{VERSION 6 1 "Windows XP" "6.1" }
{USTYLETAB {PSTYLE "note" -1 200 1 {CSTYLE "" -1 -1 "Comic Sans MS" 1 
12 128 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 2 }
{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 
2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 3 }{PSTYLE "Warning" -1 7 
1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 
0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" 
-1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 
2 -1 2 }{PSTYLE "Normal257" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Norm
al256" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 
0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 
1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 
0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" 
-1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 
2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 
0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Headin
g 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 
}1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE 
"" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 
0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 
0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Au
thor" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 
1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE 
"" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 
0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 
255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "L
eft Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 
0 1 2 2 2 2 2 2 0 0 0 1 }1 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed \+
Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 
0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 
{CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 
-1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 
-1 5 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 
255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "D
iagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 0 0 0 2 
2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 
{CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }3 1 0 0 12 
12 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 
1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }
{CSTYLE "LaTeX" -1 32 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Map
le Comment" -1 21 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "
2D Math Bold" -1 5 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "H
elp Underlined" -1 44 "Times" 1 12 0 0 0 0 0 0 1 2 2 2 0 0 0 1 }
{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 0 0 0 2 2 2 0 0 0 1 
}{CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 
1 }{CSTYLE "Page Number" -1 33 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 
1 }{CSTYLE "Maple Input Placeholder" -1 200 "Courier" 1 12 200 0 200 
1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Text" -1 201 "Times" 1 12 0 0 0 1 2 2 
2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 202 "Times" 0 1 0 
0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 
0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 
0 0 0 0 1 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 
0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier
" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times
" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle278" -1 203 "" 0 1 
0 0 0 0 2 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle277" -1 204 "" 0 1 0 0 0 
0 1 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle276" -1 205 "" 0 1 0 0 0 0 1 0 
0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle275" -1 206 "" 0 1 0 0 0 0 1 0 0 2 2 
2 0 0 0 1 }{CSTYLE "_cstyle274" -1 207 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 
0 1 }{CSTYLE "_cstyle273" -1 208 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }
{CSTYLE "_cstyle272" -1 209 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }
{CSTYLE "_cstyle271" -1 210 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }
{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 0 1 0 1 2 2 
2 0 0 0 1 }{CSTYLE "_cstyle270" -1 211 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 
0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }
{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }
{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 2 2 2 0 0 0 1 }
{CSTYLE "Plot Title" -1 27 "" 1 10 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }
{CSTYLE "2D Input" -1 19 "Times" 0 1 255 0 0 1 0 0 2 2 1 2 0 0 0 1 }
{CSTYLE "Help Maple Name" -1 35 "" 0 1 104 64 92 1 0 1 0 2 2 2 0 0 0 
1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 0 1 0 0 2 2 2 0 0 0 
1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 0 0 1 1 2 
2 2 0 0 0 1 }{CSTYLE "_cstyle269" -1 212 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 
0 0 1 }{CSTYLE "_cstyle268" -1 213 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 
}{CSTYLE "_cstyle267" -1 214 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }
{CSTYLE "Default" -1 38 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "H
elp Fixed" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "
_cstyle266" -1 215 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyl
e265" -1 216 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle264" 
-1 217 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle263" -1 
218 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle262" -1 219 ""
 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 
45 "" 0 1 147 0 15 1 2 0 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle261" -1 220 
"" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 
0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle260" -1 221 "" 0 1 
0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 0 
1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "" 0 1 0 128 128 
1 1 0 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 0 1 0 
0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "" 1 8 0 0 0 0 0 
0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "" 0 1 0 0 0 1 0 1 0 2 
2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 0 0 1 0 2 2 
2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 
0 1 }{CSTYLE "_cstyle259" -1 222 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }
{CSTYLE "_cstyle258" -1 223 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }
{CSTYLE "_cstyle257" -1 224 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }
{CSTYLE "_cstyle256" -1 225 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }
{CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }
{CSTYLE "Help Variable" -1 25 "Courier" 0 1 0 0 0 1 2 2 0 2 2 2 0 0 0 
1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 
0 0 0 1 }{CSTYLE "Help Emphasized" -1 226 "" 0 1 0 0 0 0 1 2 0 2 2 2 
0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 
0 1 }{PSTYLE "_pstyle1" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle1
" -1 227 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" 
-1 228 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle2" 
-1 204 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 
1 0 0 8 4 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle3" -1 229 "Times" 1 18 0 
0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle3" -1 205 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 
-1 1 }{CSTYLE "_cstyle4" -1 230 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 
0 1 }{CSTYLE "_cstyle5" -1 231 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 
1 }{PSTYLE "_pstyle4" -1 206 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 0 0 0 
0 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle6" 
-1 232 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle5" 
-1 207 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 
1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle7" -1 233 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle6" -1 208 1 {CSTYLE "" -1 
-1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 
2 -1 1 }{CSTYLE "_cstyle8" -1 234 "Courier" 1 12 255 0 0 1 2 1 2 2 1 
2 0 0 0 1 }{PSTYLE "_pstyle7" -1 209 1 {CSTYLE "" -1 -1 "Times" 0 1 0 
0 255 1 0 0 2 2 2 2 1 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_c
style9" -1 235 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cs
tyle10" -1 236 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cst
yle11" -1 237 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{PSTYLE "_p
style8" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 
0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle9" -1 211 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 
0 1 0 2 2 -1 1 }}
{SECT 0 {PARA 203 "" 0 "" {TEXT 227 45 "Maple Lab modified from Univer
sity of Georgia" }}{PARA 203 "" 0 "" {TEXT 227 0 "" }}{PARA 203 "" 0 "
" {TEXT 227 31 "Name:__________________________" }}{PARA 203 "" 0 "" 
{TEXT 227 0 "" }}{PARA 203 "" 0 "" {TEXT 228 20 "Maple Lab 2 Math 252"
 }{TEXT 227 0 "" }}{SECT 1 {PARA 204 "" 0 "" {TEXT 229 31 "Fundamental
 Theorem of Calculus" }}{PARA 205 "" 0 "" {TEXT 230 212 "Most function
s encountered in calculus are given by explicit formulas involving ele
mentary functions and basic algebraic operations. Another method of ge
nerating functions is by an integral in which the variable " }{TEXT 
231 1 "x" }{TEXT 230 92 " appears in at least one of the limits of int
egration. A basic example of such a function is" }{TEXT 230 0 "" }}
{PARA 206 "" 0 "" {XPPEDIT 18 0 "F(x) = Int(f(t),t = a .. x);" "6#/-%
\"FG6#%\"xG-%$IntG6$-%\"fG6#%\"tG/F.;%\"aGF'" }{TEXT 232 0 "" }}{PARA 
205 "" 0 "" {TEXT 230 28 "We assume that the function " }{TEXT 231 1 "
f" }{TEXT 230 1 "(" }{TEXT 231 1 "t" }{TEXT 230 47 ") is continuous ov
er any interval of the form [" }{TEXT 231 4 "a ,x" }{TEXT 230 20 "] fo
r all values of " }{TEXT 231 1 "x" }{TEXT 230 85 " under consideration
. In many cases the integral cannot be evaluated in closed form. " }
{TEXT 230 0 "" }}{PARA 205 "" 0 "" {TEXT 230 0 "" }}{PARA 205 "" 0 "" 
{TEXT 230 30 "As a specific example consider" }{TEXT 230 0 "" }}{PARA 
206 "" 0 "" {XPPEDIT 18 0 "F(x) = int(sin(t^2),t = 0 .. x);" "6#/-%\"F
G6#%\"xG-%$intG6$-%$sinG6#*$%\"tG\"\"#/F/;\"\"!F'" }{TEXT 232 0 "" }}
{PARA 205 "" 0 "" {TEXT 230 14 "for values of " }{TEXT 231 2 "x " }
{TEXT 230 125 "that are greater than or equal to 0. This integral cann
ot be evaluated in closed form. However for any non-negative value of \+
" }{TEXT 231 1 "x" }{TEXT 230 72 ", the integral exists and so F(x) is
 defined for non-negative values of " }{TEXT 231 1 "x" }{TEXT 230 90 "
. We define this function in Maple and then plot the graph over the in
terval from 0 to 5. " }{TEXT 230 0 "" }}{PARA 207 "" 0 "" {TEXT 233 1 
" " }{TEXT 233 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 234 28 "f1
:=x->int(sin(t^2),t=0..x);" }{MPLTEXT 1 234 0 "" }}{PARA 209 "" 1 "" 
{TEXT 235 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 234 19 "plot(f
1(x),x=0..5);" }{MPLTEXT 1 234 0 "" }}}{PARA 205 "" 0 "" {TEXT 230 52 
"We can evaluate this function at positive values of " }{TEXT 231 1 "x
" }{TEXT 230 1 "." }{TEXT 230 0 "" }}{EXCHG {PARA 208 "> " 0 "" 
{MPLTEXT 1 234 6 "f1(2);" }{MPLTEXT 1 234 0 "" }}{PARA 209 "" 1 "" 
{TEXT 235 0 "" }}}{PARA 205 "" 0 "" {TEXT 230 16 "We can evaluate " }
{TEXT 231 2 "f1" }{TEXT 230 17 "(2) as a decimal." }{TEXT 230 0 "" }}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 234 8 "f1(2.0);" }{MPLTEXT 1 
234 0 "" }}}{PARA 205 "" 0 "" {TEXT 230 202 "This function is a common
 function in physics called the Fresnel sine integral. This is why Map
le returns output involving the namd Fresnel. We can evaluate this fun
ction at a general positive value of " }{TEXT 231 1 "x" }{TEXT 230 0 "
" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 234 6 "f1(x);" }{MPLTEXT 1 
234 0 "" }}{PARA 209 "" 1 "" {TEXT 235 0 "" }}}{EXCHG {PARA 208 "> " 0
 "" {MPLTEXT 1 234 0 "" }}}{PARA 205 "" 0 "" {TEXT 230 107 "The graph \+
of F(x) suggests that this function is continuous and even differentia
ble for positive values of " }{TEXT 231 1 "x" }{TEXT 230 2 ". " }}
{PARA 205 "" 0 "" {TEXT 236 58 "Q1: Use Maple to find F'(x), where F(x
) was defined above." }}{PARA 205 "" 0 "" {TEXT 236 3 "A1:" }}{EXCHG 
{PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}{EXCHG {PARA 208 "> " 0 "" 
{MPLTEXT 1 237 0 "" }}}{PARA 205 "" 0 "" {TEXT 230 0 "" }}{PARA 205 ""
 0 "" {TEXT 236 74 "Q2: What do you notice?  Why did this happen?  Is \+
this a rule you learned?" }}{PARA 205 "" 0 "" {TEXT 236 5 "A2:  " }}
{PARA 205 "" 0 "" {TEXT 230 0 "" }}{PARA 205 "" 0 "" {TEXT 230 0 "" }}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}{EXCHG {PARA 208 "> \+
" 0 "" {MPLTEXT 1 234 0 "" }}}}{SECT 1 {PARA 204 "" 0 "" {TEXT 229 16 
"Project - Part 1" }{TEXT 229 0 "" }}{PARA 205 "" 0 "" {TEXT 236 2 "Q3
" }{TEXT 230 20 ":  For the function " }{TEXT 231 2 "f1" }{TEXT 230 1 
"(" }{TEXT 231 1 "x" }{TEXT 230 107 ") defined above determine the fir
st two local maxima and the first two local minima for positive values
 of " }{TEXT 231 1 "x" }{TEXT 230 90 ". Also determine the first three
 positive points of inflection. Be Sure to SHOW YOUR WORK!" }}{PARA 
205 "" 0 "" {TEXT 230 0 "" }}{PARA 205 "" 0 "" {TEXT 236 2 "A3" }
{TEXT 230 2 ": " }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}{EXCHG {PARA 208 "> \+
" 0 "" {MPLTEXT 1 237 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 233 0 "" 
}}}}{SECT 1 {PARA 204 "" 0 "" {TEXT 229 18 "The Error Function" }
{TEXT 229 0 "" }}{PARA 205 "" 0 "" {TEXT 230 95 "Another common functi
on that is given in terms of an indefinite integral is the error funct
ion " }{TEXT 231 3 "erf" }{TEXT 230 1 "(" }{TEXT 231 1 "x" }{TEXT 230 
34 "). This function is defined below." }{TEXT 230 0 "" }}{PARA 206 ""
 0 "" {XPPEDIT 18 0 "erf(x) = 2*int(exp(-t^2),t = 0 .. x)/sqrt(Pi);" "
6#/-%$erfG6#%\"xG*(\"\"#\"\"\"-%$intG6$-%$expG6#,$*$%\"tGF)!\"\"/F3;\"
\"!F'F*-%%sqrtG6#%#PiGF4" }{TEXT 232 0 "" }}{PARA 205 "" 0 "" {TEXT 
230 252 "The error function is used in statistics and is related to th
e normal distribution. While the integral cannot be evaluated in close
d form, this function is so common that its values have been numerical
ly tabulated. These values are well known to Maple." }{TEXT 230 0 "" }
}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 234 9 "erf(2.0);" }{MPLTEXT 1 
234 0 "" }}}{PARA 205 "" 0 "" {TEXT 230 44 "A plot of the error functi
on is shown below." }{TEXT 230 0 "" }}{EXCHG {PARA 208 "> " 0 "" 
{MPLTEXT 1 234 20 "plot(erf(x),x=0..5);" }{MPLTEXT 1 234 0 "" }}}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}{PARA 205 "" 0 "" 
{TEXT 230 13 "The constant " }{XPPEDIT 18 0 "2/sqrt(Pi);" "6#*&\"\"#\"
\"\"-%%sqrtG6#%#PiG!\"\"" }{TEXT 230 73 " is included in the definitio
n so that the error function has limit 1 as " }{TEXT 231 1 "x" }{TEXT 
230 21 " approaches infinity." }{TEXT 230 0 "" }}}{SECT 1 {PARA 204 ""
 0 "" {TEXT 229 16 "Project - Part 2" }{TEXT 229 0 "" }}{PARA 205 "" 0
 "" {TEXT 236 3 "Q4:" }{TEXT 230 21 " Define the function " }{XPPEDIT 
18 0 "f2(x) = erf(2*x)-erf(x);" "6#/-%#f2G6#%\"xG,&-%$erfG6#*&\"\"#\"
\"\"F'F.F.-F*F&!\"\"" }{TEXT 230 58 ". Plot the graph of this function
 over the interval [0,3]." }}{PARA 205 "" 0 "" {TEXT 236 3 "A4:" }
{TEXT 230 0 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}{PARA 205 "" 0 "" 
{TEXT 230 0 "" }}{PARA 205 "" 0 "" {TEXT 236 3 "Q5:" }{TEXT 230 90 " P
redict the derivative of f2(x) with respect to x, using what we have l
earned from class." }}{PARA 205 "" 0 "" {TEXT 236 3 "A5:" }}{PARA 205 
"" 0 "" {TEXT 230 0 "" }}{PARA 205 "" 0 "" {TEXT 230 0 "" }}{PARA 205 
"" 0 "" {TEXT 236 4 "Q6: " }{TEXT 230 38 " Calculate the derivative us
ing Maple." }}{PARA 205 "" 0 "" {TEXT 236 3 "A6:" }}{EXCHG {PARA 208 "
> " 0 "" {MPLTEXT 1 237 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 
237 0 "" }}}{PARA 205 "" 0 "" {TEXT 236 0 "" }}{PARA 205 "" 0 "" 
{TEXT 236 4 "Q7: " }{TEXT 230 58 "Find the maximum value of this funct
ion. At what value of " }{TEXT 231 1 "x" }{TEXT 230 30 " does the maxi
mum value occur?" }{TEXT 230 0 "" }}{PARA 205 "" 0 "" {TEXT 236 3 "A7:
" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}{EXCHG {PARA 
208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}{PARA 205 "" 0 "" {TEXT 230 0 "" 
}}{PARA 205 "" 0 "" {TEXT 236 4 "Q8: " }{TEXT 230 70 "Determine the eq
uation of the line tangent to the graph of f2(x) when " }{TEXT 231 1 "
x" }{TEXT 230 96 " = 1.5. Plot both the graph of f2(x) along with this
 tangent line on the same coordinate system." }}{PARA 205 "" 0 "" 
{TEXT 236 3 "A8:" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}}{PARA 205 "" 0 "" 
{TEXT 230 0 "" }}{PARA 205 "" 0 "" {TEXT 230 0 "" }}{EXCHG {PARA 208 "
> " 0 "" {MPLTEXT 1 237 0 "" }}}{PARA 205 "" 0 "" {TEXT 230 0 "" }}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 237 0 "" }}}{PARA 205 "" 0 "" 
{TEXT 230 0 "" }}{PARA 211 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }