{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 77 "Lab # 4: Analysis of Fun ctions using first and second derivative information" }}{PARA 256 "" 0 "" {TEXT -1 278 "In this lab we analyze functions with regards to in tervals where the function is increasing or decreasing, concave up or down, and the location of inflection points and relative extrema. So me of the Maple commands that aid in this task are demonstrated with a sample problem. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 60 "Sample Prob lem: Plotting Derivatives and Solving for Zeros. " }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 "Consider the function " }{XPPEDIT 18 0 "f(x ) = sqrt(5)*x+3*x^(2/3);" "6#/-%\"fG6#%\"xG,&*&-%%sqrtG6#\"\"&\"\"\"F' F.F.*&\"\"$F.)F'*&\"\"#F.F0!\"\"F.F." }{TEXT -1 6 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 258 4 "(1) " }{TEXT -1 43 "Plot the f, f', and f'' on the same axes. " }}{PARA 0 "" 0 "" {TEXT 256 4 "(2) " } {TEXT -1 46 "Find all relative minimum and maximum points. " }}{PARA 0 "" 0 "" {TEXT 257 3 "(3)" }{TEXT -1 33 " Find all points of inflecti on " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "First try to plot the f, f' and f'' on the same axes. " }}{PARA 0 "" 0 "" {TEXT -1 101 "In ord er to plot x^(2/3) we must use the surd function in order to get an ou tput when x is negative. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := x -> sqrt(5) * x + 3*surd(x^2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&-%%sqrtG6# \"\"&\"\"\"9$F2F2*&\"\"$F2-%%surdG6$*$)F3\"\"#F2F5F2F2F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "df := x -> D(f)(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGf*6#%\"xG6\"6$%)operatorG%&arrow GF(--%\"DG6#%\"fG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eval(df(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$\"\"&#\"\"\"\" \"#F'*(F(F'-%%surdG6$*$)%\"xGF(F'\"\"$F'F/!\"\"F'" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 85 "If you want to see what this looks like you can ev aluate this with the eval command. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "ddf := x -> D(df)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ddfGf*6#%\"xG6\"6$%)operatorG%&arrowGF(--%\"DG6#%#dfG6#9$F(F( F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eval(ddf(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"#\"\"$\"\"\"*&-%%surdG6$*$)% \"xGF&F(F'F(F/!\"#F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Now plot all three on the same axes. There are some disontinuities in the derivatives so we use discont=true. 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+F5(=^#F[]l7$$\"+:s0?%*Ffao$!+?IQb:F[]l7$$\"+>UXE7F[cn$!+Zy0%4\"F[]l7$ $\"+*QQi'=F[cn$!+&QW5D'F_cn7$$\"+kME5DF[cn$!+t(3+@%F_cn7$$\"++wA^JF[cn $!+*oo)3JF_cn7$$\"+/M[XPF[cn$!+/\"\\#pCF_cn7$$\"+o3!3O%F[cn$!+NR)f,#F_ cn7$$\"+pI;(*\\F[cn$!+Tk;\"o\"F_cn7$$\"+\\W[JcF[cn$!+VdaL9F_cn7$$\"+c> 'RG'F[cn$!+n))fQ7F_cn7$$\"+*\\k'eoF[cn$!+/))=-6F_cn7$$\"+XMk0vF[cn$!+( =\"ot(*F[cn7$$\"+\\*y_:)F[cn$!++Ek\\()F[cn7$$\"+JJK\"y)F[cn$!++C(z#zF[ cn7$$\"+D7$)\\$*F[cn$!+t&Q=H(F[cn7$$\"+MUe-5F_cn$!+aZwVmF[cn7$$\"+U7&) f5F_cn$!+3U[phF[cn7$$\"+(Qmk7\"F_cn$!+@())yo&F[cn7$$\"+x?V&=\"F_cn$!+ \\f\"QJ&F[cn7$$\"+'yE,D\"F_cn$!+*fe.&\\F[cn7$$\"+Z9t68F_cn$!+GM#Gk%F[c n7$$\"+[%4gP\"F_cn$!+CH#fN%F[cn7$$\"+=n.N9F_cn$!+`+o=TF[cn7$$\"+/cq)\\ \"F_cn$!+T:0()QF[cn7$$\"+#oR[c\"F_cn$!+OOdpOF[cn7$$\"+\"f4Ci\"F_cn$!+q F*p\\$F[cn7$$\"+rme%o\"F_cn$!+$QkfK$F[cn7$$\"+y;#)['))GF[cn7$$\"+/g(*R>F_c n$!+X@PbFF[cn7$$\"+xsj+?F_cn$!+A[aWEF[cn7$$\"+MhSl?F_cn$!+qMbMDF[cn7$$ \"+Xj4C@F_cn$!+@4hTCF[cn7$$\"+[-E)=#F_cn$!+mOiYBF[cn7$$\"+![L'[AF_cn$! +,j*HE#F[cn7$$\"+.4u6BF_cn$!+ZV+\"=#F[cn7$$\"+W*RMP#F_cn$!+u$Qd5#F[cn7 $$\"+BJ.QCF_cn$!+jHoJ?F[cn7$$\"+ISC+DF_cn$!+46ck>F[cn7$$\"+pO'Qc#F_cn$ !+LL$)**=F[cn7$$\"+0l&pi#F_cn$!+r#R#R=F[cn7$$\"+oA$\\o#F_cn$!+>yZ'y\"F [cn7$$\"+O!z8v#F_cn$!+(>&=H 0, " }{TEXT 265 36 "th ere is a relative minimum at (0,0)" }{TEXT -1 181 ". This fact is evi dent from the graph. There is also a relative maximum near x = -.7 fou nd by putting the arrow where f'(x) = 0. We can have Maple find this v alue of x exactly by " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "s olve(df(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\")\"\"\"\"# D!\"\"\"\"&#F&\"\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "So there is a relative maximum at x = " }{XPPEDIT 18 0 "-8/25*5^(1/2);" "6#,$* (\"\")\"\"\"\"#D!\"\")\"\"&*&F&F&\"\"#F(F&F(" }{TEXT -1 39 ". The rel ative maximum has a value of " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"%\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 37 "The relative maximum maximum point is" } {TEXT -1 2 " (" }{XPPEDIT 18 0 "-8/25*5^(1/2);" "6#,$*(\"\")\"\"\"\"#D !\"\")\"\"&*&F&F&\"\"#F(F&F(" }{TEXT -1 8 ", 4/5) " }{TEXT 267 31 "or approximately (-0.72, 0.80)" }{TEXT -1 32 " rounded to two decimal p laces. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "By examining the curv e describing the second derivative we see that it is undefined at x=0 \+ but negative everywhere else. So the curve does not change concavity a nd there is " }{TEXT 268 19 "no inflection point" }{TEXT -1 2 ". " }}} }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Assignment" }}{PARA 0 "" 0 "" {TEXT -1 219 "You may do the analysis for these problems by hand or wi th Maple. The above example illustrates the techniques you may use to identify the relative maximum and minimum points as well as inflectio n points. Make sure to " }{TEXT 271 24 "fit these onto one page." } {TEXT -1 5 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 11 "Graph #1 " }{TEXT 260 285 "Generate a graph of the following fun ction and its first two derivatives over the interval x = -3..3. Labe l each curve accordingly. Also label all relative maximum and minimum points as well as inflection points. You may do this by hand. Make s ure the x and y values are rounded to " }{TEXT 272 19 "two decimal pla ces." }{TEXT 273 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 269 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 1 " " }{TEXT -1 2 " " } {XPPEDIT 18 0 "f(x) = (x-1)^3*(x+3.2)+50;" "6#/-%\"fG6#%\"xG,&*&,&F'\" \"\"F+!\"\"\"\"$,&F'F+-%&FloatG6$\"#KF,F+F+F+\"#]F+" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 11 "Problem #2 " } {TEXT 270 297 "Generate a graph of the following function and its firs t two derivatives over the interval x = -3..3. Label each curve accor dingly. Also label all relative maximum and minimum points as well as inflection points. You may do this by hand. Make sure your points ar e rounded to two decimal places." }}{PARA 0 "" 0 "" {TEXT 263 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 1 " " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 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,F-F/F-\"\"%F-" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }