{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {PSTYLE "Left 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 "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 "D ash 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 "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 "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 "Author" -1 19 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 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 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 "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 "Text Ou tput" -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 "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 "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 } {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 "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 "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 "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Cou rier" 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 "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 "Li st 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 "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 "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Co urier" 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 }{CSTYLE "LaTeX" -1 32 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Ma ple 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 " Help 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 "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic Small" -1 201 "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 "_cstyle273" -1 202 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle272" -1 203 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle271" -1 204 "" 0 1 0 0 0 0 0 1 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" -1 205 "" 0 1 0 0 0 0 0 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 2 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" -1 206 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle268" -1 207 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle267" -1 208 "" 0 1 0 0 0 0 0 1 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" -1 209 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyl e265" -1 210 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle264" -1 211 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle263" -1 212 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle262" -1 213 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle261" -1 214 "" 0 1 0 0 0 0 0 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" -1 215 "" 0 1 0 0 0 0 0 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" -1 216 "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" -1 217 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle258" -1 218 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle257" -1 219 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle256" -1 220 "" 0 1 0 0 0 0 0 1 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" -1 221 "" 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" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyl e1" -1 222 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }{PSTYLE "_pstyle2 " -1 201 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 8 8 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle2" -1 223 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle3" -1 202 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" -1 224 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle4" -1 203 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 "_cstyl e4" -1 225 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle5 " -1 226 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 227 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "_pstyle5 " -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 } 3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle7" -1 228 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle6" -1 205 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 }{PSTYLE "_pstyle7" -1 206 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 {EXCHG {PARA 200 "" 0 "" {TEXT 222 77 "Lab # 4: Analysis of F unctions using first and second derivative information" }}{PARA 201 "" 0 "" {TEXT 223 278 "In this lab we analyze functions with regards to \+ intervals where the function is increasing or decreasing, concave up \+ or down, and the location of inflection points and relative extrema. \+ Some of the Maple commands that aid in this task are demonstrated with a sample problem. " }}}{SECT 1 {PARA 202 "" 0 "" {TEXT 224 60 "Sample Problem: Plotting Derivatives and Solving for Zeros. " }}{EXCHG {PARA 203 "" 0 "" {TEXT 225 29 "Consider the function " } {XPPEDIT 18 0 "f(x) = sqrt(5)*x+3*x^(2/3);" "6#/-%\"fG6#%\"xG,&*&-%%sq rtG6#\"\"&\"\"\"F'F.F.*&\"\"$F.)F'*&\"\"#F.F0!\"\"F.F." }{TEXT 225 1 " " }{TEXT 225 6 " " }}{PARA 203 "" 0 "" {TEXT 226 4 "(1) " } {TEXT 225 43 "Plot the f, f', and f'' on the same axes. " }}{PARA 203 "" 0 "" {TEXT 226 4 "(2) " }{TEXT 225 46 "Find all relative minimu m and maximum points. " }}{PARA 203 "" 0 "" {TEXT 226 3 "(3)" }{TEXT 225 33 " Find all points of inflection " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 225 54 "First try to plot the f, f' and f'' on the same axes. " }}{PARA 203 "" 0 "" {TEXT 225 101 "In order to plot x^(2/3) we must u se the surd function in order to get an output when x is negative. " } }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 227 38 "f := x -> sqrt(5) * x \+ + 3*surd(x^2,3);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6#I\"x GF%F%6$I)operatorGF%I&arrowGF%F%,&*&-I%sqrtGF%6#\"\"&\"\"\"9$F2F2-I%su rdGF%6$*$F3\"\"#\"\"$F9F%F%F%" }{TEXT 228 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 227 19 "df := x -> D(f)(x);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I#dfG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%--I\"D GF%6#I\"fGF%6#9$F%F%F%" }{TEXT 228 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 227 12 "eval(df(x));" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#, &*$\"\"&#\"\"\"\"\"#F'*&-I%surdG6$I*protectedGF-I(_syslibG6\"6$*$I\"xG F/F(\"\"$F'F2!\"\"F(" }{TEXT 228 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 225 85 "If you want to see what this looks like you can evaluate this with the eval command. " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 227 21 "ddf := x -> D(df)(x);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#>I $ddfG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%--I\"DG6$I*protectedGF 0I(_syslibGF%6#I#dfGF%6#9$F%F%F%" }{TEXT 228 1 " " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 227 13 "eval(ddf(x));" }}{PARA 204 "" 1 "" {XPPMATH 20 "6#,$*&-I%surdG6$I*protectedGF(I(_syslibG6\"6$*$I\"xGF*\" \"#\"\"$\"\"\"F-!\"##F1F/" }{TEXT 228 1 " " }}}{EXCHG {PARA 203 "" 0 " " {TEXT 225 110 "Now plot all three on the same axes. There are some d isontinuities in the derivatives so we use discont=true. " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 227 123 "plot([f(x),df(x),ddf(x)],x=-3. .3,y=-8..8,discont=true,thickness=[3,3,3],linestyle=[SOLID,DASH,DOT],c olor=[black,red,blue]);" }}{PARA 205 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6&7X7$$!\"$\"\"!$!3WrlVLY_zY!#=7$$!3!****** \\2<#pG!#<$!3#R$zH9\\;#e$F-7$$!3#)***\\7bBav#F1$!35!4@w5j+l#F-7$$!36++ ]K3XFEF1$!3l37Hs%*eG;F-7$$!3%)****\\F)H')\\#F1$!3K(=.x[=yI'!#>7$$!3#** **\\i3@/P#F1$\"35;7wsAn)H$FC7$$!3;++Dr^b^AF1$\"3i.m-d.d*=\"F-7$$!3$*** *\\7Sw%G@F1$\"3qM5hIT0Y?F-7$$!3*****\\7;)=,?F1$\"3ut8,LH'H*GF-7$$!3/++ DO\"3V(=F1$\"3wMB+i()[%p$F-7$$!3#******\\V'zV([kp984^F-7$$!3!******\\!)H%*\\\"F1$\"3iHd\"HM))G x&F-7$$!3/+++vl[p8F1$\"3_F:AoaxtjF-7$$!3\"******\\>iUC\"F1$\"3?>4]gfv# )oF-7$$!3-++DhkaI6F1$\"3sZ?p4*)RxsF-7$$!3s******\\XF`**F-$\"37$zY.ox-l (F-7$$!3u*******>#z2))F-$\"3Q\"=aiU$oqyF-7$$!3S++]7RKvuF-$\"3QeUY (z%F-7$$!3%p)*\\i:sw%)*FC$\"3S!3&z!y$\\&>%F-7$$!39$***\\(oKQm'FC$\"3'* =Z)G#z%4W$F-7$$!3O***\\(=K**zMFC$\"3F^.eV!o&>CF-7$$!3Wb+++v`hH!#?$\"3* 3-6;6!eCbFC7$$\"3%[***\\i]2=jFC$\"3gwa6rTrrhF-7$$\"3]****\\(QIKH\"F-$ \"3cNmm*3pj0\"F17$$\"38****\\7:xWCF-$\"3))>P*e;>'>e`a\"e$F17$$\"3s****\\(o3lW(F-$\"31N4\"eRY(HTF17$$\" 35*****\\A))oz)F-$\"3K\"H4,.>8s%F17$$\"3e******Hk-,5F1$\"3Xjys`cTS_F17 $$\"36+++D-eI6F1$\"3/.)f\")oPQy&F17$$\"3u***\\(=_(zC\"F1$\"3!y.E@v))zE 'F17$$\"3M+++b*=jP\"F1$\"3I@3!z=![*y'F17$$\"3g***\\(3/3(\\\"F1$\"37>$3 6+$etsF17$$\"33++vB4JB;F1$\"3IaX\"\\zSNx(F17$$\"3u*****\\KCnu\"F1$\"3[ .'QtWXpD)F17$$\"3s***\\(=n#f(=F1$\"3]8O*45ayv)F17$$\"3P+++!)RO+?F1$\"3 SWyjf2tN#*F17$$\"30++]_!>w7#F1$\"3)\\`PN2\">?(*F17$$\"3O++v)Q?QD#F1$\" 3d2![X!Rn>5!#;7$$\"3G+++5jypBF1$\"39//\")[g9j5Ffz7$$\"3<++]Ujp-DF1$\"3 za;s/>i76Ffz7$$\"3++++gEd@EF1$\"3_\\-dX\"zl:\"Ffz7$$\"39++v3'>$[FF1$\" 3M@.3s)fJ?\"Ffz7$$\"37++D6EjpGF1$\"3-7rWOE[Z7Ffz7$$\"\"$F*$\"3=3b;Sb%[ H\"Ffz-%*LINESTYLEG6#\"\"\"-%*THICKNESSG6#Fb\\l-%&COLORG6&%$RGBG$F*!\" \"F`]lF`]l-F$6'7gn7$F($\"3%[?&)zGaM\\)F-7$$!3*GyIw`3Y$HF1$\"3sw+PO&47R )F-7$$!3Ew&pex6x(GF1$\"35[K/\"zV(*H)F-7$$!3;\\Di;as8GF1$\"3GK+u^g&R>)F -7$$!3?q8D9\\J\\FF1$\"3q?%Hd.xT3)F-7$$!3yyXvV0@&o#F1$\"3)ptYJ/h9(zF-7$ $!3sWMP'exdi#F1$\"3+69#y51P'yF-7$$!3[Bl\\,#QUc#F1$\"3sOv@u-l[xF-7$$!3p 6Qi\"3%f+DF1$\"3I\"\\0r&3sDwF-7$$!3*=p]#pS:PCF1$\"3cu664B'*)\\(F-7$$!3 m.iv=#)*=P#F1$\"3)[b,RO(F-7$$!3ie67I3U9BF1$\"3@)HuwJp2C(F-7$$!3Sq0+ /\\r\\AF1$\"3GXsO/&yr4(F-7$$!3S808*GVZ=#F1$\"3a'R&3(3[t%pF-7$$!3%ytb#* 4J@7#F1$\"3u*e^+&zA(z'F-7$$!3Q`W\\KKFl?F1$\"3mHi.mgpbmF-7$$!3ksY]HPm(* >F1$\"3#pqu.W$[!['F-7$$!3y?#>@h*QS>F1$\"3D7$HGt^dK'F-7$$!3ynu(y>mP(=F1 $\"3hH:OH`\"z8'F-7$$!3t8/P(=$z9=F1$\"3!GIAekhS'fF-7$$!3=>[7[/4]C;F1$\" 3mCXx<]CY`F-7$$!3%[o%*4cd^c\"F1$\"3BkuV!yi\\8&F-7$$!3lQqur2[,:F1$\"3kt Y.9f\"[*[F-7$$!30BVv$[Q`V\"F1$\"3O:*R@;^0j%F-7$$!3=x%>wUhxP\"F1$\"3GVn I+:$pQ%F-7$$!3pY)='Hmd:8F1$\"3\"[3.9*\\33TF-7$$!3)HL(\\[OL^7F1$\"3QEW3 zE#4!QF-7$$!3IJI([U%[)=\"F1$\"39(\\wW(zOzMF-7$$!30^'p$pXnF6F1$\"35o;G$ \\+f9$F-7$$!31*oHEfb,1\"F1$\"3'[h/Ug]ju#F-7$$!3Ko-,!*y'[***F-$\"3+h#os >dsN#F-7$$!3q-eI;*)4Z$*F-$\"3&G*fgO!\\a!>F-7$$!3Ua(H([R7g()F-$\"3c%e'3 )y>%e9F-7$$!3\"p=j(o_S=\")F-$\"3=@]j()>'e@*FC7$$!3!p!3A,!)f9vF-$\"3&p_ '))>)[4i$FC7$$!3G5J[FaW$)oF-$!3<:2psB-2HFC7$$!3uWsYA%yjE'F-$!31lb%)[V1 65F-7$$!37p#4]Xm.i&F-$!3=P&z@1^U(=F-7$$!3MRO+],=)*\\F-$!35KDM@\"*zSGF- 7$$!3#>>w()y/>O%F-$!3?/j#ef35,%F-7$$!3;.#)y3\")*3t$F-$!33fy=]L-@aF-7$$ !3]iyp.&o5:$F-$!3c(f@(*Ht+.(F-7$$!3Okp_U$=l[#F-$!3DB!z**3;YW*F-7$$!3%) Hd@cn8#*=F-$!39'*z:;FnZ7F17$$!3G&>LP,-%e7F-$!3Wn?yD)4]v\"F17$$!3/7w*33 &>^&*FC$!3-y^YY9ER@F17$$!3TrKYC+P=lFC$!3K<&pe>\"[LFF17$$!3[`uMowx))[FC $!3()=)[mdGOB$F17$$!3cN;B7`=fKFC$!3!pG0CWr^-%F17$$!3eEP)>)))f_cF17$$!38jo3n[>A7FC$!3g'\\')3-Ll W'F17$$!3C)3z0ynz9)Fdt$!35cQaM#GIq(F17$$!3#eJMaLx46'Fdt$!3*\\c$3)*>K.( )F17$$!3GW&*G!*o)R2%Fdt$!3v2>K[0kG5Ffz7$$!3HsZ9Xk*p.#Fdt$!3u'z,(=_7a8F fz7$$!3'***************f!#E$!3')42K[jS)4\"!#97gn7$$\"3'*************** fFh_m$\"3I4dF%[yG5\"F[`m7$$\"3NL!Q!*>*[V?Fdt$\"3+(4R;2m'*z\"Ffz7$$\"3Q mg2)Rsp3%Fdt$\"3U+rc%[EXZ\"Ffz7$$\"3U*49rfb/8'Fdt$\"3#=cht(fQ;8Ffz7$$ \"3YK@:'zQR<)Fdt$\"39DD2aEY;7Ffz7$$\"3')>GU>04E7FC$\"3gFE()[j%44\"Ffz7 $$\"3YE/BfryM;FC$\"3I\\sa4rj65Ffz7$$\"3/Sc%)Q/=_CFC$\"32VUH=r9?\"*F17$ $\"3C`3Y=PdpKFC$\"33/Ju*3p1\\)F17$$\"3O!G\"px-O/\\FC$\"3OXCGNa'**p(F17 $$\"3z1<#p$o9RlFC$\"3H'pS>K[.?(F17$$\"3kI?)\\#\\)RQ*FC$\"3]y)Q\"y?BPmF 17$$\"3XNUI,B)GA\"F-$\"3w#[G_0>aE'F17$$\"353Xx$*eui=F-$\"3M&)Rn!eN!QdF 17$$\"3^)H'[<4&o]#F-$\"3oP1KF`(zS&F17$$\"3O5UXAY*y9$F-$\"3;J\"QDtIh<&F 17$$\"39^bE'>CAu$F-$\"3F,S)foK9,&F17$$\"3YjZ.X!=wN%F-$\"3)**)e2!H,T([F 17$$\"3O\")=wV#fS*\\F-$\"3)4qOI&)3pv%F17$$\"3i\"3$\\n$f%GcF-$\"3o9))ff *)ReYF17$$\"3/lzVsy,\"G'F-$\"3u`Fh\">D9d%F17$$\"3_;%)ye5&G](F-$\"3+>&f:&R2PWF17$$\"3Il[porc_\")F-$\"3C *)e<%eypP%F17$$\"35AEWn!*oy()F-$\"3rqm\\q%>[K%F17$$\"3[ka0NxEZ$*F-$\"3 -P%)pO(y:G%F17$$\"3UF`\\wiL-5F1$\"3FL!)H\")G^MUF17$$\"3Ez2)QR5'f5F1$\" 3M)ei1CPy>%F17$$\"3FKD73QBE6F1$\"3Ce+*=#\\OeTF17$$\"3K'eH'=o?&=\"F1$\" 3i\"yuO0Qf7%F17$$\"3)3=vyb4*\\7F1$\"3'f(\\D\"H[F4%F17$$\"3Nc>]m=_68F1$ \"3)HU[-a1K1%F17$$\"3K%Q)*p%y!eP\"F1$\"3!G*=2JJHMSF17$$\"3@:`+XC%[V\"F 1$\"3;op2J^G4SF17$$\"3ShHDM#>&)\\\"F1$\"37I?v/U!Q)RF17$$\"3+xcCA:mk:F1 $\"3!*G`>\"p@)eRF17$$\"3)G_!Qy&QAi\"F1$\"37yf%p;%>QRF17$$\"3O`6QwLU%o \"F1$\"3i8(4E([)p\"RF17$$\"33nE]djm[F1$\"3Ef_q*)4rRQF17$$\"3At*)*p@80+#F1$\"3r84\\sKLBQF17$$\"3y>%pV6!Hl ?F1$\"3`q1Dp?c1QF17$$\"3hCq76w)R7#F1$\"33JMvw&f>z$F17$$\"3O\"oB\"z%f\" )=#F1$\"3IIdS5')fwPF17$$\"3O>z(e?S&[AF1$\"3u5MYgQoiPF17$$\"3.*o^KYb;J# F1$\"3_:<4+=+DF1$\"3:qQz?[k4PF17$$\"3'3QAr_4Qc#F1$\"3 KK.yhBN(p$F17$$\"3uz67&>5pi#F1$\"3qDMdlpb&o$F17$$\"3!Q@Ic:$*[o#F1$\"3I +D+[l/vOF17$$\"3i.tur\"[8v#F1$\"3%Rjs\\[mLm$F17$$\"32F%y.L'y5GF1$\"3[I y(o&RB`OF17$$\"3_!oEY!)fT(GF1$\"38)og!o0uUOF17$$\"3Mn`v0j\"[$HF1$\"3*p \\6`.#)Hj$F17$Fa\\l$\"3!*e5?E0zAOF1-Ff\\lF[]lFi\\l-F]]l6&F_]l$\"#5Fa]l F`]lF`]l-F$6'7gn7$F($!+KG!3a\"!#57$$!+%Q6Y$H!\"*$!+Uzu'e\"F]cn7$$!+)4< x(GFacn$!+$=8(G;F]cn7$$!+CNt8GFacn$!+C9Gy;F]cn7$$!+DeK\\FFacn$!+Js!4t \"F]cn7$$!+XUA&o#Facn$!+@)=iy\"F]cn7$$!+uQzDEFacn$!+t^KS=F]cn7$$!+orDk DFacn$!+BYW**=F]cn7$$!+=eh+DFacn$!+n;F]cn7$$!+n&yrV#Facn$!+`GjK?F]c n7$$!+cb#>P#Facn$!+[6`2@F]cn7$$!+p1X9BFacn$!+4.gx@F]cn7$$!+gvu\\AFacn$ !+PA]hAF]cn7$$!+s(yZ=#Facn$!+F3h^BF]cn7$$!+3$p@7#Facn$!+4vcWCF]cn7$$!+ ;RJl?Facn$!+jVqMDF]cn7$$!+btq(*>Facn$!+`vr\\EF]cn7$$!+JdVS>Facn$!+x<]a FF]cn7$$!+;_\"Q(=Facn$!+'QXe)GF]cn7$$!+sZ%[\"=Facn$!+#\\X:,$F]cn7$$!+ \\[9]C[VF]cn7$$!+V*\\cJ\"Fa cn$!+r^RCYF]cn7$$!+e(49D\"Facn$!+/GfV\\F]cn7$$!+pKc)=\"Facn$!+^*e^H&F] cn7$$!+hgvF6Facn$!+tw@zcF]cn7$$!+B+Cg5Facn$!+(*zYmhF]cn7$$!*fQd***Facn $!+tkXqmF]cn7$$!*\"y*zM*Facn$!+$\\XPH(F]cn7$$!*R[5w)Facn$!+$HWC&zF]cn7 $$!*ld$>\")Facn$!+`OH,))F]cn7$$!*mwb^(Facn$!+g8Yc(*F]cn7$$!*b^W)oFacn$ !+Fwo'4\"Facn7$$!*T6uE'Facn$!+f='HC\"Facn7$$!*cF9i&Facn$!+*HmpV\"Facn7 $$!*Lo#**\\Facn$!+dDA!o\"Facn7$$!*m?IO%Facn$!+xkh9?Facn7$$!*XT?t$Facn$ !+RW6\"[#Facn7$$!*3PA:$Facn$!+/8a2JFacn7$$!*%er([#Facn$!+.D'4E%Facn7$$ !*8gL*=Facn$!+B#Q>8'Facn7$$!*(Hlf7Facn$!+S=yb5!\")7$$!++m$Qc*!#6$!+`RG C:Feao7$$!)N9JlFacn$!+9)zY`#Feao7$$!+\"3A;!\\Fiao$!+&y)R;PFeao7$$!+i15 sKFiao$!+rj+qjFeao7$$!+_*RtX#Fiao$!+$*ydJ$*Feao7$$!+V#zDk\"Fiao$!+f)Rm f\"!\"(7$$!+))))>N7Fiao$!+YI&[L#Feco7$$!+N`=y#)!#7$!+Ly-\")RFeco7$$!+g NGTiF^do$!+&3V:!eFeco7$$!+(y\"Q/UF^do$!+`kRC)*Feco7$$!+8+[n@F^do$!+w%o mP#!\"'7$$!+%R#y08!#8$!+$yYj+\"!\"%7gn7$$\"+yVplQFdeo$!+OASnB!\"&7$$\" +E$*yHCF^do$!+d*y3/#F`eo7$$\"+9#4IZ%F^do$!+ZNqX!*Feco7$$\"++\"Hi^'F^do $!+[MZxaFeco7$$\"+!**[%f&)F^do$!+ezc2QFeco7$$\"+w))ek7Fiao$!+DJyiAFeco 7$$\"+aGBt;Fiao$!+f)exb\"Feco7$$\"+43_!\\#Fiao$!+tE=m\"*Feao7$$\"+k(3y I$Fiao$!+<`[yiFeao7$$\"+uYQU\\Fiao$!+!z'evOFeao7$$\"+%egpd'Fiao$!+pS;6 DFeao7$$\"++@V@%*Fiao$!+)Q!3b:Feao7$$\"+i.fE7F]cn$!+Sf*Q4\"Feao7$$\"+0 :Pm=F]cn$!+K*\\/D'Facn7$$\"+LNR5DF]cn$!+Nzr4UFacn7$$\"+PYN^JF]cn$!+\"f ,(3JFacn7$$\"+HwgXPF]cn$!+8*R\"pCFacn7$$\"+\"=A4O%F]cn$!+h\"4f,#Facn7$ $\"+s8G(*\\F]cn$!+vL6\"o\"Facn7$$\"+](*fJcF]cn$!+4m]L9Facn7$$\"+qT2%G' F]cn$!+x$p&Q7Facn7$$\"+%*RxeoF]cn$!+V`;-6Facn7$$\"+z)\\d](F]cn$!+$R' \\t(*F]cn7$$\"+4BQb\")F]cn$!+ZZ\\\\()F]cn7$$\"+HNU\"y)F]cn$!+Z:&y#zF]c n7$$\"+L*G*\\$*F]cn$!+tpt\"H(F]cn7$$\"+%o$f-5Facn$!+i7oVmF]cn7$$\"+A/' )f5Facn$!+hHTphF]cn7$$\"+^_ZE6Facn$!+\\!Hyo&F]cn7$$\"+j1W&=\"Facn$!+KY w8`F]cn7$$\"+l]8]7Facn$!+))[J]\\F]cn7$$\"+M%R5)*R>Facn$!+YE ObFF]cn7$$\"+0?k+?Facn$!+*[OXk#F]cn7$$\"+c0Tl?Facn$!+NiaMDF]cn7$$\"+*[ +T7#Facn$!+qXgTCF]cn7$$\"+)3k#)=#Facn$!+v\"=mM#F]cn7$$\"+Nqj[AFacn$!+J :*HE#F]cn7$$\"+fTu6BFacn$!+^-+\"=#F]cn7$$\"+4HWtBFacn$!+m[t0@F]cn7$$\" +\"yN!QCFacn$!+4+oJ?F]cn7$$\"+&RY-]#Facn$!+J'eX'>F]cn7$$\"+Kd'Qc#Facn$ !+&HJ)**=F]cn7$$\"+p#epi#Facn$!+DwBR=F]cn7$$\"+fP$\\o#Facn$!+(\\wky\"F ]cn7$$\"+7-Q^FFacn$!+7U=H 0, " }{TEXT 226 36 "there is a relative minimum at (0,0)" }{TEXT 225 181 ". This fact is evident from the graph. There is also a relat ive maximum near x = -.7 found by putting the arrow where f'(x) = 0. W e can have Maple find this value of x exactly by " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 227 17 "solve(df(x)=0,x);" }}{PARA 204 "" 1 " " {XPPMATH 20 "6#,$*$\"\"&#\"\"\"\"\"##!\")\"#D" }{TEXT 228 1 " " }}} {EXCHG {PARA 203 "" 0 "" {TEXT 225 38 "So there is a relative maximum \+ at x = " }{XPPEDIT 18 0 "-8/25*5^(1/2);" "6#,$*(\"\")\"\"\"\"#D!\"\") \"\"&*&F&F&\"\"#F(F&F(" }{TEXT 225 1 " " }{TEXT 225 39 ". The relativ e maximum has a value of " }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 227 5 "f(%);" }}{PARA 204 "" 1 "" {XPPMATH 20 "6##\"\"%\"\"&" }{TEXT 228 1 " " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 226 37 "The relative maxim um maximum point is" }{TEXT 225 2 " (" }{XPPEDIT 18 0 "-8/25*5^(1/2);" "6#,$*(\"\")\"\"\"\"#D!\"\")\"\"&*&F&F&\"\"#F(F&F(" }{TEXT 225 1 " " }{TEXT 225 8 ", 4/5) " }{TEXT 226 31 "or approximately (-0.72, 0.80) " }{TEXT 225 32 " rounded to two decimal places. " }}}{EXCHG {PARA 203 "" 0 "" {TEXT 225 173 "By examining the curve describing the secon d derivative we see that it is undefined at x=0 but negative everywher e else. So the curve does not change concavity and there is " }{TEXT 226 19 "no inflection point" }{TEXT 225 2 ". " }}}}{SECT 1 {PARA 202 " " 0 "" {TEXT 224 10 "Assignment" }}{PARA 203 "" 0 "" {TEXT 225 219 "Yo u may do the analysis for these problems by hand or with Maple. The a bove example illustrates the techniques you may use to identify the re lative maximum and minimum points as well as inflection points. Make s ure to " }{TEXT 226 24 "fit these onto one page." }{TEXT 225 5 " " }}{EXCHG {PARA 203 "" 0 "" {TEXT 226 11 "Graph #1 " }{TEXT 225 285 "Generate a graph of the following function and its first two derivati ves over the interval x = -3..3. Label each curve accordingly. Also \+ label all relative maximum and minimum points as well as inflection po ints. You may do this by hand. Make sure the x and y values are round ed to " }{TEXT 226 19 "two decimal places." }{TEXT 225 1 " " }}{PARA 203 "" 0 "" {TEXT 225 1 " " }}{PARA 203 "" 0 "" {TEXT 225 1 " " } {TEXT 225 2 " " }{XPPEDIT 18 0 "f(x) = (x-1)^3*(x+3.2)+50;" "6#/-%\"f G6#%\"xG,&*&,&F'\"\"\"F+!\"\"\"\"$,&F'F+-%&FloatG6$\"#KF,F+F+F+\"#]F+" }{TEXT 225 1 " " }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" }}{EXCHG {PARA 203 "" 0 "" {TEXT 226 11 "Problem #2 " }{TEXT 225 297 "Generate \+ a graph of the following function and its first two derivatives over t he interval x = -3..3. Label each curve accordingly. Also label all \+ relative maximum and minimum points as well as inflection points. You may do this by hand. Make sure your points are rounded to two decimal places." }}{PARA 203 "" 0 "" {TEXT 226 1 " " }}{PARA 203 "" 0 "" {TEXT 225 1 " " }}{PARA 203 "" 0 "" {TEXT 225 2 " " }{XPPEDIT 18 0 "f (x) = (x^2-2*x)/(x^2-2*x+2)+4;" "6#/-%\"fG6#%\"xG,&*&,&*$F'\"\"#\"\"\" *&F,F-F'F-!\"\"F-,(F+F-F.F/F,F-F/F-\"\"%F-" }{TEXT 225 1 " " }}{PARA 203 "" 0 "" }{PARA 203 "" 0 "" }}{PARA 203 "" 0 "" }}{PARA 206 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }