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{CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 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 "Time s" 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 "Bullet Item" -1 15 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 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 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 "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 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 "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 54 "Maximum and minimum problems invo lving area and volume" }}{PARA 0 "" 0 "" {TEXT -1 58 "by Peter Stone, \+ School of Life and Physical Sciences, RMIT" }}{PARA 0 "" 0 "" {TEXT -1 25 "peter.stone@rmit.edu.au " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "Version: 7.9.2004" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 36 "Advice for solving max/min problems " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "S uggestions specific to the solution of max/min problems " }}{PARA 15 " " 0 "" {TEXT -1 99 "Define any symbols you wish to use which are not a lready specified in the statement of the problem." }}{PARA 15 "" 0 "" {TEXT -1 29 "Make a sketch if appropriate." }}{PARA 15 "" 0 "" {TEXT -1 98 "Express the quantity, say Q, to be maximized or minimized as a \+ function of one or more variables. " }}{PARA 15 "" 0 "" {TEXT -1 211 " If Q depends on more than one variable (say n variables) find n-1 equa tions relating these variables (constraints).\n(If this cannot be done , the problem cannot be solved by single variable calculus techniques. ) " }}{PARA 15 "" 0 "" {TEXT -1 98 "Use the constraints to eliminate v ariables and hence express Q as a function of one variable, say " } {TEXT 283 1 "x" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 47 "Deter mine the interval I in which the variable " }{TEXT 285 1 "x" }{TEXT -1 40 " must lie for the problem to make sense." }}{PARA 15 "" 0 "" {TEXT -1 115 "Find all the local local maximum or minimum values of Q, which may occur at a critical point (where the derivative " } {XPPEDIT 18 0 "dQ/dx" "6#*&%#dQG\"\"\"%#dxG!\"\"" }{TEXT -1 46 " is ze ro) or at an endpoint of the interval I." }}{PARA 15 "" 0 "" {TEXT -1 183 "Give some justification that one particular value is the required extreme value. Probably the safest and most instructive way to do thi s is to make a sketch of the graph of Q against " }{TEXT 284 1 "x" } {TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 58 "Make a concluding stat ement answering the given question. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "General advice for problem sol ving " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 153 "Adapted from pages 144-145 (beginning of Section 3.7) \+ of \"Calculus with Analytic Geometry\", by Harley Flanders and Justin \+ J. Price, Academic Press, 1978." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 167 "A major part of your time in Calculus an d other courses is devoted to solving problems. It is worth your while to develop sound techniques. Here are a few suggestions." }}{PARA 15 "" 0 "" {TEXT 259 5 "Think" }{TEXT -1 350 ": Before plunging into a pr oblem, take a moment to think. Read the problem again. Think about it. What are its essential features? Have you seen a problem like it befo re? What techniques are needed?\nTry to make a rough estimate of the a nswer. It will help you understand the problem and will serve as a che ck against unreasonable answers. A car will " }{TEXT 260 3 "not" } {TEXT -1 61 " go 1000 km in 3 hr; a weight dropped from 3,000 metres w ill " }{TEXT 260 3 "not" }{TEXT -1 65 " hit the earth at 5 km. per hou r; the volume of a petrol tank is " }{TEXT 260 3 "not" }{TEXT -1 13 " \+ 1000 litres." }}{PARA 15 "" 0 "" {TEXT 259 16 "Examine the data" } {TEXT -1 232 ": Be sure you understand what is given. Translate the da ta into mathematical language. Whenever possible, make a clear diagram and label it accurately. Place axes to simplify computations. If you \+ get stuck, check that you are using " }{TEXT 260 3 "all" }{TEXT -1 11 " the data. " }}{PARA 15 "" 0 "" {TEXT 259 16 "Avoid sloppiness" } {TEXT -1 7 ": \n(a) " }{TEXT 261 28 "Avoid sloppiness in language" } {TEXT -1 84 ". \nMathematics is written in English sentences. A typica l mathematical sentence is \"" }{XPPEDIT 18 0 "y = 4*x + 1" "6#/%\"yG, &*&\"\"%\"\"\"%\"xGF(F(F(F(" }{TEXT -1 163 "\". The equal sign is the \+ verb in this sentence; it means \"equals\" or \"is equal to\". The equ al sign is not to be used in place of \"and\", nor as a punctuation ma rk. " }{TEXT 287 1 "*" }{TEXT -1 60 "Quantities on opposite sides of a n equal sign must be equal." }{TEXT 286 1 "*" }{TEXT -1 282 "\nUse sho rt simple sentences. Avoid pronouns such as \"it\" and \"which\". Give names and use them. Otherwise you may write gibberish like the follow ing: \"To find the minimum of it, differentiate it and set it equal to zero, then solve it which if you substitute it, it is the minimum.\" \n" }{TEXT 257 6 "Better" }{TEXT -1 26 ": \"To find the minimum of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 ", set its derivat ive " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 20 " equa l to zero. Let " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 49 " be the solution of the resulting equation. Then " }{XPPEDIT 18 0 "f( x[0])" "6#-%\"fG6#&%\"xG6#\"\"!" }{TEXT -1 25 " is the minimum value o f " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{XPPEDIT 18 0 ")" "6#%#% ?G" }{TEXT -1 7 ".\"\n(b) " }{TEXT 261 31 "Avoid sloppiness in computa tion" }{TEXT -1 274 ". \nDo calculations in a sequence of neat, orderl y steps. Include all steps except utterly trivial ones. This will help eliminate errors, or at least make errors easier to find. Check any n umbers used; be sure that you have not dropped a minus sign or transpo sed digits.\n(c) " }{TEXT 261 25 "Avoid sloppiness in units" }{TEXT -1 199 ". \nIf you start out measuring lengths in metres, all lengths \+ must be in metres, all areas in square metres, and all volumes in cubi c metres. Do not mix metres and millimetres, seconds and years.\n(d) \+ " }{TEXT 261 30 "Avoid sloppiness in the answer" }{TEXT -1 77 ". \nBe \+ sure to answer the question that is asked. If the problem asks for the " }{TEXT 260 13 "maximum value" }{TEXT -1 6 " of f(" }{TEXT 289 1 "x " }{TEXT -1 47 "), the answer is not the point on the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 53 " where the maximu m occurs. If the problem asks for a " }{TEXT 260 7 "formula" }{TEXT -1 30 ", the answer is not a number.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "Example 1 \+ - rectangular enclosure fenced on 3 sides " }}{PARA 0 "" 0 "" {TEXT 281 8 "Question" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 278 "A f armer wants to construct a rectangular enclosure. Fencing is needed fo r just three sides of the enclosure because one side is formed by the \+ bank of a long straight river. If 300 metres of fencing is available f or the job, what is the largest possible area for the enclosure? 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" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A=x*y" "6#/%\"AG*&%\"xG\"\"\"%\"yGF'" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 101 "Since 300 metres \+ of fencing is available, the area will be a maximum when all of the fe ncing is used." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+2*y = 300;" "6#/,&%\"xG\"\"\"*& \"\"#F&%\"yGF&F&\"$+$" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 28 "This equation constitutes a " }{TEXT 259 10 "constrain t" }{TEXT -1 17 " for the problem." }}{PARA 0 "" 0 "" {TEXT -1 45 "We \+ can use equation (ii) to express the area " }{TEXT 293 1 "A" }{TEXT -1 41 " as a function of a single variable, say " }{TEXT 296 1 "x" } {TEXT -1 27 ", since we can substitute " }{XPPEDIT 18 0 "y = (300-x)/ 2;" "6#/%\"yG*&,&\"$+$\"\"\"%\"xG!\"\"F(\"\"#F*" }{TEXT -1 19 " in eq uation (i). " }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = x*``((300-x)/2);" "6#/%\"AG *&%\"xG\"\"\"-%!G6#*&,&\"$+$F'F&!\"\"F'\"\"#F.F'" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "A = 150*x-x^2/2;" "6#/%\"AG,&*&\"$]\"\"\"\"%\"xGF(F(*&F )\"\"#F+!\"\"F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "This \+ equation is valid for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "``<=300" "6#1%!G\"$+$" }{TEXT -1 7 ". Let " }{XPPEDIT 18 0 "f(x) = 150*x-x^2/2;" "6#/-%\"fG6#%\"xG,&*&\"$]\"\"\"\"F'F+F+*&F' \"\"#F-!\"\"F." }{TEXT -1 25 ". The following graph of " }{XPPEDIT 18 0 "A=f(x)" "6#/%\"AG-%\"fG6#%\"xG" }{TEXT -1 11 " shows how " }{TEXT 353 1 "A" }{TEXT -1 13 " varies for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"! %\"xG" }{XPPEDIT 18 0 "``<=300" "6#1%!G\"$+$" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f := x -> 150*x-x^2/2;\nplot(f(x),x=0..300,A);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"$]\"\" \"\"9$F/F/*&#F/\"\"#F/*$)F0F3F/F/!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3% *******\\i9Rl!#<$\"39abg1s\"\\f*!#:7$$\"3*****\\PC#)GA\"!#;$\"3?[5_;8b ffS*\\F4$\"3Rb5 4Hu0WiF77$$\"30++v=$f%GcF4$\"3)Q\"Q)HE6(eoF77$$\"3q+++Dy,\"G'F4$\"3!Qh ,;\\n*[uF77$$\"3I++]7&)\\\"F0$\"3]a>m.*)*\\7\"Fhp7$$\"3-+]P>:mk:F0$\"3Re g&RW4H7\"Fhp7$$\"3,+]iv&QAi\"F0$\"3`&e9`')Gv6\"Fhp7$$\"3%****\\PPBWo\" F0$\"3$QL2'4S*z5\"Fhp7$$\"3-+++bjm[_#3%4\"Fhp7$$\"3!)***\\ (yb^6=F0$\"3KE!)4A!zk2\"Fhp7$$\"3)***\\PMaKs=F0$\"3uL(ea)oob5Fhp7$$\"3 *)***\\7TW)R>F0$\"3iqg%pWo#G5Fhp7$$\"3s*****\\@80+#F0$\"3=A=03EV(***F7 7$$\"3'*****\\7,Hl?F0$\"3+&o[NWNAl*F77$$\"3')**\\P4w)R7#F0$\"36jBt;t>. $*F77$$\"3?++]x%f\")=#F0$\"3!=jMwm#=#)))F77$$\"3q**\\P/-a[AF0$\"3-n6<7 yV[%)F77$$\"38+](=Yb;J#F0$\"3#*)RUg0xg&zF77$$\"3y****\\i@OtBF0$\"3:@mc lE>OuF77$$\"3v**\\PfL'zV#F0$\"3qD)\\BoB6&oF77$$\"35+++!*>=+DF0$\"3--#) RM*z\"[iF77$$\"3?++DE&4Qc#F0$\"3C[()Hfka\"f&F77$$\"3<+]P%>5pi#F0$\"3x& \\o!pqO+\\F77$$\"3K+++bJ*[o#F0$\"3YB2fEF77$$\"3C+]P/)fT(GF0$\"39- .?gTU3=F77$$\"3:+]i0j\"[$HF0$\"3aV8jb%3^c*F07$$\"$+$F)F(-%'COLOURG6&%$ RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q\"AFb[l-%%VIEWG6$;F(Fez%(DEF AULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "It looks as though the ma ximum value of " }{TEXT 294 1 "A" }{TEXT -1 14 " occurs where " } {XPPEDIT 18 0 "x =150" "6#/%\"xG\"$]\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 170 "To check this, we note that the maximum point occur s where the tangent line to the graph is horizontal, that is, where th e gradient, as given by the derivative, is zero. " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{GLPLOT2D 278 126 126 {PLOTDATA 2 "6,-%'CURVESG6$7S7 $$\"$5\"\"\"!$\"&]/\"F*7$$\"3RLLLBxV<6!#:$\"3i3eB_I#=0\"!#87$$\"3imm;$ >5E8\"F0$\"39$[<\\O7v0\"F37$$\"3MLLLAKn\\6F0$\"3!\\%[Uubjj5F37$$\"3QLL Lc$\\o;\"F0$\"35VAJC`]p5F37$$\"3gmm;bQ%R=\"F0$\"3[XllcU0v5F37$$\"3ILL$ Qk#z*>\"F0$\"395;6F37$$\"3cmmmqFc#Q\"F0$\"39bKe[U5= 6F37$$\"3(*****\\9!H.S\"F0$\"3IHZKZG.?6F37$$\"3pmmm]^0;9F0$\"37-_8JmZ@ 6F37$$\"30++]9#4LV\"F0$\"3Z'pk&phxA6F37$$\"3'*****\\j\"R(\\9F0$\"39gr: CptB6F37$$\"3mmmm\"4#)oY\"F0$\"3K7Q2-;XC6F37$$\"3wmm;^Yi#[\"F0$\"3\")f ki[!\\[7\"F37$$\"3CLLLG^g*\\\"F0$\"3H$=Q?#***\\7\"F37$$\"3PLL$=2Vs^\"F 0$\"3QvoBQ8&[7\"F37$$\"3:++]`pfK:F0$\"3Mf7J>(oW7\"F37$$\"3YLLLm&z\"\\: F0$\"3zKw7&o!zB6F37$$\"33+++G5Jm:F0$\"3h\"z#yB9!G7\"F37$$\"3-+++@32$e \"F0$\"3%G>\\$>'\\:7\"F37$$\"3=++]#y'G*f\"F0$\"3z)R5u1r+7\"F37$$\"3')* ****H%=H<;F0$\"3iJGy687=6F37$$\"3hmmm!>qMj\"F0$\"3'>q,T&G4;6F37$$\"3=+ ++ISu];F0$\"3[z'4(=\"QO6\"F37$$\"3ILL$ep'Rm;F0$\"3_uy!)pg:66F37$$\"35+ ++%>4No\"F0$\"3;N(ey=i\"36F37$$\"3cmm;@2h*p\"F0$\"3'=m(**zx206F37$$\"3 8++]c9W;P4\"F37$$\"3aLLL(>:nw\"F0$\"3Y?dv,:V*3\"F3 7$$\"3eLLLSDo$y\"F0$\"3E8]:3@w%3\"F37$$\"3!pmm^Q40!=F0$\"3-wLraq%)z5F3 7$$\"34+++3:(f\"=F0$\"3A.h1.53v5F37$$\"3$pmmc%GpL=F0$\"3!3WvBaC$p5F37$ $\"3GLLL@Ia\\=F0$\"3U#e=\"Q)4R1\"F37$$\"3))****\\9EWm=F0$\"3C-#R^!*fy0 \"F37$$\"3A++]\"oy,\\&=!=0\"F37$$\"$!>F*F+-%'COLOURG6&%$RG BG$\"*++++\"!\")$F*F*F^[l-F$6$7$7$$\"$I\"F*$\"&]7\"F*7$$\"$q\"F*Fe[l-% &COLORG6&FjzF*$\"\"(!\"\"F*-F$6%7$7$$\"$:\"F*$\"3++++++vo5F37$$\"$N\"F *$\"3++++++v=6F3-Fhz6&FjzF[[lF^[lF[[l-%*THICKNESSG6#\"\"#-F$6%7$7$$\"$ l\"F*F[]l7$$\"$&=F*Ff\\l-Fhz6&FjzF^[lF^[lF[[lF_]l-%%TEXTG6%7$$\"$;\"F* $\"&+5\"F*Q*f'(x)~>~06\"F]]l-F_^l6%7$$\"$]\"F*$\"&]8\"F*Q*f'(x)~=~0Fg^ lFj[l-F_^l6%7$$\"$%=F*Fd^lQ*f'(x)~<~0Fg^lF\\^l-%+AXESLABELSG6%Q\"xFg^l Q!Fg^l-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;F(FezF^`l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 47.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dA/dx=150-x" "6#/*&%#dAG\"\"\"%#dxG!\"\",&\"$]\"F& %\"xGF(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The derivativ e " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*&%#dAG\"\"\" %#dxG!\"\"" }{TEXT -1 11 " is 0 when " }{XPPEDIT 18 0 "x=150" "6#/%\"x G\"$]\"" }{TEXT -1 15 ", as suggested." }}{PARA 0 "" 0 "" {TEXT -1 21 "The maximum value of " }{TEXT 330 1 "A" }{TEXT -1 14 " is therefore \+ " }{XPPEDIT 18 0 "f(150) = 150*``((300-150)/2);" "6#/-%\"fG6#\"$]\"*&F '\"\"\"-%!G6#*&,&\"$+$F)F'!\"\"F)\"\"#F0F)" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "``(150)*`.`*``(75) = 11250;" "6#/*(-%!G6#\"$]\"\"\"\"% \".GF)-F&6#\"#vF)\"&]7\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "We can check that " }{XPPEDIT 18 0 "x=150" "6#/%\"xG\"$]\"" } {TEXT -1 39 " gives a maximum point on the graph of " }{XPPEDIT 18 0 " A = 150*x-x^2/2" "6#/%\"AG,&*&\"$]\"\"\"\"%\"xGF(F(*&F)\"\"#F+!\"\"F, " }{TEXT -1 35 " from the fact that the derivative " }{XPPEDIT 18 0 "d A/dx = 150-x;" "6#/*&%#dAG\"\"\"%#dxG!\"\",&\"$]\"F&%\"xGF(" }{TEXT -1 18 " is positive for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "`` < 150;" "6#2%!G\"$]\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "dA/dx" "6#*&%#dAG\"\"\"%#dxG!\"\"" }{TEXT -1 18 " is n egative for " }{XPPEDIT 18 0 "150\"!\"*$!*1wbV$FjoFco7$$!+C.*p5\"Fjo$!* %RUkYFjo-%&STYLEG6#%,PATCHNOGRIDGF0-%*THICKNESSG6#\"\"#-F$6'7$7$FAFdo7 $$FipF4$!#5Fbo7%7$$\"+w'4I*=Fjo$!+ggdN$)FaqF^q7$$\"+C.*p!=Fjo$!+SRUk&* FaqFbpF0Ffp-%%TEXTG6&7$$FeoF4$\"#vF`oQ\"x6\"F0-%%FONTG6$%*HELVETICAG\" #5-F^r6&7$$!#:FboFbrQ)~x~<~150FerF0Ffr-F^r6&7$$F4F4FbrQ$150FerF0Ffr-F^ r6&7$$\"#:FboFbrQ(x~>~150FerF0Ffr-F^r6&7$$\"\"$F4FbrQ\"3FerF0Ffr-F^r6& 7$Far$FipFboQ'f~'(x)FerF0Ffr-F^r6&7$FdsFetQ\"0FerF0Ffr-F^r6&7$F_tFetFj tF0Ffr-F^r6&7$F^s$F`tFboQ\"+FerF0-Fgr6$Fir\"#9-F^r6&7$Fis$\"\"%FboQ\"_ FerF0Fcu-%+AXESLABELSG6$Q!FerF_v-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!#NF bo$\"#DFbo;$!#8Fbo$\"#6Fbo" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 21 "Alterna tively, since " }{XPPEDIT 18 0 "d^2*A/(d*x^2)=-1" "6#/*(%\"dG\"\"#%\"A G\"\"\"*&F%F(*$%\"xGF&F(!\"\",$F(F," }{TEXT -1 31 " is negative for al l values of " }{TEXT 358 1 "x" }{TEXT -1 36 ", any turning point on th e graph of " }{XPPEDIT 18 0 "A = 150*x-x^2/2" "6#/%\"AG,&*&\"$]\"\"\" \"%\"xGF(F(*&F)\"\"#F+!\"\"F," }{TEXT -1 22 " is a maximum point. " } }{PARA 0 "" 0 "" {TEXT -1 147 "The maximum area is 11250 square metres , and it is obtained when the length of the rectangular enclosure is 1 50 metres and its width is 75 metres. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 22 ": Since the graph of " }{XPPEDIT 18 0 "A = 150*x -x^2/2" "6#/%\"AG,&*&\"$]\"\"\"\"%\"xGF(F(*&F)\"\"#F+!\"\"F," }{TEXT -1 174 " is a parabola with a vertical axis of symmetry which passes \+ through the maximum point, we do not need to use the derivative to see that the maximum value of A occurs where " }{XPPEDIT 18 0 "x=150" "6# /%\"xG\"$]\"" }{TEXT -1 31 ", which is mid-way between the " }{TEXT 295 1 "x" }{TEXT -1 12 " intercepts " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\" \"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=300" "6#/%\"xG\"$+$" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f := x -> 150*x-x^2/2;\nDiff(f(x),x);\nvalue(%); \nxmax := solve(%);\nAmax = f(xmax);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"$]\"\"\"\"9$F/F/*&#F/ \"\"#F/*$)F0F3F/F/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Di ffG6$,&*&\"$]\"\"\"\"%\"xGF)F)*&#F)\"\"#F)*$)F*F-F)F)!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"$]\"\"\"\"%\"xG!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%xmaxG\"$]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%% AmaxG\"&]7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Examp le 2 - rectangle under a semi-circle" }}{PARA 0 "" 0 "" {TEXT 273 8 "Q uestion" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 90 "Find the dim ensions of the rectangle of largest area which has two of its vertices on the " }{TEXT 272 1 "x" }{TEXT -1 151 " axis, and two of its vertic es on the semi-circle of radius 1 lying in the 1st and 2nd quadrants, \+ with its centre at the origin, and joining the points" }{XPPEDIT 18 0 "``(-1,0)" "6#-%!G6$,$\"\"\"!\"\"\"\"!" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 64 "You can play the fol lowing animation to illustrate this problem." }}{PARA 0 "" 0 "" {TEXT -1 76 "(Click on the graphic and use the controls in the context bar. \+ The buttons: " }{TEXT 262 12 "->|, <-, ->" }{TEXT -1 50 " are useful \+ for closing in on the maximum point.) 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'Fe[lFc^mF`_mFe_mF\\`mFe`m-F(6$7&7$$!3\")*****HmE3.&F.F/7$Fjev$\"3#*** ***\\d4f%yF67$$!3%******RLFjp$\"3!******HrWVc\"F.FccmFgcmF]dm-F(6&FjgvFbdmFgcmF]dm- F(6&FjgvFgdmFgcmF]dmFjdmFgemF\\fmFafmFifmF_gm-F[em6%FdgmQ,x~=~.996917F cemFjgm-F[em6%F`hmQ/y~=~.784591e-1FcemFjgmF\\an-F[em6%7$$!*jWNe$F[[n$ \"+GIoS;F_[nFg]nFjgmFghmF_im7=F'Fe[lFc^mF`_mFe_mF\\`mFe`m-F(6$7&7$$!3p *****zj4x+&F.F/7$Ffiv$\"3y*****pd\")f#RF67$$!3-+++O!H#*\\#FjpFiiv7$F\\ jvF/Feam-Fabm6%7&7$$!+Q'4x+&F_[nF/7$Fcjv$\"+x:)f#RF\\gv7$$!+O!H#*\\#F[ [nFfjv7$FijvF/FhbmF\\cm-F(6&7#FhivFccmFjgmF]dm-F(6&F^[wFbdmFjgmF]dm-F( 6&F^[wFgdmFjgmF]dm-F(6&7#7$$\"3-+++s!e%)*>Fjp$\"3Y+++/0\"f%yF6FccmFgcm F]dm-F(6&Fe[wFbdmFgcmF]dm-F(6&Fe[wFgdmFgcmF]dmFjdmFgemF\\fmFafmFifmF_g m-F[em6%FdgmQ,x~=~.999229FcemFjgm-F[em6%F`hmQ/y~=~.392598e-1FcemFjgmF] ]n-F[em6%7$$!*S-#fNF[[n$\"+c,h_7F_[nFg]nFjgmFghmF_im79F'Fe[lFc^mF`_mFe _mF\\`mFe`m-F(6$7&F+F+7$FizF/F`]wFeam-Fabm6%7&7$$!\"&FjamF/Fd]w7$$!#DF jamF/Fg]wFhbmF\\cm-F(6&7#F`^mFccmFgcmF]dm-F(6&F\\^wFbdmFgcmF]dm-F(6&F \\^wFgdmFgcmF]dmFjdmFgemF\\fmFafmFifmF_gm-F[em6%FdgmQ'x~=~1.FcemFjgm-F [em6%F`hmQ'y~=~0.FcemFjgmFbhmFghmF_im-%(SCALINGG6#%.UNCONSTRAINEDG-%*A XESSTYLEG6#%%NONEG-%*AXESTICKSG6$F0F0F_im" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT 274 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 30 "The semi-circle has equation: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1-x^2)" "6 #/%\"yG-%%sqrtG6#,&\"\"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 13 " ------- ( i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "If we take a general point " }{XPPEDIT 18 0 "P(x,y);" "6#-%\"PG6$%\"xG% \"yG" }{TEXT -1 72 " on the section of the semi-circle in the 1st quad rant then the numbers " }{TEXT 276 1 "x" }{TEXT -1 5 " and " }{TEXT 277 1 "y" }{TEXT -1 23 " satisfy equation (i). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 300 300 {PLOTDATA 2 "63-%'CURVESG6$7S7$$ \"\"\"\"\"!$F*F*7$$\"3%y\\W\"HJcw**!#=$\"31q+d@uUUo!#>7$$\"3Mr#\\#p_6= **F/$\"3uZ%)4&*>5x7F/7$$\"3pyC-I#[.\")*F/$\"3a)>O?m:$Q>F/7$$\"3K)HHf&* )Rd'*F/$\"35)>yf`>^f#F/7$$\"3E'*R4Kvch%*F/$\"3#3seB<)3PKF/7$$\"3B7SL)R 3>C*F/$\"3/,zuO+J>QF/7$$\"3y%)o(eEjn(*)F/$\"3!=9zKNalS%F/7$$\"3'R3uzy\\F/$\"39]$RB!3Ys')F/7$$\"3%zeb@\"f+]WF/$\"3e%e+@v-`&*)F/7$$\"3Lb -rmU*[\"QF/$\"3$[ICT-KPC*F/7$$\"3k?>94^7PKF/$\"3[IP#[*[bh%*F/7$$\"3s3B (*3`5*e#F/$\"3;Ij+\"38!f'*F/7$$\"3'R#zu,%[4'>F/$\"3OB7VW$\\e!)*F/7$$\" 3a.xP3T(oH\"F/$\"3)pIb6G\\b\"**F/7$$\"3D3BW%>!*z\"oF2$\"3+nE@J/tw**F/7 $$\"3%[*=b#ec1b\"!#?$\"2!)e8t(z)*****!#<7$$!3s5SKN^;mnF2$\"3[4'*zXK3x* *F/7$$!3c?vEKjew7F/$\"3()pAta;==**F/7$$!3)>@&fjQH>>F/$\"3Ew$4\"yt39)*F /7$$!3Sr=G7zpuDF/$\"3-OkT\"[jGm*F/7$$!36S>V-7j/KF/$\"3cK5WG*4EZ*F/7$$! 38`xMXL%4!QF/$\"3e6Y_lsZ\\#*F/7$$!3L'*HrKB*[W%F/$\"3W\"y7g$>%y&*)F/7$$ !3s\"4]_i`Y+&F/$\"3!y!)Q%)flvl)F/7$$!31k#eVXs*zbF/$\"3=@)z#3!G%)H)F/7$ $!3uVZ0b**>zgF/$\"3CexilG)*RzF/7$$!3Uk1Y5)[')f'F/$\"3]i=%o\"G%Q^(F/7$$ !3G3'\\U0]-1(F/$\"354j1X!p=3(F/7$$!38IyW%*Qc7vF/$\"3cErLF[5+mF/7$$!3YE dsL13BzF/$\"3yjGuUC@,hF/7$$!3V/esK4R<$)F/$\"3S\"Q*\\Vrm^bF/7$$!3')Qloz n?h')F/$\"3e-i^R%\\$)*\\F/7$$!3Y!3Z7B%yu*)F/$\"3R<$R^j$e5WF/7$$!3))GHo [=VY#*F/$\"3d\"o>S%eM3QF/7$$!3;R1516\\g%*F/$\"3.bj`7FBSKF/7$$!3kb)4yk- Hm*F/$\"3#)3`(f\"4buDF/7$$!3(RbB'=gL/)*F/$\"3)>EF/7$$!3dIA3TjH8 **F/$\"3O)f2vk%)RJ\"F/7$$!3w_8!pw6n(**F/$\"34(R(G%y?2#oF27$$!\"\"F*$!3 0BmIq&o?5%!#F-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-F$6$7&7$$\"3a+++SSDg ')F/F+7$Ff[l$\"3++++++++]F/7$$!3a+++SSDg')F/Fi[l7$F\\\\lF+-%&COLORG6&F ^[l$\"\"*Fgz$\"#:!\"#$\"#&)Ff\\l-%)POLYGONSG6%7&7$$\"+SSDg')!#5F+7$F^] l$\"\"&Fgz7$$!+SSDg')F`]lFb]l7$Fe]lF+-F`\\l6&F^[lFb\\l$\"#vFf\\lFg\\l- %&STYLEG6#%,PATCHNOGRIDG-F$6&7$Fh[lF[\\l-%'SYMBOLG6#%&CROSSG-F`\\l6&F^ [l$\"\"'FgzF*F)-F]^l6#%&POINTG-F$6&Fb^l-Fd^l6#%'CIRCLEGFg^lF[_l-F$6&Fb ^l-Fd^l6#%(DIAMONDGFg^lF[_l-%%TEXTG6%7$$\"#7Fgz$!\"'Ff\\lQ\"x6\"-F`\\l 6&F^[l$F)Ff\\lFd`lFd`l-Fi_l6%7$F^`l$\"$D\"Ff\\lQ\"yFa`lFb`l-Fi_l6%7$$! \"%Ff\\l$\"$3\"Ff\\lQ\"1Fa`lFb`l-Fi_l6%7$F(F^`lFbalFb`l-Fi_l6%7$FfzF^` lQ#-1Fa`lFb`l-Fi_l6%7$F($\"#eFf\\lQ'P(x,y)Fa`l-F`\\l6&F^[lFb]lF*Fb\\l- Fi_l6%7$FfzF]blQ(Q(-x,y)Fa`lF`bl-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q!F a`lF\\cl-%%FONTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!#6 FgzF\\`l;$Ff\\lFgzFh`l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" }}}{PARA 257 "" 0 "" {TEXT -1 16 "We need to find " }{TEXT 278 1 "x" }{TEXT -1 5 " and " }{TEXT 279 1 "y" }{TEXT -1 51 " such tha t the area of the rectangle with vertices " }{XPPEDIT 18 0 "``(x,0),`` (x,y),``(-x,y);" "6%-%!G6$%\"xG\"\"!-F$6$F&%\"yG-F$6$,$F&!\"\"F*" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "``(-x,0);" "6#-%!G6$,$%\"xG!\"\"\" \"!" }{TEXT -1 19 " has maximum area. " }}{PARA 257 "" 0 "" {TEXT -1 9 "The area " }{TEXT 275 1 "A" }{TEXT -1 30 " of the rectangle is give n by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = 2*x*y;" "6#/%\"AG*(\"\"#\"\"\"%\"xGF'%\"yGF'" }{TEXT -1 15 " ------- (ii). " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The equ ation (i) provides a constraint for the problem which can be used with equation (ii) to express the area " }{TEXT 280 1 "A" }{TEXT -1 13 " i n terms of " }{TEXT 332 1 "x" }{TEXT -1 15 " only, namely: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A=2*x*sqrt(1-x^2)" "6#/%\"AG*(\"\"#\"\"\"%\"xGF'-%%sqrtG6#,&F'F' *$F(F&!\"\"F'" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "This equation is valid fo r " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 1;" "6 #1%!G\"\"\"" }{TEXT -1 7 ". Let " }{XPPEDIT 18 0 "f(x)=2*x*sqrt(1-x^2 )" "6#/-%\"fG6#%\"xG*(\"\"#\"\"\"F'F*-%%sqrtG6#,&F*F**$F'F)!\"\"F*" } {TEXT -1 33 ". The following graph shows how " }{TEXT 333 1 "A" } {TEXT -1 13 " varies for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f := x \+ -> 2*x*sqrt(1-x^2);\nplot(f(x),x=0..1,labels=[`x`,`A`]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*(\" \"#\"\"\"9$F/-%%sqrtG6#,&F/F/*$)F0F.F/!\"\"F/F/F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 371 216 216 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$\"\"!F)F(7 $$\"3emmm;arz@!#>$\"3=%pxF4&ReVF-7$$\"3[LL$e9ui2%F-$\"33FJ>MBxX\")F-7$ $\"3nmmm\"z_\"4iF-$\"3d7$)*fRM%R7!#=7$$\"3[mmmT&phN)F-$\"3-ozHW*)Ql;F: 7$$\"3CLLe*=)H\\5F:$\"3YLZ?\\8,(3#F:7$$\"3gmm\"z/3uC\"F:$\"3k()eQ#**H` Z#F:7$$\"3%)***\\7LRDX\"F:$\"37l@'ReoU(GF:7$$\"3]mm\"zR'ok;F:$\"3713\" GATukKhesF:7$$\"3A++D\"=lj;%F :$\"3K\\%)3H:1vvF:7$$\"31++vV&RY2aF:$\"3K_#*z(Rgt4*F:7$$\"39mm;zXu9cF:$\"3ep3!)ocL#H*F:7$$\"3l**** **\\y))GeF:$\"3M?vcE$[DZ*F:7$$\"3'*)***\\i_QQgF:$\"3*z!3GoUZE'*F:7$$\" 3@***\\7y%3TiF:$\"3;N%Gcq*y_(*F:7$$\"35****\\P![hY'F:$\"3aW)QpAr\\')*F :7$$\"3kKLL$Qx$omF:$\"3p9eEU*)eQ**F:7$$\"3!)*****\\P+V)oF:$\"3CCx`GSS' )**F:7$$\"3?mm\"zpe*zqF:$\"3G'zQ5Mo*****F:7$$\"3%)*****\\#\\'QH(F:$\"3 #3GxP,$\\z**F:7$$\"3GKLe9S8&\\(F:$\"3vl3'))3&RB**F:7$$\"3R***\\i?=bq(F :$\"3`L25:ckA)*F:7$$\"3\"HLL$3s?6zF:$\"3%\\1>>kQzn*F:7$$\"3a***\\7`Wl7 )F:$\"3EgN\"yR?9Z*F:7$$\"3#pmmm'*RRL)F:$\"3+YG4#y#*>@*F:7$$\"3Qmm;a<.Y &)F:$\"3a,yE?Yfv))F:7$$\"3=LLe9tOc()F:$\"3mm(oswI\"e%)F:7$$\"3u****** \\Qk\\*)F:$\"3'\\DGY3Xb)zF:7$$\"3CLL$3dg6<*F:$\"3g*)fIC9i6tF:7$$\"3y** *\\(oTAq#*F:$\"3o9(>3&yx_pF:7$$\"3ImmmmxGp$*F:$\"3w%e6\\F8&\\lF:7$$\"3 sK$eRA5\\Z*F:$\"3svy-W%=)fgF:7$$\"3A++D\"oK0e*F:$\"3%)HHQ5LP\"\\&F:7$$ \"3C+++]oi\"o*F:$\"3/6d8x\\/Z[F:7$$\"3A++v=5s#y*F:$\"3qh#oQ62k0%F:7$$ \"3;+D1k2/P)*F:$\"3^'[fl6/t`$F:7$$\"35+]P40O\"*)*F:$\"3O`J&Rr=\"3HF:7$ $\"3k]7.#Q?&=**F:$\"3`JChnN9FDF:7$$\"31+voa-oX**F:$\"3/*z\\qWk/2#F:7$$ \"3ACc,\">g#f**F:$\"3/jKIDU8'z\"F:7$$\"3[\\PMF,%G(**F:$\"3Qid\"ykO!p9F :7$$\"357y]&4I'z**F:$\"3OKeUDcIt7F:7$$\"3uu=nj+U')**F:$\"3r0V;9Q`S5F:7 $$\"3OPf$=.5K***F:$\"3Oj.B_$QRO(F-7$$\"\"\"F)F(-%'COLOURG6&%$RGBG$\"#5 !\"\"F(F(-%+AXESLABELSG6$%\"xG%\"AG-%%VIEWG6$;F(F^^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The maximum value of " }{TEXT 297 1 "A" }{TEXT -1 14 " occurs where " }{TEXT 299 1 "x" }{TEXT -1 1 " " }{TEXT 298 1 "~" }{TEXT -1 6 " 0.7. " }}{PARA 0 "" 0 "" {TEXT -1 24 "To locate this value of " }{TEXT 300 1 "x" }{TEXT -1 171 " more pr ecisely we note that the maximum point occurs where the tangent line t o the graph is horizontal, that is, where the gradient, as given by th e derivative, is zero. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 308 133 133 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$\"3%************ **>'!#=$\"39$y#)>Wi!H(*F*7$$\"3rmmmYa([B'F*$\"3iK!eIFm#\\(*F*7$$\"3MLL L'Q?_E'F*$\"3Ef9Sh>Fm(*F*7$$\"3#pmmYWY$*H'F*$\"3W.,w#[^Zy*F*7$$\"3immm 7()pLjF*$\"3i#zbBabE!)*F*7$$\"37LLL5x)yO'F*$\"3M0\"[zjq(>)*F*7$$\"3-nm m(G&e*R'F*$\"31-vm5O+N)*F*7$$\"3L+++$H1CV'F*$\"35RC8s\\7])*F*7$$\"35mm mB)\\jY'F*$\"3%*\\?1iy0l)*F*7$$\"3,+++(\\%=+lF*$\"3)>uE3&*>#z)*F*7$$\" 3ALLLomF*$\"3c(\\WiGN&Q**F*7$$\"31nmmV4_)p'F*$\"3WM]#=sIs%**F* 7$$\"3eLLLX$zXt'F*$\"3!)oka5iuc**F*7$$\"3KLLLTb7lnF*$\"3MGHQzw4k**F*7$ $\"3g******G!e1!oF*$\"3kg[FN&>=(**F*7$$\"3(RLL8I5@$oF*$\"391DbS1!z(**F *7$$\"3')******G%=m'oF*$\"3Wx$[g!pu$)**F*7$$\"3S+++F$y%**oF*$\"3#)fK[j -]))**F*7$$\"3+LLL$=kP$pF*$\"3aVG5l?g#***F*7$$\"3'RLLBI\\_'pF*$\"3O(p$ =\"y'e&***F*7$$\"3'pmmmD5#**pF*$\"3u6BPB`&z***F*7$$\"3DnmmVh[MqF*$\"3+ e18kuY****F*7$$\"3G+++2R>lqF*$\"2)z#\\.@')*****!#<7$$\"3GnmmK\"f$)4(F* $\"3%*4=2=4q****F*7$$\"3x*****f0AE8(F*$\"3+5lD\"4r%)***F*7$$\"39+++U;9 mrF*$\"3Ipu1Q\\L'***F*7$$\"3Y+++lNd)>(F*$\"3%fR'QFpP$***F*7$$\"38+++'o $eMsF*$\"3!pm>:G]!*)**F*7$$\"3ILLL\"QSpE(F*$\"3I84pkI@%)**F*7$$\"3%)** ****f!)[,tF*$\"3Ua=X13/y**F*7$$\"3bmmm\"R$zKtF*$\"33Pr)o\"e`r**F*7$$\" 3k+++)Q=qO(F*$\"3CbG.5nTj**F*7$$\"3cLLLU9A*R(F*$\"31srEqj(F*$\"3A7]NN40j)*F*7$$\"3!QLL8p&QnwF*$\"3Wsw%[x,V%)*F*7$$\"3S nmmUg3*p(F*$\"3??G0$o2k#)*F*7$$\"3`+++H_)Gt(F*$\"3M<)>FKQh!)*F*7$$\"3Q +++j`BlxF*$\"3/V.9dqc&y*F*7$$\"3E+++++++yF*$\"3i(4HmTg@w*F*-%'COLOURG6 &%$RGBG$\"*++++\"!\")$\"\"!F`[lF_[l-F$6$7$7$$\"3!*******4y1rnF*$\"\"\" F`[l7$$\"3W+++5y1rtF*Fg[l-%&COLORG6&F[[lF`[l$\"\"(!\"\"F`[l-F$6%7$7$$ \"3-+++++++jF*$\"3-+++!pevz*F*7$$\"3S+++++++nF*$\"3w******42tg**F*-Fiz 6&F[[lF\\[lF_[lF\\[l-%*THICKNESSG6#\"\"#-F$6%7$7$$\"3!**************R( F*$\"33+++(p\\V(**F*7$Fdz$\"3a+++.~06\"F_]l-Fc^l6%7$$\"#rF[_l$\"& N+\"!\"%Q*f'(x)~=~0F]_lF\\\\l-Fc^l6%7$$\"$w(Fh^lFi^lQ*f'(x)~<~0F]_lF`^ l-%+AXESLABELSG6%Q\"xF]_lQ!F]_l-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NO NEG-%%VIEWG6$;$\"#iF[_l$\"#yF[_lFe`l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 43.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}{PARA 0 "" 0 "" {TEXT -1 27 "We can find th e derivative " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*& %#dAG\"\"\"%#dxG!\"\"" }{TEXT -1 46 " by using the product rule and th e chain rule." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = \+ 2*x*(1-x^2)^(1/2);" "6#/%\"AG*(\"\"#\"\"\"%\"xGF'),&F'F'*$F(F&!\"\"*&F 'F'F&F,F'" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 4 "so " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dA/dx = 2*`.`*(1-x^2) ^(1/2)+2*x*`.`" "6#/*&%#dAG\"\"\"%#dxG!\"\",&*(\"\"#F&%\".GF&),&F&F&*$ %\"xGF+F(*&F&F&F+F(F&F&*(F+F&F0F&F,F&F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff([(1-x^2)^(1/2)],x)" "6#-%%DiffG6$7#),&\"\"\"F)*$%\"xG\"\"#!\" \"*&F)F)F,F-F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=2*`.`*(1-x^2)^(1/2 )+2*x*`.`" "6#/%!G,&*(\"\"#\"\"\"%\".GF(),&F(F(*$%\"xGF'!\"\"*&F(F(F'F .F(F(*(F'F(F-F(F)F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\" \"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-x^2)^(-1/2)*`.`*(-2 *x)" "6#*(),&\"\"\"F&*$%\"xG\"\"#!\"\",$*&F&F&F)F*F*F&%\".GF&,$*&F)F&F (F&F*F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*sqrt(1-x^2)-2*x^2/sqrt(1- x^2);" "6#/%!G,&*&\"\"#\"\"\"-%%sqrtG6#,&F(F(*$%\"xGF'!\"\"F(F(*(F'F(* $F.F'F(-F*6#,&F(F(*$F.F'F/F/F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ (2*(1-x^2)-2*x^2)/sqrt(1-x^2);" "6#/%!G*&,&*&\"\"#\"\"\",&F)F)*$%\"xGF (!\"\"F)F)*&F(F)*$F,F(F)F-F)-%%sqrtG6#,&F)F)*$F,F(F-F-" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=(2-4*x^2)/sqrt(1-x^2)" "6#/%!G*&,&\"\"#\"\"\"*&\"\"% F(*$%\"xGF'F(!\"\"F(-%%sqrtG6#,&F(F(*$F,F'F-F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 6 " w hen " }{XPPEDIT 18 0 "2-4*x^2=0" "6#/,&\"\"#\"\"\"*&\"\"%F&*$%\"xGF%F& !\"\"\"\"!" }{TEXT -1 16 ", that is, when " }{XPPEDIT 18 0 "x^2=1/2" " 6#/*$%\"xG\"\"#*&\"\"\"F(F&!\"\"" }{TEXT -1 14 ", which gives " } {XPPEDIT 18 0 "x=``" "6#/%\"xG%!G" }{TEXT 354 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1/sqrt(2) = ``;" "6#/*&\"\"\"F%-%%sqrtG6#\"\"#!\"\"%!G " }{TEXT 357 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2)/2" "6#*&-%% sqrtG6#\"\"#\"\"\"F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "It is the positive value " }{XPPEDIT 18 0 "x[max] = sqrt(2)/2; " "6#/&%\"xG6#%$maxG*&-%%sqrtG6#\"\"#\"\"\"F,!\"\"" }{TEXT -1 1 " " } {TEXT 356 1 "~" }{TEXT -1 46 " 0.70711 which is applicable to this pro blem. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Although it is obvious from the graph of " }{XPPEDIT 18 0 "A=f(x)" "6 #/%\"AG-%\"fG6#%\"xG" }{TEXT -1 20 " that this value of " }{TEXT 380 1 "x" }{TEXT -1 114 " gives a maximum point on the graph, we can verif y this independently by investigating the sign of the derivative " } {XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "0 < x;" "6#2\"\"!%\"xG" }{XPPEDIT 18 0 "`` < 1;" "6#2%! G\"\"\"" }{TEXT -1 9 ". Since " }{XPPEDIT 18 0 "sqrt(1-x^2)" "6#-%%sq rtG6#,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 24 " is always positive fo r " }{XPPEDIT 18 0 "0 < x;" "6#2\"\"!%\"xG" }{XPPEDIT 18 0 "`` < 1;" " 6#2%!G\"\"\"" }{TEXT -1 15 ", the sign of " }{XPPEDIT 18 0 "`f '`(x) \+ = (2-4*x^2)/sqrt(1-x^2);" "6#/-%$f~'G6#%\"xG*&,&\"\"#\"\"\"*&\"\"%F+*$ )F'F*F+F+!\"\"F+-%%sqrtG6#,&F+F+*$F/F+F0F0" }{TEXT -1 45 " is determi ned by the sign of the numerator " }{XPPEDIT 18 0 "2-4*x^2" "6#,&\"\"# \"\"\"*&\"\"%F%*$%\"xGF$F%!\"\"" }{TEXT -1 19 ". Thus we see that " } {XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 14 " positive fo r " }{XPPEDIT 18 0 "0\"!\"*$!*1wbV$FjoFco7$$!+C.*p5\"Fjo$!*%RUkYFjo-%&STYLEG 6#%,PATCHNOGRIDGF0-%*THICKNESSG6#\"\"#-F$6'7$7$FAFdo7$$FipF4$!#5Fbo7%7 $$\"+w'4I*=Fjo$!+ggdN$)FaqF^q7$$\"+C.*p!=Fjo$!+SRUk&*FaqFbpF0Ffp-%%TEX TG6&7$$FeoF4$\"#vF`oQ\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-F^r6&7$$!#:Fbo FbrQ*~x~<~xmaxFerF0Ffr-F^r6&7$$F4F4FbrQ%xmaxFerF0Ffr-F^r6&7$$\"#:FboFb rQ)x~>~xmaxFerF0Ffr-F^r6&7$$\"\"$F4FbrQ\"3FerF0Ffr-F^r6&7$Far$FipFboQ' f~'(x)FerF0Ffr-F^r6&7$FdsFetQ\"0FerF0Ffr-F^r6&7$F_tFetFjtF0Ffr-F^r6&7$ F^s$F`tFboQ\"+FerF0-Fgr6$Fir\"#9-F^r6&7$Fis$\"\"%FboQ\"_FerF0Fcu-F^r6& 7$$\"#XFboFetFbuF0Fcu-%+AXESLABELSG6$Q!FerFdv-%*AXESSTYLEG6#%%NONEG-%% VIEWG6$;$!#NFbo$\"#DFbo;$!#8Fbo$\"#6Fbo" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" }}{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The maxim um area " }{XPPEDIT 18 0 "A[max]" "6#&%\"AG6#%$maxG" }{TEXT -1 31 " i s obtained by substituting " }{XPPEDIT 18 0 "x=1/sqrt(2)" "6#/%\"xG*& \"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 16 " in the formula " } {XPPEDIT 18 0 "A=2*x*sqrt(1-x^2)" "6#/%\"AG*(\"\"#\"\"\"%\"xGF'-%%sqrt G6#,&F'F'*$F(F&!\"\"F'" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 " A[max] = 2/sqrt(2);" "6#/&%\"AG6#%$maxG*&\"\"#\"\"\"-%%sqrtG6#F)!\"\" " }{TEXT -1 2 " " }{TEXT 382 1 "." }{TEXT -1 2 " " }{XPPEDIT 18 0 "1 /sqrt(2) = 1;" "6#/*&\"\"\"F%-%%sqrtG6#\"\"#!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "The maximum area is therefore " }{TEXT 301 13 "1 square unit" }{TEXT -1 36 ", and it is obtained when the poi nt " }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "``(1/sqrt(2),1/sqrt(2))" "6#-%!G6$*&\"\"\"F'-%%sqrtG6# \"\"#!\"\"*&F'F'-F)6#F+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "Note that the maximum occurs when the line joinining " } {XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 33 " to the ori gin makes an angle of " }{XPPEDIT 18 0 "45^o" "6#)\"#X%\"oG" }{TEXT -1 36 " with the positive direction of the " }{TEXT 334 1 "x" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f := x -> 2*x*sqrt (1-x^2);\nDiff(f(x),x);\nvalue(%);\nnormal(%);\ndf := unapply(%,x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,$*(\"\"#\"\"\"9$F/-%%sqrtG6#,&F/F/*$)F0F.F/!\"\"F/F/F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,$*(\"\"#\"\"\"%\"xGF),&F)F) *$)F*F(F)!\"\"#F)F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\" \"\",&F&F&*$)%\"xGF%F&!\"\"#F&F%F&*(F%F&F*F%F'#F+F%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\",&F&!\"\"*&F%F&)%\"xGF%F&F&F&,&F&F& *$F*F&F(#F(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGf*6#%\"xG6\"6 $%)operatorG%&arrowGF(,$*(\"\"#\"\"\",&F/!\"\"*&F.F/)9$F.F/F/F/,&F/F/* $F3F/F1#F1F.F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(df(x)=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&\"\"#!\"\"F%#\"\"\"F%F&,$*&F%F&F%F'F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f(1/s qrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Example 3 - printed page with mar gins" }}{PARA 0 "" 0 "" {TEXT 263 8 "Question" }{TEXT -1 3 ": " }} {PARA 0 "" 0 "" {TEXT -1 226 "A printed page must contain 60 square cm . of printed material. There are to be margins of 5 cm on each side an d margins of 3 cm at the top and bottom. Find the outside dimensions o f the page needed to minimise its total area. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 264 8 "Solution " }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 37 "Let the overall wid th of the page be " }{TEXT 265 1 "x" }{TEXT -1 31 " cm. and the overal l height be " }{TEXT 266 1 "y" }{TEXT -1 5 " cm. " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{GLPLOT2D 359 369 369 {PLOTDATA 2 "6>-%'CURVESG6$7'7 $$\"\"!F)F(7$F($\"#@F)7$$\"#=F)F+7$F.F(F'-%'COLOURG6&%$RGBG$\")#)eqk! \")$\"))eqk\"F7F8-F$6$7'7$$\"\"&F)$\"\"$F)7$F>F.7$$\"#8F)F.7$FDF@F=-F2 6&F4$\")!\\DP\"F7FI$\")viobF7-%)POLYGONSG6$7&F=FBFCFF-%&COLORG6&F4$\" \")!\"\"FT$\"$&o!\"$-F$6%7$F07$F+F(-FR6&F4F)$\"\"%FVF)-%*LINESTYLEG6# \"\"#-F$6%7$F07$F.$FYF)FhnF\\o-F$6%7$F-7$F+F+FhnF\\o-F$6%7$F'7$F(FdoFh nF\\o-F$6&7$7$$\"#?F)$\"$:\"FV7$Fap$\"$5#FV7%7$$\"++++v>F7$\"++++C?F7F ep7$$\"++++D?F7F\\q-%&STYLEG6#%,PATCHNOGRIDGFhn-F$6&7$7$Fap$\"#&*FV7$F apF(7%7$F_q$\")+++wF7F[r7$FjpF^rFaqFhn-F$6&7$7$$\"#5F)$!\"#F)7$F.Fgr7% 7$$\"++++OFct7 %7$$\"+++++YF`s$\"++++v5F7Fet7$Fht$\"++++D5F7FaqF1-F$6&7$7$$F_oF)Fct7$ F(Fct7%7$$\"*++++%F`sF]uFdu7$FguFjtFaqF1-F$6&7$7$$\"#;F)Fct7$F.Fct7%7$ $\"++++gFVQ\"3Fc[lF1-F[[l6%7$Ffw$F[wFVF^\\lF1-F [[l6%7$Fap$\"$2\"FVQ\"yFc[lFhn-F[[l6%7$Ffw$!#=FVQ\"xFc[lFhn-%*AXESSTYL EG6#%%NONEG-%+AXESLABELSG6%Q!Fc[lFf]l-%%FONTG6#%(DEFAULTG-%%VIEWG6$Fj] lFj]l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve \+ 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "C urve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" }}}{PARA 0 "" 0 "" {TEXT -1 31 "The total area of the page is " }{XPPEDIT 18 0 "x*y;" "6 #*&%\"xG\"\"\"%\"yGF%" }{TEXT -1 67 " square cm. and the width and le ngth of the printed rectangle are " }{XPPEDIT 18 0 "``(x - 10)" "6#-%! G6#,&%\"xG\"\"\"\"#5!\"\"" }{TEXT -1 9 " cm. and " }{XPPEDIT 18 0 "``( y - 6)" "6#-%!G6#,&%\"yG\"\"\"\"\"'!\"\"" }{TEXT -1 19 " cm. respectiv ely. " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{GLPLOT2D 330 352 352 {PLOTDATA 2 "6D-%'CURVESG6$7'7$$\"\"!F)F(7$F($\"#@F)7$$\"#=F)F+7$F.F(F '-%'COLOURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"F7F8-F$6$7'7$$\"\"&F)$\"\"$F )7$F>F.7$$\"#8F)F.7$FDF@F=-F26&F4$\")!\\DP\"F7FI$\")viobF7-%)POLYGONSG 6$7&F=FBFCFF-%&COLORG6&F4$\"\")!\"\"FT$\"$&o!\"$-F$6%7$F07$F+F(-FR6&F4 F)$\"\"%FVF)-%*LINESTYLEG6#\"\"#-F$6%7$F07$F.$FYF)FhnF\\o-F$6%7$F-7$F+ F+FhnF\\o-F$6%7$F'7$F(FdoFhnF\\o-F$6&7$7$$\"#?F)$\"$:\"FV7$Fap$\"$5#FV 7%7$$\"++++v>F7$\"++++C?F7Fep7$$\"++++D?F7F\\q-%&STYLEG6#%,PATCHNOGRID GFhn-F$6&7$7$Fap$\"#&*FV7$FapF(7%7$F_q$\")+++wF7F[r7$FjpF^rFaqFhn-F$6& 7$7$$\"#5F)$!\"#F)7$F.Fgr7%7$$\"++++OFct7%7$$\"+++++YF`s$\"++++v5F7Fet7$Fht$\"++++D5 F7FaqF1-F$6&7$7$$F_oF)Fct7$F(Fct7%7$$\"*++++%F`sF]uFdu7$FguFjtFaqF1-F$ 6&7$7$$\"#;F)Fct7$F.Fct7%7$$\"++++gFVQ\"3F]_lF1-Fe^l6%7$Ffw$F[wFVFh_lF1-Fe^l6%7$F ap$\"$2\"FVQ\"yF]_lFhn-Fe^l6%7$Ffw$!#=FVQ\"xF]_lFhn-Fe^l6%7$FfwFDQ'x~- ~10F]_lF\\\\l-Fe^l6%7$F`]lFctQ&y~-~6F]_lF\\\\l-%*AXESSTYLEG6#%%NONEG-% +AXESLABELSG6%Q!F]_lFhal-%%FONTG6#%(DEFAULTG-%%VIEWG6$F\\blF\\bl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9 " "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "C urve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28 " "Curve 29" "Curve 30" "Curve 31" }}}{PARA 0 "" 0 "" {TEXT -1 53 "Thi s means that the area of the printed rectangle is " }{XPPEDIT 18 0 "(x -10)*(y-6)" "6#*&,&%\"xG\"\"\"\"#5!\"\"F&,&%\"yGF&\"\"'F(F&" }{TEXT -1 20 " square cm. so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(x-10)*(y-6) = 60;" "6#/*&,&%\"xG\"\"\"\"#5!\"\"F',&%\" yGF'\"\"'F)F'\"#g" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 28 "This equation constitutes a " }{TEXT 259 10 "constraint" }{TEXT -1 18 " for the problem. " }}{PARA 0 "" 0 "" {TEXT -1 37 "The q uantity to minimise is the area " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "A=x*y" "6#/%\"AG*&%\"xG\"\"\"%\"yGF'" }{TEXT -1 16 " - ------ (ii). " }}{PARA 0 "" 0 "" {TEXT -1 24 "We can eliminate either \+ " }{TEXT 267 1 "x" }{TEXT -1 4 " or " }{TEXT 268 1 "y" }{TEXT -1 43 " \+ from equation (ii) by using equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "isolate((x-10)*(y- 6)=60,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*&\"\"\"F',&%\"xG F'\"#5!\"\"F+\"#g\"\"'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 23 "Thus we can substitute " }{XPPEDIT 18 0 "y = 60/(x -10)+6" "6#/%\"yG,&*&\"#g\"\"\",&%\"xGF(\"#5!\"\"F,F(\"\"'F(" }{TEXT -1 25 " in equation (ii) to get " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "A = x*(60/(x-10)+6)" "6#/%\"AG*&%\"xG\"\"\",&*&\"#gF',& F&F'\"#5!\"\"F-F'\"\"'F'F'" }{XPPEDIT 18 0 "`` = x*(60+6*x-60)/(x-10) " "6#/%!G*(%\"xG\"\"\",(\"#gF'*&\"\"'F'F&F'F'F)!\"\"F',&F&F'\"#5F,F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 256 " " 0 "" {TEXT -1 5 " A = " }{XPPEDIT 18 0 "6*x^2/(x-10)" "6#*(\"\"'\"\" \"*$%\"xG\"\"#F%,&F'F%\"#5!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Maple can do this. We als o construct a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 43 " to give the area in terms of the variable " }{TEXT 269 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "isolate((x-10)*(y-6)=60,y);\nsubs(%,x*y); \nnormal(%);\nf := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"yG,&*&\"#g\"\"\",&%\"xGF(\"#5!\"\"F,F(\"\"'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&*&\"#gF%,&F$F%\"#5!\"\"F+F%\"\"'F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"'\"\"\"%\"xG\"\"#,&F'F&\"#5!\" \"F+F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operat orG%&arrowGF(,$*(\"\"'\"\"\"9$\"\"#,&F0F/\"#5!\"\"F4F/F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Note that the appropriate inter val to look at is " }{XPPEDIT 18 0 "10 < x " "6#2\"#5%\"xG" }{TEXT -1 71 ", since the page cannot be narrower than 10 cm. (Each margin is 5 cm.)" }}{PARA 0 "" 0 "" {TEXT -1 10 "The graph " }{XPPEDIT 18 0 "A = \+ f(x);" "6#/%\"AG-%\"fG6#%\"xG" }{TEXT -1 25 " indicates that the area \+ " }{TEXT 352 1 "A" }{TEXT -1 19 " is a minimum when " }{TEXT 271 1 "x " }{TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT -1 4 " 20." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot(f(x), x=10..50,A=0..500);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7gp7$$\"33+++WYs-5!#;$\"3cJ#)Q+*QT@#!#87$$ \"3(******zG\\a+\"F*$\"3%=:!*>(R486F-7$$\"30+++KR<35F*$\"3ElULW*)*3Y(! #97$$\"3.+++x&)*3,\"F*$\"3)4z!z!zffi&F87$$\"3,+++AKi85F*$\"3t7&*y^=1DX F87$$\"3\"******f'yM;5F*$\"3E@R:xY=\"z$F87$$\"3********4D2>5F*$\"3!zph 9QLqE$F87$$\"31+++arz@5F*$\"3U&RDtwgR(GF87$$\"3'******zz@X-\"F*$\"3y_m XWRFoDF87$$\"3/+++UkCF5F*$\"3^5&*RxrvBBF87$$\"3%******f3r*H5F*$\"3\\zJ xCxsB@F87$$\"3#******4t&pK5F*$\"3)pg$e-P1d>F87$$\"33+++w.UN5F*$\"39%4$ Gca1;=F87$$\"3)*******>]9Q5F*$\"3!)[3j;KB&p\"F87$$\"30+++k'p3/\"F*$\"3 q:'QytL0f\"F87$$\"3&******zI%fV5F*$\"3C.F_F@%*)\\\"F87$$\"3/+++_*=j/\" F*$\"3_KBiK_9=9F87$$\"3#******ffV!\\5F*$\"3YE$[S$RMY8F87$$\"3++++S#o<0 \"F*$\"3%Qt&fuy6#G\"F87$$\"3)******\\)G\\a5F*$\"3'=;M$[2LC7F87$$\"3'** *****Hv@d5F*$\"39**QvGF1s6F87$$\"3/+++u@%*f5F*$\"3_3*[oAhX7\"F87$$\"3% ******z\"omi5F*$\"3SOSY0X?\"3\"F87$$\"3-+++i9Rl5F*$\"3(3!=3nWZT5F87$$ \"35+++1h6o5F*$\"39B-y4f$\\+\"F87$$\"33+++^2%32\"F*$\"3Z7_xQd?7(*!#:7$ $\"31+++'RlN2\"F*$\"3u)*QX/y9+%*Ffs7$$\"3'*******R+Hw5F*$\"3WWU:,m\\5 \"*Ffs7$$\"3#******zKR<3\"F*$\"3m$4$z'Q^%*e)Ffs7$$\"3)******ph)=(3\"F* $\"3@.a[Oa%R8)Ffs7$$\"33+++9+$>4\"F*$\"3'pWdh6i=y(Ffs7$$\"33+++59n'4\" F*$\"3]s_nY[fkuFfs7$$\"3*******p!GT,6F*$\"3u*46T>gs<(Ffs7$$\"34+++/U:1 6F*$\"3JFv%>QZe\"pFfs7$$\"3\"******z*pj:6F*$\"3#)H!G*zH.ekFfs7$$\"3#** ****>z>^7\"F*$\"3gEs`miZqgFfs7$$\"3$******fe-Y8\"F*$\"3JaO!Q@G$QdFfs7$ $\"3.+++z`3W6F*$\"3kj3y+*[1X&Ffs7$$\"3'******> &Ffs7$$\"3'******f'40j6F*$\"3C'3(4%ehw(\\Ffs7$$\"33+++_(zV=\"F*$\"3!eX >#o7ykXFfs7$$\"3#*******Q&3d?\"F*$\"3`s9kqKlf*yRFfs7$$\"33+++7hO[7F*$\"3!>]P'y\"3[w$Ffs7$$\"3-+++ZkI\"H \"F*$\"34z$*Q=0ZMMFfs7$$\"3'******>yYUL\"F*$\"3owGbE*Gc>$Ffs7$$\"3)*** ****GI)pP\"F*$\"3L(*4L+Mx(>9F*$\"33GSMw%e8)GFfs7$$\" 3++++>K'*)\\\"F*$\"3k/A6w8(=q#Ffs7$$\"3-+++Kd,\"e\"F*$\"3!HqKXr$G\"e#F fs7$$\"3))******eX(em\"F*$\"3p&R;GN&f+DFfs7$$\"33+++U7Y]c0T\">F*$\"3) ec47nU[S#Ffs7$$\"3#*******H,Q+?F*$\"3dXjm'3++S#Ffs7$$\"3!*******\\*3q3 #F*$\"3Mwh6Y(yTS#Ffs7$$\"3/+++q=\\q@F*$\"3U&)ozM,!\\T#Ffs7$$\"3))***** *eBIYAF*$\"3s!f7A_0#HCFfs7$$\"3))*****HO[kL#F*$\"3_\">Qd9?3X#Ffs7$$\"3 6+++`Q\"GT#F*$\"3!)*yT$4FPsCFfs7$$\"3<+++s]k,DF*$\"3B\"\\>z$)[0]#Ffs7$ $\"35+++`dF!e#F*$\"3zX.lAg%y_#Ffs7$$\"3))*****>2Ylm#F*$\"3#*=H`$p`*fDF fs7$$\"37+++\"RV'\\$F*$\"3 '*Rxn!=.#QHFfs7$$\"3++++:#fke$F*$\"3)y:hl*G&Q)HFfs7$$\"3q*****H&4NnOF* $\"3]wC&*\\GNDIFfs7$$\"3?+++],s`PF*$\"39:D_a\">,2$Ffs7$$\"3;+++zM)>$QF *$\"3?)QP,tb56$Ffs7$$\"3w******pfa$Ffs7$$\"3E+++#G2A3%F*$\"35[,%o0!*RC$Ffs7$$\"3;++ +$)G[kTF*$\"3[!*\\CoTH)G$Ffs7$$\"39+++7yh]UF*$\"3?:$RQ)4&\\L$Ffs7$$\"3 p*****p)fdLVF*$\"3M*zb)*[K,Q$Ffs7$$\"3$)*****>q7%=WF*$\"3!