PARA 
4 "" 0 "" {TEXT -1 16 "D. Tangent Lines" }}{PARA 0 "" 0 "" {TEXT -1 
83 "__________________________________________________________________
_________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Evaluating the derivative
 at x=a gives the slope of the tangent line at (a,f(a)). To find the \+
" }{TEXT 263 8 "equation" }{TEXT -1 281 " of this tangent line require
s a little more information. If we begin with the point-slope formula:
 y-y1 = m(x -x1 ) where x1  a, y1 = f(a), and m = f'(a), and then solv
e for y, we get y = f(a) + f'(a)*(x-a) as the linear function which re
presents the equation of the tangent line." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
119 "Here we define a target value, a, a function, f(x), the tangent l
ine as a function, T(x), and then graph them together." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a :=2;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "f :=x -> 2+ sin(x^2)/(x-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"#\"\"\"*&
-%$sinG6#*$)9$F-F.F.,&F5F.F.!\"\"F7F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 32 "T := x -> f(a) + D(f)(a) *(x-a);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"TGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%\"fG6#%
\"aG\"\"\"*&--%\"DG6#F.F/F1,&9$F1F0!\"\"F1F1F(F(F(" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 56 "plot( \{f(x), T(x)\}, x=-2..5, y=-10..10, d
iscont = true);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 83 "_______________________________________________________________
____________________" }}{PARA 4 "" 0 "" {TEXT -1 27 "E. Higher Order D
erivatives" }}{PARA 0 "" 0 "" {TEXT -1 83 "___________________________
________________________________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 179 "There are occasions when we wish to find the 2nd deriv
ative of a function or even higher order derivatives such as the 3rd o
r 4th derivative. There is a way to do this either the " }{TEXT 264 4 
"diff" }{TEXT -1 4 " or " }{TEXT 265 2 "D " }{TEXT -1 9 "commands." }}
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" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"\"\"$*&F'F(F&F(F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"'\"\"#\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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"" {TEXT -1 181 "We can also create tables which show the various deri
vatives of a function. In this case, we let f(x) = sin(7x), and then c
reate a table showing f(x) and its first four derivatives." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f :=
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0 "" {MPLTEXT 1 0 54 "transpose(array( [ seq( [k,(D@@k)(f)(x) ], k=0..
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82 "Here is another table showing the square root of x and its first f
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}{PARA 0 "" 0 "" {TEXT -1 83 "________________________________________
___________________________________________" }}{PARA 4 "" 0 "" {TEXT 
-1 40 "F. Graphing A Function & Its Derivatives" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "_________________________________________________________
__________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 240 "The derivative o
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