><^/onlU$Ffs 7$$\"38+++Epa-XF*$\"3=nWMq?$GZ$Ffs7$$\"3s******Rv&)zXF*$\"3-Kg*)e*=b^$ Ffs7$$\"3')*****zAk%oYF*$\"3#z')**zrMYc$Ffs7$$\"3!)*****p5:xu%F*$\"35* e/ChE(3OFfs7$$\"3q*****>28A$[F*$\"3=Z=%zL&*el$Ffs7$$\"30+++2%)38\\F*$ \"36%e7qh%=,PFfs7$$\"#]\"\"!$\"$v$Fjgl-%'COLOURG6&%$RGBG$\"#5!\"\"$Fjg lFjglFdhl-%+AXESLABELSG6$Q\"x6\"Q\"AFihl-%%VIEWG6$;$FbhlFjglFhgl;Fdhl$ \"$+&Fjgl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "To find this value \+ more precisely we note that the minimum point occurs where the tangent line to the graph is horizontal, that is, where the gradient, as give n by the derivative, is zero. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 305 153 153 {PLOTDATA 2 "6,-%'CURVESG6$7Z7$$\"32++++++I8!#;$ \"3o<=====;K!#:7$$\"3z;a8c\\#=M\"F*$\"3ImraN$z.;$F-7$$\"3_L3F7*\\ON\"F *$\"3n6vgWIy3JF-7$$\"31]iSo[Zl8F*$\"3mKqyqY)41$F-7$$\"3ym;aC)*Hx8F*$\" 3?PRXAri;IF-7$$\"3+DJv'[xyR\"F*$\"3+MY*pPGn%HF-7$$\"3S$ek*[^X=9F*$\"3G VF?7&=\\)GF-7$$\"3&\\(=F#)ofT9F*$\"3o2\\4uLmBGF-7$$\"3om\"zbhQZY\"F*$ \"37rVBq9*)pFF-7$$\"3emmJZP.)[\"F*$\"34*4m^UVAs#F-7$$\"3mmT0z)G8^\"F*$ \"31?U#F-7$$\"3-+]_?iqs=F*$\"3'3p1oJS6T#F-7$$\"3.+]PbBq>>F*$\"3l([q -R1US#F-7$$\"3I+]ZR=*\\'>F*$\"3!Qi*o>?w+CF-7$$\"3%o\"z%)H!>h+#F*$\"38c Y))GB-+CF-7$$\"3CL$3rBB]0#F*$\"3bZhA&z@/ZRim4T#F-7$$\"3ELL='f*H(=#F*$\"3vlzw)> GxT#F-7$$\"31]7LW75MAF*$\"3q&G?-_WmU#F-7$$\"3;]P*4[n'yAF*$\"3[v`;!)*Qk V#F-7$$\"3fLektw;DBF*$\"3YL9?'\\tyW#F-7$$\"3_$e*Gm$pyO#F*$\"3eNwm(**f$ fCF-7$$\"3fmTg5*GRT#F*$\"3G6pzFoqsCF-7$$\"3q;zMK=xhCF*$\"3#[gq\\(Q_([# F-7$$\"3B](ojB>M]#F*$\"3j(yg\">=9,DF-7$$\"3'p;zOd*R[DF*$\"3)QAdgH.k#F*$\"3_b')Q1z( *\\DF-7$$\"3-]7`(R:Vo#F*$\"3#G$*zr0tYIF*$\"3l]6)[V*=@FF-7$$\"3#)\\7Gj,Y$4$F*$\"3-fG)p$ HoUFF-7$$\"3'pmmF(\\YQJF*$\"3UjJjsSljFF-7$$\"3%p;a1*))[%=$F*$\"3)=oW`7 d`y#F-7$$\"3q$eksqJ,B$F*$\"3d#Rg-V@r!GF-7$$\"3q***\\ars?F$F*$\"3UVwa%o >t#GF-7$$\"3/L3(Q%=9?LF*$\"3gj*y\\)**o]GF-7$$\"3qmmOXa8jLF*$\"3]HLJH7y rGF-7$$\"3a]7$=fv*3MF*$\"33//<\")QX%*GF-7$$\"3e](o5Y]GX$F*$\"3AABbROK; HF-7$$\"#N\"\"!$\"$%HFi\\l-%'COLOURG6&%$RGBG$\"*++++\"!\")$Fi\\lFi\\lF c]l-F$6$7$7$$\"#;Fi\\l$\"$S#Fi\\l7$$\"#CFi\\lFj]l-%&COLORG6&F_]lFi\\l$ \"\"(!\"\"Fi\\l-F$6$7$7$$\"#8Fi\\l$\"$1$Fi\\l7$$\"#~06\"Fb_l-Ff`l6%7$$\"#?Fi\\lF`_lQ*f'(x)~=~0F^al F_^l-Ff`l6%7$$\"#IFi\\l$\"$a#Fi\\lQ*f'(x)~<~0F^alF_`l-%+AXESLABELSG6%Q \"xF^alQ!F^al-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$\"$ L\"Fd^lFg\\lFebl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 179 "To find this val ue more precisely we note that the minimum point occurs where the tang ent line to the graph is horizontal, that is, where the gradient, as g iven by the derivative " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~' G6#%\"xG*&%#dAG\"\"\"%#dxG!\"\"" }{TEXT -1 10 ", is zero." }}{PARA 0 " " 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "A=6*x^2/(x-10)" "6#/%\"AG*( \"\"'\"\"\"*$%\"xG\"\"#F',&F)F'\"#5!\"\"F-" }{TEXT -1 14 ", we can fin d " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*&%#dAG\"\"\" %#dxG!\"\"" }{TEXT -1 26 " using the quotient rule. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#% \"xG*&%#dAG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (12* x*`.`*(x-10)-6*x^2*`.`*1)/((x-10)^2);" "6#/%!G*&,&**\"#7\"\"\"%\"xGF)% \".GF),&F*F)\"#5!\"\"F)F)**\"\"'F)*$F*\"\"#F)F+F)F)F)F.F)*$,&F*F)F-F.F 2F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(12*x^2-120*x-6*x^2)/(x-10)^2" "6#/%!G*&,(*&\"#7\"\"\"*$%\"xG\" \"#F)F)*&\"$?\"F)F+F)!\"\"*&\"\"'F)*$F+F,F)F/F)*$,&F+F)\"#5F/F,F/" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=( 6*x^2-120*x)/(x-10)^2" "6#/%!G*&,&*&\"\"'\"\"\"*$%\"xG\"\"#F)F)*&\"$? \"F)F+F)!\"\"F)*$,&F+F)\"#5F/F,F/" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(6 *x*(x-20))/(x-10)^2" "6#/%!G**\"\"'\"\"\"%\"xGF',&F(F'\"#?!\"\"F'*$,&F (F'\"#5F+\"\"#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "For \+ " }{XPPEDIT 18 0 "x > 10" "6#2\"#5%\"xG" }{TEXT -1 17 ", the derivativ e " }{XPPEDIT 18 0 "`f '`(x) = 6*x*(x-20)/((x-10)^2);" "6#/-%$f~'G6#% \"xG**\"\"'\"\"\"F'F*,&F'F*\"#?!\"\"F**$),&F'F*\"#5F-\"\"#F*F-" } {TEXT -1 15 " is zero when " }{XPPEDIT 18 0 "x=20" "6#/%\"xG\"#?" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 118 "Although it is already obvious from the graph drawn above that we do indeed have a minimum p oint on the graph of A = " }{XPPEDIT 18 0 "6*x^2/(x-10)" "6#*(\"\"'\" \"\"*$%\"xG\"\"#F%,&F'F%\"#5!\"\"F+" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "x=20" "6#/%\"xG\"#?" }{TEXT -1 143 " ( and not a maximum point or a stationary point of inflection ), we can verify this independently by investigating the sign of the derivative " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*&%#dAG\"\"\"%#dxG!\"\"" }{TEXT -1 2 ". " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 317 145 145 {PLOTDATA 2 " 68-%'CURVESG6$7$7$$!3++++++++N!#<$!3+++++++]7F*7$$\"3++++++++bF*F+-%'C OLOURG6&%$RGBG\"\"!F4F4-F$6$7$7$F($\"3++++++++]!#=7$F.F9F0-F$6$7$7$F($ \"\"\"F47$F.FAF0-F$6$7$F'F@F0-F$6$7$7$$!3++++++++DF*F+7$FKFAF0-F$6$7$7 $$!3++++++++]F;F+7$FRFAF0-F$6$7$7$F9F+7$F9FAF0-F$6$7$7$$\"3++++++++DF* F+7$FhnFAF0-F$6'7$7$$!\"#F4$!\"$!\"\"7$$FcoF4$!#5Fco7%7$$!+C.*p5\"!\"* $!+ggdN$)FgoFdo7$$!+w'4I>\"F\\p$!+SRUk&*Fgo-%&STYLEG6#%,PATCHNOGRIDGF0 -%*THICKNESSG6#\"\"#-F$6'7$7$FAFeo7$$F[qF4Fao7%7$$\"+C.*p!=F\\p$!*1wbV $F\\pF`q7$$\"+w'4I*=F\\p$!*%RUkYF\\pFdpF0Fhp-%%TEXTG6&7$$FboF4$\"#vF`o Q\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-F^r6&7$$!#:FcoFbrQ-~10~<~x~<~20Fer F0Ffr-F^r6&7$$F4F4FbrQ#20FerF0Ffr-F^r6&7$$\"#:FcoFbrQ'x~>~20FerF0Ffr-F ^r6&7$$\"\"$F4FbrQ\"3FerF0Ffr-F^r6&7$Far$F[qFcoQ'f~'(x)FerF0Ffr-F^r6&7 $FdsFetQ\"0FerF0Ffr-F^r6&7$F^s$\"\"%FcoQ\"_FerF0-Fgr6$Fir\"#9-F^r6&7$F is$F`tFcoQ\"+FerF0Fau-%+AXESLABELSG6$Q!FerF\\v-%*AXESSTYLEG6#%%NONEG-% %VIEWG6$;$!#NFco$\"#DFco;$!#8Fco$\"#6Fco" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "It is now clear that the minimum point occurs where " }{TEXT 359 1 "x" }{TEXT -1 43 " is exactly 20. The corresponding value of " } {TEXT 374 1 "A" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "60/(20-10)+6=12" " 6#/,&*&\"#g\"\"\",&\"#?F'\"#5!\"\"F+F'\"\"'F'\"#7" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 133 "The outer dimensions of the printed page which give the minimum printed area are therefore 20 cm. horizontally and 12 cm. vertically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f := x \+ -> 6*x^2/(x-10);\nDiff(f(x),x);\nvalue(%);\nnormal(%);\nsolve(%,x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,$*(\"\"'\"\"\"9$\"\"#,&F0F/\"#5!\"\"F4F/F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%%DiffG6$,$*(\"\"'\"\"\"%\"xG\"\"#,&F*F)\"#5!\"\"F.F )F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"#7\"\"\"%\"xGF&,&F'F&\"#5 !\"\"F*F&*(\"\"'F&F'\"\"#F(!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $**\"\"'\"\"\"%\"xGF&,&F'F&\"#?!\"\"F&,&F'F&\"#5F*!\"#F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"\"!\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 31 "A procedure for drawing a box: " }{TEXT 0 7 "drawbox" } {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "d rawbox: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 304 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 305 2 " " }{TEXT -1 36 " drawbox( length, width, height ) " } {TEXT 306 1 "\n" }{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Param eters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 27 " length, width, height - " }{TEXT -1 37 "the dimensions of th e box to be drawn" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 " " {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proced ure " }{TEXT 0 7 "drawbox" }{TEXT -1 53 " draws an open or closed box \+ of the given dimensions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 307 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 15 "" 0 "" {TEXT 262 16 "open=true/false\n" }{TEXT -1 123 "This option determines whether an open or closed box is drawn.\nThe default is \"open=false \", in which a closed box is drawn." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 15 "" 0 "" {TEXT 262 24 "colour/color=[c1,c2,c3]\n" }{TEXT -1 73 "Colours c1, c2, c3 can be chose for the base, sides and top respec tively." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 15 "" 0 "" {TEXT 262 44 "lightmodel=none/light1/light2/light3/light4\n" }{TEXT -1 68 "T his option chooses a predefined light model to illuminate the plot." } }{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 15 "How to activate" }{TEXT -1 1 ":" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the subsec tion, place the cursor anywhere after the prompt [ > and press [Enter ].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "drawbox: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "drawbox" {MPLTEXT 1 0 2074 "drawbox := proc(l ength,width,height)\n local a,b,c,A1,B1,C1,D1,A2,B2,C2,D2,base,sides ,p1,p2,p3,\n Options,opn,lightmdl,clr,baseclr,sideclr,topclr;\n\n \+ sideclr := green;\n baseclr := blue;\n topclr := red;\n opn := f alse;\n lightmdl := light2;\n if nargs>=4 then\n Options:=[ar gs[4..nargs]];\n if not type(Options,list(equation)) then\n \+ error \"each optional argument must be an equation\"\n end if; \n if hasoption(Options,open,'opn','Options') then\n if o pn<>true then opn := false end if;\n end if;\n hasoption(Opt ions,lightmodel,'lightmdl','Options');\n if hasoption(Options,col or,'clr','Options') or\n hasoption(Options,colour,'clr','Optio ns') then\n if type(clr,list) then\n if nops(clr)>= 1 then baseclr := clr[1] end if;\n if nops(clr)>=2 then sid eclr := clr[2] end if;\n if nops(clr)>=3 then topclr := clr [3] end if; \n else \n baseclr := clr;\n \+ sideclr := clr;\n topclr := clr;\n end if;\n \+ end if; \n if nops(Options)>0 then\n error \"%1 is n ot a valid option for %2\",op(1,Options),procname;\n end if;\n \+ end if;\n\n Digits := 10;\n a := evalf(length);\n b := evalf(wid th);\n c := evalf(height);\n if not type([a,b,c],list(numeric)) th en\n error \"could not evaluate the dimensions\"\n end if;\n \+ if a<0 or b<0 or c<0 then\n error \"the dimensions cannot be nega tive\"\n end if;\n A1 := [0,0,0]: B1 := [a,0,0]: \n C1 := [a,b,0 ]: D1 := [0,b,0]:\n A2 := [0,0,c]: B2 := [a,0,c]:\n C2 := [a,b,c]: D2 := [0,b,c]:\n base:=[A1,B1,C1,D1]:\n sides:=[[A1,D1,D2,A2],[A1 ,B1,B2,A2],[B1,C1,C2,B2],[C1,D1,D2,C2]]: \n p1:=plots[polygonplot3d ](sides,color=sideclr):\n p2:=plots[polygonplot3d](base,color=basecl r):\n if opn then\n plots[display]([p1,p2],style=patch,scaling= constrained,\n lightmodel=lightmdl)\n else\n p3:=plots [polygonplot3d]([A2,B2,C2,D2],color=topclr):\n plots[display]([p1 ,p2,p3],style=patch,scaling=constrained,\n lightmodel=lightmd l);\n end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 48 "Example 4 - maximising the volume of an open box" }}{PARA 0 "" 0 "" {TEXT 308 8 "Question" }{TEXT -1 3 ": " }}{PARA 0 " " 0 "" {TEXT -1 284 "An open rectangular box is to be made from a rect angular sheet of cardboard 10 cm. by 8 cm. by cutting equal squares fr om the four corners and bending the resulting four flaps to form the s ides of the box. Find the dimensions of the box needed to ensure that \+ its volume is a maximum. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 409 302 302 {PLOTDATA 2 "65-%)POLYGONSG6$7&7$$\"\"#\"\"!F(7$ $\"\")F*F(7$F,$\"\"'F*7$F(F/-%&COLORG6&%$RGBG$\"\"&!\"\"$F-F8$\"\"*F8- F$6'7&7$$F*F*F(F'F17$F@F/7&F+7$$\"#5F*F(7$FDF/F.7&7$F(F@7$F,F@F+F'7&F1 F.7$F,F,7$F(F,-F36&F5F6F:F9-%'CURVESG6%7%FA7$F@F,FL-%'COLOURG6&F5$\")! \\DP\"!\")FW$\")viobFY-%*LINESTYLEG6#\"\"$-FP6%7%FK7$FDF,FFFTFfn-FP6%7 %FC7$FDF@FIFTFfn-FP6%7%F?7$F@F@FHFTFfn-FP6%7$F]o7$$\"3+++++++]6!#;F,-F U6&F5$\")#)eqkFY$\"))eqk\"FYFap-Fgn6#F)-FP6%7$Fao7$FjoF@F]pFcp-FP6%7$F S7$F@$\"3++++++++&*!#F<7$F'-F(6'7&F+F47%$!+nY$Q*> !\"*F2$\"+:>=p:!#57%FXF,Fen7&F.F17%$\"+nY$Q*zFZF2Fen7%F[oF,Fen7&F+F.7% F/FXFen7%F,FXFen7&F1F47%F,$\"+nY$Q*fFZFen7%F/FcoFenFQ7$F'-F(6'7&F+F47% $!+\"ow`(>FZF2$\"+-$*oGJFgn7%F[pF,F]p7&F.F17%$\"+\"ow`(zFZF2F]p7%FbpF, F]p7&F+F.7%F/F[pF]p7%F,F[pF]p7&F1F47%F,$\"+\"ow`(fFZF]p7%F/FjpF]pFQ7$F '-F(6'7&F+F47%$!+T)RZ%>FZF2$\"+ws!*oYFgn7%FbqF,Fdq7&F.F17%$\"+T)RZ%zFZ F2Fdq7%FiqF,Fdq7&F+F.7%F/FbqFdq7%F,FbqFdq7&F1F47%F,$\"+T)RZ%fFZFdq7%F/ FarFdqFQ7$F'-F(6'7&F+F47%$!+LI6->FZF2$\"+)))R.='Fgn7%FirF,F[s7&F.F17%$ \"+LI6-zFZF2F[s7%F`sF,F[s7&F+F.7%F/FirF[s7%F,FirF[s7&F1F47%F,$\"+LI6-f FZF[s7%F/FhsF[sFQ7$F'-F(6'7&F+F47%$!+l!fx%=FZF2$\"+]'oOl(Fgn7%F`tF,Fbt 7&F.F17%$\"+l!fx%yFZF2Fbt7%FgtF,Fbt7&F+F.7%F/F`tFbt7%F,F`tFbt7&F1F47%F ,$\"+l!fx%eFZFbt7%F/F_uFbtFQ7$F'-F(6'7&F+F47%$!+[I,#y\"FZF2$\"+'**4)z! *Fgn7%FguF,Fiu7&F.F17%$\"+[I,#y(FZF2Fiu7%F^vF,Fiu7&F+F.7%F/FguFiu7%F,F guFiu7&F1F47%F,$\"+[I,#y&FZFiu7%F/FfvFiuFQ7$F'-F(6'7&F+F47%$!+H.G0\"3_\"FZF2$\"+(4'*))H\"FZ7%F\\zF,F^z7&F.F17%$\"+J>\"3_(FZ F2F^z7%FczF,F^z7&F+F.7%F/F\\zF^z7%F,F\\zF^z7&F1F47%F,$\"+J>\"3_&FZF^z7 %F/F[[lF^zFQ7$F'-F(6'7&F+F47%$!+iN@99FZF2$\"+jN@99FZ7%Fc[lF,Fe[l7&F.F1 7%$\"+iN@9uFZF2Fe[l7%Fj[lF,Fe[l7&F+F.7%F/Fc[lFe[l7%F,Fc[lFe[l7&F1F47%F ,$\"+iN@9aFZFe[l7%F/Fb\\lFe[lFQ7$F'-F(6'7&F+F47%$!+(4'*))H\"FZF2$\"+J> \"3_\"FZ7%Fj\\lF,F\\]l7&F.F17%$\"+(4'*))H(FZF2F\\]l7%Fa]lF,F\\]l7&F+F. 7%F/Fj\\lF\\]l7%F,Fj\\lF\\]l7&F1F47%F,$\"+(4'*))H&FZF\\]l7%F/Fi]lF\\]l FQ7$F'-F(6'7&F+F47%$!+/0dv6FZF2$\"+*)R.=;FZ7%Fa^lF,Fc^l7&F.F17%$\"+/0d vrFZF2Fc^l7%Fh^lF,Fc^l7&F+F.7%F/Fa^lFc^l7%F,Fa^lFc^l7&F1F47%F,$\"+/0dv ^FZFc^l7%F/F`_lFc^lFQ7$F'-F(6'7&F+F47%$!+Gr*\\/\"FZF2$\"+H.G0FZ7%FdelF,Ffel7&F.F17%$\"+F2*oY'FZF2Ffel7%F[flF, Ffel7&F+F.7%F/FdelFfel7%F,FdelFfel7&F1F47%F,$\"+F2*oY%FZFfel7%F/FcflFf elFQ7$F'-F(6'7&F+F47%$!+/$*oGJFgnF2$\"+\"ow`(>FZ7%F[glF,F]gl7&F.F17%$ \"+I*oGJ'FZF2F]gl7%FbglF,F]gl7&F+F.7%F/F[glF]gl7%F,F[glF]gl7&F1F47%F,$ \"+I*oGJ%FZF]gl7%F/FjglF]glFQ7$F'-F(6'7&F+F47%$!+/>=p:FgnF2$\"+oY$Q*>F Z7%FbhlF,Fdhl7&F.F17%$\"+!>=p:'FZF2Fdhl7%FihlF,Fdhl7&F+F.7%F/FbhlFdhl7 %F,FbhlFdhl7&F1F47%F,$\"+!>=p:%FZFdhl7%F/FailFdhlFQ7$F'-F(6'7&F+F47%$ \"+;w1-T!#>F2$\"\"#F-7%FiilF,F\\jl7&F.F17%$\"+++++gFZF2F\\jl7%FajlF,F \\jl7&F+F.7%F/FiilF\\jl7%F,FiilF\\jl7&F1F47%F,$\"+++++SFZF\\jl7%F/Fijl F\\jlFQ-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%%NONEG-%+PROJECTIONG 6%$!$N\"F-$\"#]F-\"\"\"" 1 2 0 1 10 0 2 1 1 1 1 1.000000 51.000000 -137.000000 1 0 "Curve 1" "Curve 2" }}}{PARA 257 "" 0 "" {TEXT -1 0 " " }{TEXT 309 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Let the length of a side of eac h square to be cut from the corners be " }{TEXT 310 1 "x" }{TEXT -1 5 " cm. " }}{PARA 0 "" 0 "" {TEXT -1 54 "Then the rectangle which forms \+ the base of the box is " }{XPPEDIT 18 0 "``(10-2*x);" "6#-%!G6#,&\"#5 \"\"\"*&\"\"#F(%\"xGF(!\"\"" }{TEXT -1 8 " cm. by " }{XPPEDIT 18 0 "`` (8-2*x);" "6#-%!G6#,&\"\")\"\"\"*&\"\"#F(%\"xGF(!\"\"" }{TEXT -1 50 " \+ cm. and the height of the resulting rectangle is " }{TEXT 312 1 "x" } {TEXT -1 5 " cm. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 300 300 {PLOTDATA 2 "6C-%)POLYGONSG6$7&7$$\"\"#\"\"!F(7$$\"\")F*F(7$F, $\"\"'F*7$F(F/-%&COLORG6&%$RGBG$\"\"&!\"\"$F-F8$\"\"*F8-F$6'7&7$$F*F*F (F'F17$F@F/7&F+7$$\"#5F*F(7$FDF/F.7&7$F(F@7$F,F@F+F'7&F1F.7$F,F,7$F(F, -F36&F5F6F:F9-%'CURVESG6%7%FA7$F@F,FL-%'COLOURG6&F5$\")!\\DP\"!\")FW$ \")viobFY-%*LINESTYLEG6#\"\"$-FP6%7%FK7$FDF,FFFTFfn-FP6%7%FC7$FDF@FIFT Ffn-FP6%7%F?7$F@F@FHFTFfn-FP6%7$F]o7$$\"3+++++++]6!#;F,-FU6&F5$\")#)eq kFY$\"))eqk\"FYFap-Fgn6#F)-FP6%7$Fao7$FjoF@F]pFcp-FP6%7$FS7$F@$\"3++++ ++++&*!# " 0 "" {MPLTEXT 1 0 132 "xx := 1.5;\nLength := 10-2*xx;\nWidth := 8-2*xx;\nHeight := xx; \ndrawbox(Length,Width,Height,open=true);\n'Volume'=Length*Width*Hei ght;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#:!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'LengthG$\"#q!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&WidthG$\"#]!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%'HeightG$\"#:!\"\"" }}{PARA 13 "" 1 "" {GLPLOT3D 413 253 253 {PLOTDATA 3 "6'-%)POLYGONSG6'7&7%$\"\"!F)F(F(7%F($\"#]!\"\"F(7%F(F+$\" #:F-7%F(F(F/7&F'7%$\"#qF-F(F(7%F4F(F/F17&F37%F4F+F(7%F4F+F/F67&F8F*F.F 9-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-F$6$7&F'F3F8F*-F<6&F>F(F(F?-%&ST YLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 4 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG$\"&+D&!\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Taking a \+ smaller value of " }{TEXT 318 1 "x" }{TEXT -1 123 " results in a shall ower box, but this is counterbalanced, to some extent, by the fact tha t the area of the base increases. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "xx := .5; \nLength := 10-2* xx;\nWidth := 8-2*xx;\nHeight := xx; \ndrawbox(Length,Width,Height,op en=true);\n'Volume'=Length*Width*Height;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'LengthG$ \"#!*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&WidthG$\"#q!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'HeightG$\"\"&!\"\"" }}{PARA 13 "" 1 "" {GLPLOT3D 341 191 191 {PLOTDATA 3 "6'-%)POLYGONSG6'7&7%$\"\"!F)F( F(7%F($\"#q!\"\"F(7%F(F+$\"\"&F-7%F(F(F/7&F'7%$\"#!*F-F(F(7%F4F(F/F17& F37%F4F+F(7%F4F+F/F67&F8F*F.F9-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-F$6 $7&F'F3F8F*-F<6&F>F(F(F?-%&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G- %(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 4 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %'VolumeG$\"&+:$!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "For a value of " }{TEXT 319 1 "x" }{TEXT -1 51 " close to zero the volume is significantly reduced." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "xx := .05; \+ \nLength := 10-2*xx;\nWidth := 8-2*xx;\nHeight := xx; \ndrawbox(Lengt h,Width,Height,open=true);\n'Volume'=Length*Width*Height;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"&!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'LengthG$\"$!**!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Widt hG$\"$!z!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'HeightG$\"\"&!\"#" }}{PARA 13 "" 1 "" {GLPLOT3D 315 185 185 {PLOTDATA 3 "6'-%)POLYGONSG6' 7&7%$\"\"!F)F(F(7%F($\"$!z!\"#F(7%F(F+$\"\"&F-7%F(F(F/7&F'7%$\"$!**F-F (F(7%F4F(F/F17&F37%F4F+F(7%F4F+F/F67&F8F*F.F9-%'COLOURG6&%$RGBGF($\"*+ +++\"!\")F(-F$6$7&F'F3F8F*-F<6&F>F(F(F?-%&STYLEG6#%&PATCHG-%+LIGHTMODE LG6#%(LIGHT_2G-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 4 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG$\"(+0\"R!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "On the other hand, when " } {TEXT 313 1 "x" }{TEXT -1 102 " is close to 4, one of the dimensions o f the base is close to zero, so that the volume is again small." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "xx := 3.8; \nLength := 10-2*xx;\nWidth := 8-2*xx;\nHeight := xx; \+ \ndrawbox(Length,Width,Height,open=true);\n'Volume'=Length*Width*Heig ht;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#Q!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'LengthG$\"#C!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&WidthG$\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'HeightG$\"#Q!\"\"" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)POLYGONSG6'7&7%$\"\"!F)F(F(7%F($\"\"%!\"\"F(7%F(F+$ \"#QF-7%F(F(F/7&F'7%$\"#CF-F(F(7%F4F(F/F17&F37%F4F+F(7%F4F+F/F67&F8F*F .F9-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-F$6$7&F'F3F8F*-F<6&F>F(F(F?-%& STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G-%(SCALINGG6#%,CONSTRAINEDG " 1 2 0 1 10 0 2 1 4 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG$\"%[O!\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can se t up a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " to calculate the volume " }{TEXT 337 1 "V" }{TEXT -1 26 " for val ues of the length " }{TEXT 314 1 "x" }{TEXT -1 86 " of the side of the squares cut from each corner of the original cardboard sheet when " } {TEXT 316 1 "x" }{TEXT -1 18 " between 0 and 4. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "f := x -> x* (10-2*x)*(8-2*x);\nplot(f(x),x=0..4,labels=[`x`,`V`]); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*(9$\" \"\",&\"#5F.*&\"\"#F.F-F.!\"\"F.,&\"\")F.*&F2F.F-F.F3F.F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7X7$ $\"\"!F)F(7$$\"39LLLL3VfV!#>$\"3[CeQq$f%>M!#<7$$\"3Hmmmm;')=()F-$\"3\" o[J'QP2/nF07$$\"3-++]7z>^7!#=$\"3!y>VXkRQX*F07$$\"3RLLLe'40j\"F9$\"3[1 _0GMV57!#;7$$\"3/++](Q&3d?F9$\"3M\\tv/E\"o\\\"FA7$$\"3mmmm;6m$[#F9$\"3 mCIJR&))4x\"FA7$$\"3jmmmmW18HF9$\"3Dk)f'4c%[.#FA7$$\"3fmmm;yYULF9$\"3Y #)H_:fr'G#FA7$$\"3/++](GI)pPF9$\"3^mi_agnDDFA7$$\"3%HLL$eF>(>%F9$\"3Ki E')['QJv#FA7$$\"3Qmmm\">K'*)\\F9$\"3AjS*[!R7XJFA7$$\"3P*****\\Kd,\"eF9 $\"3kV;I'*fH6NFA7$$\"3-mmm\"fX(emF9$\"3[/P]$*G*)[QFA7$$\"3.*****\\U7Y] (F9$\"3q6i\"[**f_9%FA7$$\"3'QLLLV!pu$)F9$\"3*>X!z]:#)4WFA7$$\"3xmmm;c0 T\"*F9$\"3K')ox&y]-h%FA7$$\"3#*******H,Q+5F0$\"3-6[A8*f2![FA7$$\"3)*** ****\\*3q3\"F0$\"3awryW?6c\\FA7$$\"3)*******p=\\q6F0$\"3kIv)35/K2&FA7$ $\"3mmm;fBIY7F0$\"3)o7)pJe)H:&FA7$$\"3GLLLj$[kL\"F0$\"3)[\"\\TpnX;_FA7 $$\"3?LLL`Q\"GT\"F0$\"3Kvlgb4xW_FA7$$\"3!*****\\s]k,:F0$\"3'fWcKiI)\\_ FA7$$\"39LLL`dF!e\"F0$\"3gj'Gfzz0B&FA7$$\"33++]sgam;F0$\"3Gqo-ebE&=&FA 7$$\"3/++]^FA7$$\"3QLLLe/TM=F0$\"3\")[/(=@B-.&FA7$ $\"3JLL$eDBJ\">F0$\"3P`q(\\U$oH\\FA7$$\"3immmTc-)*>F0$\"3VZea#HaJ![FA7 $$\"3Mmm;f`@'3#F0$\"3o\"*=4e?R`YFA7$$\"3y****\\nZ)H;#F0$\"3e(>mQMz!4XF A7$$\"3YmmmJy*eC#F0$\"33R'*3N>&*RVFA7$$\"3')******R^bJBF0$\"3eSu(y=w@: %FA7$$\"3f*****\\5a`T#F0$\"3]&3%H)=tq&RFA7$$\"3o****\\7RV'\\#F0$\"3o?? Uqv!*ePFA7$$\"3k*****\\@fke#F0$\"3F28\"[OD'HNFA7$$\"3/LLL`4NnEF0$\"3# \\%H!pc$p;LFA7$$\"3#*******\\,s`FF0$\"3_B#*)*f%3O3$FA7$$\"3[mm;zM)>$GF 0$\"3#y]Pf/\\&oGFA7$$\"3$*******pfaFA7$$\"3#)****\\7yh]KF0$\"3o6&RbdmXq\"FA7$$\"3xmmm' )fdLLF0$\"3#p*Q[HW$3[\"FA7$$\"3bmmm,FT=MF0$\"3#y]eU!\\ud7FA7$$\"3FLL$e #pa-NF0$\"3SyR7%eOO/\"FA7$$\"3!*******Rv&)zNF0$\"3C/ErQ>'Qa)F07$$\"3IL LLGUYoOF0$\"3Y)Hk/m*zxkF07$$\"3_mmm1^rZPF0$\"3RDfDA45OZF07$$\"34++]sI@ KQF0$\"3Cw+4Ae_.IF07$$\"34++]2%)38RF0$\"3#)>+l))Hgy9F07$$\"\"%F)F(-%'C OLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%\"xG%\"VG-%%VIEWG6$;F(F] \\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "A maximum value of the volume " }{TEXT 349 1 "V" }{TEXT -1 13 " occurs when " }{TEXT 315 1 "x" }{TEXT -1 16 " is about 1.45. " }}{PARA 0 "" 0 "" {TEXT -1 179 "To find this value more precisely we note that the maximum point \+ occurs where the tangent line to the graph is horizontal, that is, whe re the gradient, as given by the derivative " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*&%#dAG\"\"\"%#dxG!\"\"" }{TEXT -1 10 ", \+ is zero." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 310 157 157 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$\"3A+++++++!*!#=$\"3-+++++gvX!#;7$$\"3 /nm;/IO$G*F*$\"3\"oQBzt*>WYF-7$$\"3%RLe*Qc\"*H&*F*$\"3;D$pG2Q1q%F-7$$ \"3hnm\"H')*=2)*F*$\"3cUPY()GagZF-7$$\"3\"omTS?I'35!#<$\"3#f]$=7>3<[F- 7$$\"3JLekk(3k.\"F@$\"3%=q/\"*Rb'p[F-7$$\"3om\"Hi/j@1\"F@$\"3s$*zP\"*z 9:\\F-7$$\"3++D18,$))3\"F@$\"3Ak!G%3F+f\\F-7$$\"3tm\"H\"F@$\"3.\"[jRN4S5&F-7$$\"3(****\\AaB^A\"F@ $\"3z;Hb\"QxJ8&F-7$$\"37++v3zF`7F@$\"3KlTS?c4f^F-7$$\"3/++vd)4/G\"F@$ \"3qt4Tzy#4=&F-7$$\"3um\"HnE[]I\"F@$\"3%4HVUI?\")>&F-7$$\"3NLL3=dMM8F@ $\"3SJu\"Rq_`@&F-7$$\"3ULLL-X;f8F@$\"3Ce$HS$QEF_F-7$$\"39+Dc[Y.)Q\"F@$ \"36e(zM&)p!Q_F-7$$\"3`LL$)>'*e89F@$\"3Qt=;*zU\\C&F-7$$\"34+DctuiT9F@$ \"3/*oK5YC'\\_F-7$$\"37+voShKo9F@$\"3;NvOb)[8D&F-7$$\"3_L$e*)R$='\\\"F @$\"3ME\"o#=aN]_F-7$$\"3_Le9e]w@:F@$\"3qbAUE\\(pC&F-7$$\"3um;aL$e$\\:F @$\"3t5H3DqsS_F-7$$\"3im\"H<**>!y:F@$\"34dJA`6VJ_F-7$$\"35+vV\\+(Hg\"F @$\"3P,ks3Y0@_F-7$$\"3*om\"H&z;*H;F@$\"3Fn6]hV]2_F-7$$\"3-++]?avd;F@$ \"3%G&=Tw')*4>&F-7$$\"31+]7%3!*\\o\"F@$\"351,;Q[Vs^F-7$$\"3B+Dc@5M67M)>F @$\"3Q:;%\\Z3i#[F-7$$\"3rm;/GT)4,#F@$\"3%eW\\R9\"G#y%F-7$$\"3tLe*3vF$Q ?F@$\"3@.%>/INpt%F-7$$\"3A++]+PXj?F@$\"3qU`&ziWPp%F-7$$\"3mL$3U(3D#4#F @$\"3')R)\\HO+Dk%F-7$$\"3zmmm4u+=@F@$\"3GiUm#eM^f%F-7$$\"3A+Dc[#pa9#F@ $\"3^\\N1np3VXF-7$$\"3C+vVKPvr@F@$\"3/v()ej9#=\\%F-7$$\"3;+++++++AF@$ \"3o**********>NWF--%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fa[lF`[l-F$6$ 7$7$$\"3'******zwuC2\"F@$\"35+++N/Q^_F-7$$\"3++++oZZs=F@Fh[l-%&COLORG6 &F\\[lFa[l$\"\"(!\"\"Fa[l-F$6$7$7$F($\"3)***********z!y%F-7$$\"3++++++ ++:F@$\"3')**********fLaF--Fjz6&F\\[lF][lF`[lF][l-F$6$7$7$Fj\\l$\"3k** ********R!R&F-7$$\"33+++++++@F@$\"39+++++?ZZF--Fjz6&F\\[lF`[lF`[lF][l- %%TEXTG6%7$$\"$0\"!\"#$\"#^Fa[lQ*f'(x)~>~06\"F^]l-F^^l6%7$$\"$v\"Fc^l$ \"$N&Fb\\lQ*f'(x)~=~0Fg^lF]\\l-F^^l6%7$$\"$&>Fc^lFd^lQ*f'(x)~<~0Fg^lF[ ^l-%+AXESLABELSG6%Q\"xFg^lQ!Fg^l-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%N ONEG-%%VIEWG6$;$\"\"*Fb\\l$\"#AFb\\lF^`l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V =x*(10-2*x)*(8-2*x)" "6#/%\"VG*(%\"xG\"\"\",&\"#5F'*&\"\"#F'F&F'!\"\"F ',&\"\")F'*&F+F'F&F'F,F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=x*(80-36*x+4*x^2)" "6#/%!G*&%\"xG\"\"\",( \"#!)F'*&\"#OF'F&F'!\"\"*&\"\"%F'*$F&\"\"#F'F'F'" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=80*x-36*x^2+4*x^3 " "6#/%!G,(*&\"#!)\"\"\"%\"xGF(F(*&\"#OF(*$F)\"\"#F(!\"\"*&\"\"%F(*$F) \"\"$F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 " ``=4*x^3-36*x^2+80*x" "6#/%!G,(*&\"\"%\"\"\"*$%\"xG\" \"$F(F(*&\"#OF(*$F*\"\"#F(!\"\"*&\"#!)F(F*F(F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dV/dx=12*x^2-72*x+80" "6#/*&%#dVG\"\"\"%#dxG!\"\",(*&\" #7F&*$%\"xG\"\"#F&F&*&\"#sF&F-F&F(\"#!)F&" }{XPPEDIT 18 0 " ``=4*(3*x^ 2-18*x+20)" "6#/%!G*&\"\"%\"\"\",(*&\"\"$F'*$%\"xG\"\"#F'F'*&\"#=F'F,F '!\"\"\"#?F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The de rivative " }{XPPEDIT 18 0 "`f '`(x) = dV/dx;" "6#/-%$f~'G6#%\"xG*&%#dV G\"\"\"%#dxG!\"\"" }{TEXT -1 14 " is zero when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "3*x^2-18*x+20=0" "6#/,(*&\"\"$\"\"\"*$% \"xG\"\"#F'F'*&\"#=F'F)F'!\"\"\"#?F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Using the formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x =-b/(2*a)" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"#F(% \"aGF(!\"\"F," }{TEXT -1 1 " " }{TEXT 362 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(b^2-4*a*c)/(2*a)" "6#*&-%%sqrtG6#,&*$%\"bG\"\"#\" \"\"*(\"\"%F+%\"aGF+%\"cGF+!\"\"F+*&F*F+F.F+F0" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 44 "for the solutions of the quadratic equati on " }{XPPEDIT 18 0 "a*x^2+b*x+c=0" "6#/,(*&%\"aG\"\"\"*$%\"xG\"\"#F'F '*&%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 8 " gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x= 18/(2*`.`*3)" "6#/%\"xG*&\"#=\"\" \"*(\"\"#F'%\".GF'\"\"$F'!\"\"" }{TEXT -1 1 " " }{TEXT 363 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(18^2-4*`.`*3*`.`*20)/(2*`.`*3)" "6 #*&-%%sqrtG6#,&*$\"#=\"\"#\"\"\"*,\"\"%F+%\".GF+\"\"$F+F.F+\"#?F+!\"\" F+*(F*F+F.F+F/F+F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 " ``= 3" "6#/%!G\" \"$" }{TEXT -1 1 " " }{TEXT 364 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sqrt(324-240)/6" "6#*&-%%sqrtG6#,&\"$C$\"\"\"\"$S#!\"\"F)\"\"'F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 3" "6#/%!G\"\"$" }{TEXT -1 1 " " } {TEXT 365 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(84)/6" "6#*&-%%sq rtG6#\"#%)\"\"\"\"\"'!\"\"" }{XPPEDIT 18 0 " ``= 3" "6#/%!G\"\"$" } {TEXT -1 1 " " }{TEXT 366 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2 1)/3;" "6#*&-%%sqrtG6#\"#@\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The maximum poi nt must be given by the value " }{TEXT 369 1 "x" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "x[max] = 3-sqrt(21)/3;" "6#/&%\"xG6#%$maxG,&\"\"$\"\"\" *&-%%sqrtG6#\"#@F*F)!\"\"F0" }{TEXT -1 1 " " }{TEXT 367 1 "~" }{TEXT -1 9 " 1.4725. " }}{PARA 0 "" 0 "" {TEXT -1 98 "We can use the second \+ derivative test to verify that the corresponding turning point on the \+ graph " }{XPPEDIT 18 0 "V = x*(10-2*x)*(8-2*x);" "6#/%\"VG*(%\"xG\"\" \",&\"#5F'*&\"\"#F'F&F'!\"\"F',&\"\")F'*&F+F'F&F'F,F'" }{TEXT -1 21 " \+ is a maximum point. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "`f '`(x) = dV/dx;" "6#/-%$f~'G6#%\"xG*&%#dVG\"\"\"%#dxG!\"\"" } {XPPEDIT 18 0 "``=12*x^2-72*x+80" "6#/%!G,(*&\"#7\"\"\"*$%\"xG\"\"#F(F (*&\"#sF(F*F(!\"\"\"#!)F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`f ''`(x) = d^2*V/(d*x^2);" "6#/-%%f~''G6#%\"xG*(%\"dG\"\"#%\"VG\"\"\"*&F)F,*$) F'F*F,F,!\"\"" }{XPPEDIT 18 0 "``=24*x-72" "6#/%!G,&*&\"#C\"\"\"%\"xGF (F(\"#s!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " } {XPPEDIT 18 0 "`f ''`(3-sqrt(21)/3);" "6#-%%f~''G6#,&\"\"$\"\"\"*&-%%s qrtG6#\"#@F(F'!\"\"F." }{TEXT -1 103 " is negative as required for a m aximum point. It is sufficient to use the rough approximation 1.5 for " }{XPPEDIT 18 0 "3-sqrt(21)/3" "6#,&\"\"$\"\"\"*&-%%sqrtG6#\"#@F%F$! \"\"F+" }{TEXT -1 14 " to see this. " }}{PARA 0 "" 0 "" {TEXT -1 97 "T his test could be regarded as being unecessary if the graph drawn abov e is known to be correct. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The height of the box with maximum volume is " } {XPPEDIT 18 0 "3-sqrt(21)/3;" "6#,&\"\"$\"\"\"*&-%%sqrtG6#\"#@F%F$!\" \"F+" }{TEXT -1 1 " " }{TEXT 368 1 "~" }{TEXT -1 11 " 1.4725 cm." }} {PARA 0 "" 0 "" {TEXT -1 29 "The corresponding length is " }{XPPEDIT 18 0 "10-2*x[max] = 4+2*sqrt(21)/3;" "6#/,&\"#5\"\"\"*&\"\"#F&&%\"xG6# %$maxGF&!\"\",&\"\"%F&*(F(F&-%%sqrtG6#\"#@F&\"\"$F-F&" }{TEXT -1 1 " \+ " }{TEXT 370 1 "~" }{TEXT -1 13 " 7.0551 cm. " }}{PARA 0 "" 0 "" {TEXT -1 28 "The corresponding width is " }{XPPEDIT 18 0 "8-2*x[max] \+ = 2+2*sqrt(21)/3;" "6#/,&\"\")\"\"\"*&\"\"#F&&%\"xG6#%$maxGF&!\"\",&F( F&*(F(F&-%%sqrtG6#\"#@F&\"\"$F-F&" }{TEXT -1 1 " " }{TEXT 371 1 "~" } {TEXT -1 12 " 5.0551 cm. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "f := x -> x*(10-2*x)*(8-2*x);\nDiff(f(x),x);\nvalue(%);\nsimplify(%); \ndf := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\" xG6\"6$%)operatorG%&arrowGF(*(9$\"\"\",&\"#5F.*&\"\"#F.F-F.!\"\"F.,&\" \")F.*&F2F.F-F.F3F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6 $*(%\"xG\"\"\",&\"#5F(*&\"\"#F(F'F(!\"\"F(,&\"\")F(*&F,F(F'F(F-F(F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&\"#5\"\"\"*&\"\"#F'%\"xGF'!\"\" F',&\"\")F'*&F)F'F*F'F+F'F'*(F)F'F*F'F,F'F+*(F)F'F*F'F%F'F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(\"#!)\"\"\"*&\"#sF%%\"xGF%!\"\"*&\"#7F%)F( \"\"#F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,(\"#!)\"\"\"*&\"#sF.9$F.!\"\"*&\"#7F.)F1\"\"#F.F.F(F( F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Th e derivative is a quadratic function and so we obtain two values of " }{TEXT 317 1 "x" }{TEXT -1 30 " where the derivative is zero." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(df(x)=0,x);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,& \"\"$\"\"\"*&F$!\"\"\"#@#F%\"\"#F%,&F$F%*&F$F'F(F)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+K__FX!\"*$\"+oZZs9F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "We can draw the box." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "x x := 3-sqrt(21)/3; \nLength := 10-2*xx;\n``=evalf(%);\nWidth := 8-2*xx ;\n``=evalf(%);\nHeight := xx; \n``=evalf(%); \ndrawbox(Length,Width,H eight,open=true);\nVolume:=Length*Width*Height;\n``=evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,&\"\"$\"\"\"*&#F'F&F'*$-%%sqrtG 6#\"#@F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'LengthG,&\"\"%\" \"\"*&#\"\"#\"\"$F'-%%sqrtG6#\"#@F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+j/0bq!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&WidthG,&\" \"#\"\"\"*&#F&\"\"$F'-%%sqrtG6#\"#@F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+j/0b]!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'HeightG, &\"\"$\"\"\"*&#F'F&F'*$-%%sqrtG6#\"#@F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+oZZs9!\"*" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)POLYGONSG6'7&7%$\"\"!F)F(F(7%F($\"+j/0b]!\"* F(7%F(F+$\"+oZZs9F-7%F(F(F/7&F'7%$\"+j/0bqF-F(F(7%F4F(F/F17&F37%F4F+F( 7%F4F+F/F67&F8F*F.F9-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-F$6$7&F'F3F8F *-F<6&F>F(F(F?-%&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G-%(SCALINGG 6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 4 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'VolumeG* (,&\"\"%\"\"\"*&#\"\"#\"\"$F(-%%sqrtG6#\"#@F(F(F(,&F+F(*&F*F(F-F(F(F(, &F,F(*&#F(F,F(*$F-F(F(!\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$ \"+J/Q^_!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Examp le 5 - most economical open box " }}{PARA 0 "" 0 "" {TEXT 320 8 "Quest ion" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 244 "A rectangular b ox with a square base and open top is required to have a volume of 5 c ubic metres. Find the dimensions of the box that will minimise the sur face area in order to ensure that the box can be manufactured in the m ost economical way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 321 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 " " 0 "" {TEXT -1 49 "Let the length of one side of the square base be \+ " }{TEXT 322 1 "x" }{TEXT -1 27 " metres. and the height be " }{TEXT 323 1 "h" }{TEXT -1 9 " metres. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 374 303 303 {PLOTDATA 2 "69-%'CURVESG6$7+7$$\"\"#\"\"!$F*F*7 $F+F+7$F+$\"\"$F*7$$\"3'******4\\DN@\"!#<$\"3;+++zyn\")QF37$$\"3'***** *4\\DN@$F3F47$F7$\"3?+++'yyn\"))!#=F'7$F(F.F--%'COLOURG6&%$RGBG$\")#)e qk!\")$\"))eqk\"FDFE-F$6$7$F=F6F>-F$6%7%F,7$F1F:F9-F?6&FA$\")`B)e)FD$ \")fqkdFD$\")p:#R%FD-%*LINESTYLEG6#F/-F$6%7$FMF0FNFV-%)POLYGONSG6$7&F, F-F=F'-%&COLORG6&FA\"\"\"$\"\")!\"\"$\"#l!\"#-Fgn6$7&7$$\"+\"\\DN@$!\" *$\"+'yyn\"))!#57$Fho$\"+zyn\")QFjoF=F'-F[o6&FAF]o$\"#xFco$\"#iFco-Fgn 6$7&F-7$$\"+\"\\DN@\"FjoF_pF^pF=-F[o6&FAF]o$\"#&)Fco$\"\"(F`o-F$6%7%7$ $!3w**************fF-FW6#F)-F$6%7$7$FgqF.F-F>Fjq-F$6%7%7 $F(FgqF'7$$\"33+++++++EF3F+F>Fjq-F$6%7$F97$$\"30+++\"\\DN\"QF3F:F>Fjq- F$6&7$7$$!\"$F`o$\"#8F`o7$FasF+7%7$$!+++++DF]p$\"*+++I\"FjoFes7$$!++++ +NF]pFjs-%&STYLEG6#%,PATCHNOGRIDGFN-F$6&7$7$Fas$\"#-Ffy6%7$$F]oF*FasQ\"xF\\zF>-Ffy6%7$$\"+YFw1HFjo$\"+$R*Q3WF]pFazF>- %+AXESLABELSG6%Q!F\\zF\\[l-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-% %VIEWG6$F`[lF`[l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 46.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "C urve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "The volume of the box is " }{XPPEDIT 18 0 "x^2*h" "6#*&%\"xG\"\"#% \"hG\"\"\"" }{TEXT -1 60 " cubic metres, and we are given that this is 4 cubic metres." }}{PARA 0 "" 0 "" {TEXT -1 25 "This gives the equati on " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*h = 5;" " 6#/*&%\"xG\"\"#%\"hG\"\"\"\"\"&" }{TEXT -1 14 " ------- (i), " }} {PARA 0 "" 0 "" {TEXT -1 20 "which constitutes a " }{TEXT 259 10 "cons traint" }{TEXT -1 17 " for the problem." }}{PARA 0 "" 0 "" {TEXT -1 17 "The surface area " }{TEXT 325 1 "A" }{TEXT -1 48 " is made up from the area of the base, which is " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"# " }{TEXT -1 76 " square metres and the area of the four sides each of \+ which has an area of " }{XPPEDIT 18 0 "x*h" "6#*&%\"xG\"\"\"%\"hGF%" }{TEXT -1 16 " square metres." }}{PARA 0 "" 0 "" {TEXT -1 36 "The tot al surface area is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "A=x^2+4*x*h" "6#/%\"AG,&*$%\"xG\"\"#\"\"\"*(\"\"%F)F'F) %\"hGF)F)" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can eliminate the variable " } {TEXT 324 1 "h" }{TEXT -1 36 " from equation (ii) by substituting " } {XPPEDIT 18 0 "h = 5/(x^2);" "6#/%\"hG*&\"\"&\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 20 " from equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 11 "Thi s gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = x^2+4 *x*`.`;" "6#/%\"AG,&*$%\"xG\"\"#\"\"\"*(\"\"%F)F'F)%\".GF)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "5/(x^2)" "6#*&\"\"&\"\"\"*$%\"xG\"\"#!\"\"" } {TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "A = x^2+20/x;" "6#/%\"AG,&*$%\"xG\"\" #\"\"\"*&\"#?F)F'!\"\"F)" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We can experiment \+ by calculating the surface area for different choices of " }{TEXT 328 1 "x" }{TEXT -1 51 ", and draw the resulting boxes using the procedure " }{TEXT 0 7 "drawbox" }{TEXT -1 1 " " }{HYPERLNK 17 "drawbox" 1 "" " drawbox" }{TEXT -1 27 " from an earlier section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "xx := 3.;\n hh := 5/xx^2; \ndrawbox(xx,xx,hh,\n open=true,color=[COLOR(RGB,1 ,.6,.6),coral]);\n'Area'=xx^2+4*xx*hh;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hhG$\"+b bbbb!#5" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)PO LYGONSG6'7&7%$\"\"!F)F(F(7%F($\"\"$F)F(7%F(F+$\"+bbbbb!#57%F(F(F.7&F'7 %F+F(F(7%F+F(F.F17&F37%F+F+F(7%F+F+F.F47&F6F*F-F7-%'COLOURG6&%$RGBG$\" *++++\"!\")$\")AR!)\\F?F(-F$6$7&F'F3F6F*-%&COLORG6&F<\"\"\"$\"\"'!\"\" FI-%(SCALINGG6#%,CONSTRAINEDG-%&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGH T_2G" 1 2 0 1 10 0 2 1 4 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%AreaG$\"+mmmm:!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "xx := 1.;\nhh := 4/xx^2; \+ \ndrawbox(xx,xx,hh,open=true,color=[COLOR(RGB,1,.6,.6),coral]);\n'Area '=xx^2+4*xx*hh;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"\"\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hhG$\"\"%\"\"!" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)POLYGONSG6'7&7%$\"\"!F)F( F(7%F($\"\"\"F)F(7%F(F+$\"\"%F)7%F(F(F.7&F'7%F+F(F(7%F+F(F.F07&F27%F+F +F(7%F+F+F.F37&F5F*F-F6-%'COLOURG6&%$RGBG$\"*++++\"!\")$\")AR!)\\F>F(- F$6$7&F'F2F5F*-%&COLORG6&F;F,$\"\"'!\"\"FG-%(SCALINGG6#%,CONSTRAINEDG- %&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G" 1 2 0 1 10 0 2 1 4 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%AreaG$\"#<\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "To determine the value of " }{TEXT 329 1 "x" }{TEXT -1 28 " for which the surface area " }{TEXT 375 1 "A " }{TEXT -1 35 " is a minimum we set up a function " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 56 " to calculate the surface area in terms of the variable " }{TEXT 326 1 "x" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f := \+ x -> x^2+20/x;\nplot(f(x),x=0..5,A=0..45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\" \"\"F1*&\"#?F1F/!\"\"F1F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7dp7$$\"3(*******R`!eS$!#?$\"3yS#*4KiKs e!#97$$\"3&*******z1h6oF*$\"3Y0zkcJ;OHF-7$$\"3*******>gT<-\"!#>$\"3W+b *yF-7$$\"3++++O@Ki8F6$\"3+$4<2u\"3o9F-7$$\"3;+++qE!Hq\"F6$\"3Q'p( 3MbYu6F-7$$\"3*******R?$[V?F6$\"3-cJKVX@(y*!#:7$$\"3;+++QP1%Q#F6$\"3Mk mhTJ/*Q)FH7$$\"3)******>FWYs#F6$\"3Ys5J*>:/M(FH7$$\"3;+++1[AlIF6$\"3cy %psI;[_'FH7$$\"3K+++S`!eS$F6$\"3;V(4+#yLseFH7$$\"39+++ueQYPF6$\"3D)fzw T#\\Q`FH7$$\"3)******zSmp3%F6$\"3m&>Wq)=i$*[FH7$$\"3z*****>%paFWF6$\"3 KZ4%*=O>%FH7$$\"38+++5!3(3^F6$\"3=_t *eC5\\\"RFH7$$\"3/+++X&)G\\aF6$\"3aeW)=eL-n$FH7$$\"3y*****z2p)*y&F6$\" 3%)pV0kIMaMFH7$$\"3I+++7'\\/8'F6$\"3)*GcCQ5WiKFH7$$\"3W*****f9I5Z'F6$ \"3!G[?]))R24$FH7$$\"3k+++!o5;\"oF6$\"3z?**G3&4i$HFH7$$\"3[+++97>_rF6$ \"3mZZ7tpR'z#FH7$$\"3G+++[* F6$\"3[C,l(>?]<#FH7$$\"3J*****H&\\DO&*F6$\"3I$z-WI]t4#FH7$$\"3.+++-;u@ 5!#=$\"37O+4pkad>FH7$$\"3++++4x&)*3\"Fdt$\"3[#*>bA2AN=FH7$$\"31+++q^7 \\6Fdt$\"332R<&['eS\"> !z7FH7$$\"3$******HBn5!=Fdt$\"3=HQ***4x26\"FH7$$\"3))*****R2P\"Q?Fdt$ \"3fRf?'oOq\")*!#;7$$\"3'******\\!pu/BFdt$\"39@b&[GaIo)Fdw7$$\"3%***** *\\tc8d#Fdt$\"3=s8LUpg%y(Fdw7$$\"3u*****fcmz$GFdt$\"3S3'=oU``0(Fdw7$$ \"35+++(RwX5$Fdt$\"3'fe'*z4T !p#[a1GFdw7$$\"3&******>*>VB$)Fdt$\"3E=Mg)\\M@Z#Fdw7$$\"3c*****R`l2Q*F dt$\"3*zR.C#4-?AFdw7$$\"3-+++0j$o/\"!#<$\"37PBY.]5??Fdw7$$\"3!******>& >jU6F_[l$\"3oN7Bpc!4)=Fdw7$$\"3%******H;v/D\"F_[l$\"3;H()zN3wb_ll\"Fdw7$$\"35+++Q[6j9F_[l$\"3OM@C.s, \"e\"Fdw7$$\"35+++\\z(yb\"F_[l$\"3jO*e$))f\\E:Fdw7$$\"3%******\\Xg0n\" F_[l$\"3o\\.KI-Gw9Fdw7$$\"3)******pJpW`(>F_[l$\"3]EW5(=!o-9Fdw 7$$\"3#******4f#=$3#F_[l$\"3kI\"QyVMSR\"Fdw7$$\"3%)*****Hxpe=#F_[l$\"3 )e^\\sDqFR\"Fdw7$$\"35+++uI,$H#F_[l$\"3QT\\.Ad+)R\"Fdw7$$\"3=+++rSS\"R #F_[l$\"3F5:^-,@39Fdw7$$\"3-+++`?`(\\#F_[l$\"3X3U+frbC9Fdw7$$\"3!***** ***>pxg#F_[l$\"3erKWd])pW\"Fdw7$$\"38+++g4t.FF_[l$\"33u'H1oM2Z\"Fdw7$$ \"3*********Gst!GF_[l$\"36olH%)Qa+:Fdw7$$\"38+++ERW9HF_[l$\"3w)*o!\\fN c`\"Fdw7$$\"3@+++KE>>IF_[l$\"3-\"e1X;\")Rd\"Fdw7$$\"3%******>RU07$F_[l $\"3([WBUq#p9;Fdw7$$\"36+++?S2LKF_[l$\"3AZr8vI)Qm\"Fdw7$$\"3?+++$p)=ML F_[l$\"3sP8\"))\\F:r\"Fdw7$$\"3\"*******)=]@W$F_[l$\"31:,(ew\"Fdw7$ $\"3')******\\$z*RNF_[l$\"3GYG%pU?\"==Fdw7$$\"3#)*****RYKpk$F_[l$\"3?q -3PxTy=Fdw7$$\"3))*****z+nvu$F_[l$\"3T/OXJa5Q>Fdw7$$\"3)******R5fF&QF_ [l$\"3q4r^rP[.?Fdw7$$\"3')*****\\g.c&RF_[l$\"3+JCoB=Hq?Fdw7$$\"3K+++nA FjSF_[l$\"31jjqvABV@Fdw7$$\"3q*****\\)*pp;%F_[l$\"3Qw0'e%*Gj@#Fdw7$$\" 3#)*****z(e,tUF_[l$\"3+-dkg*>RH#Fdw7$$\"3G+++fO=yVF_[l$\"3yNbNa'fOP#Fd w7$$\"3u*****f#>#[Z%F_[l$\"3O2Rc'G[$\\CFdw7$$\"3)******pG!e&e%F_[l$\"3 x`]>tW!*QDFdw7$$\"3%)*****\\)Qk%o%F_[l$\"3!36D!Gc^@EFdw7$$\"3y*****>Mm -z%F_[l$\"3`#3n=[y@r#Fdw7$$\"3C+++60O\"*[F_[l$\"30De$)e\\U,GFdw7$$\"\" &\"\"!$\"#HF^gl-%'COLOURG6&%$RGBG$\"#5!\"\"$F^glF^glFhgl-%+AXESLABELSG 6$Q\"x6\"Q\"AF]hl-%%VIEWG6$;FhglF\\gl;Fhgl$\"#XF^gl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The minimum value of the surface area " }{TEXT 338 1 "A" }{TEXT -1 13 " occurs when " }{TEXT 327 1 "x" }{TEXT -1 15 " is about 2.1. " }}{PARA 0 "" 0 "" {TEXT -1 189 "To find this value mo re precisely we note that the minimum point occurs where the tangent l ine to the graph is horizontal, that is, where the gradient, as given \+ by the derivative, is zero. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 292 168 168 {PLOTDATA 2 "6,-%'CURVESG6$7U7$$\"3!************ **R\"!#<$\"3cG9dG9dC;!#;7$$\"3HLLLBxV<9F*$\"3\"38s^q4>h\"F-7$$\"3mmmmY a([V\"F*$\"3M;$>$fft*f\"F-7$$\"3)*****\\;z/]9F*$\"3-x*[=mG&*e\"F-7$$\" 3ILLL'Q?_Y\"F*$\"39Yt6^%p'z:F-7$$\"3mmmmWkM*\\\"F*$\"3S\")zu+&=(e:F-7$ $\"3dmmm7()pL:F*$\"3zP$z+Eg#R:F-7$$\"3ILLL5x)yc\"F*$\"3@=)\\8uG9_\"F-7 $$\"3_mmm(G&e*f\"F*$\"3O\\=E(Q\">1:F-7$$\"3%)*****HH1Cj\"F*$\"3_g`\"=@ g;\\\"F-7$$\"3]mmmB)\\jm\"F*$\"3q(*)>QM+zZ\"F-7$$\"3'******p\\%=+o=F*$\"3hh7=kpc>9F-7$$\"3cmmmV4_) *=F*$\"3\"*)>pk\")*)QT\"F-7$$\"3JLLLX$zX$>F*$\"3Zw!39F-7$$\"3GL LLTb7l>F*$\"3C6Ei@&=RS\"F-7$$\"3y******G!e1+#F*$\"3Y`^h[V$**R\"F-7$$\" 3ELLL,.6K?F*$\"3YYswrd9(R\"F-7$$\"3/+++H%=m1#F*$\"3ccl*ppb[R\"F-7$$\"3 #******pKy%*4#F*$\"3#Q;;@_)R$R\"F-7$$\"3=LLL$=kP8#F*$\"3!e%*f)ecg#R\"F -7$$\"3CLLL-$\\_;#F*$\"3)yW0+i6DR\"F-7$$\"3Ymmmc-@*>#F*$\"3s#)Q!=%*pIR \"F-7$$\"3ammmVh[MAF*$\"3?-230KN%R\"F-7$$\"3++++2R>lAF*$\"3sO@ljp.'R\" F-7$$\"3emmmK\"f$)H#F*$\"3mj;G'\\J%)R\"F-7$$\"3%******f0AEL#F*$\"3.F/) zy;:S\"F-7$$\"3m*****>kThO#F*$\"31Foui0709F-7$$\"3'******\\ct&)R#F*$\" 30t^[KW949F-7$$\"3')*****fo$eMCF*$\"3:UFOga@99F-7$$\"3/LLL\"QSpY#F*$\" 3M`%[LL+$>9F-7$$\"3z******f!)[,DF*$\"3OQp\\b$o_U\"F-7$$\"3^mmm\"R$zKDF *$\"3l:!Q4AY6V\"F-7$$\"3;+++)Q=qc#F*$\"3z29KaB2Q9F-7$$\"31LLLU9A*f#F*$ \"3Qo)f8Rc]W\"F-7$$\"3*******H\"H)Gj#F*$\"3SP0fW3$GX\"F-7$$\"3WLLL`Jzl EF*$\"3g2Ewp5*3Y\"F-7$$\"3v*****\\7Z-q#F*$\"3gs()>-k!)p9F-7$$\"3rmmm%R IMt#F*$\"3'oERr]X)y9F-7$$\"3ammm!3ltw#F*$\"39PKTv+a)[\"F-7$$\"38LLLq(= 5!GF*$\"3Q_jfLlf)\\\"F-7$$\"3()*****f,V>$GF*$\"3A8b@C!>#3:F-7$$\"3KLLL \"p&QnGF*$\"3v(f*GS&*o>:F-7$$\"3CmmmUg3**GF*$\"35\\@NzDMI:F-7$$\"3/+++ H_)G$HF*$\"3KHH'3$R5U:F-7$$\"3o*****HON_'HF*$\"3+i')Q\"zWPb\"F-7$$\"\" $\"\"!$\"3immmmmmm:F--%'COLOURG6&%$RGBG$\"*++++\"!\")$F`[lF`[lFj[l-F$6 $7$7$$\"33+++!pMWv\"F*$\"31+++]mZ#R\"F-7$$\"3!********oMWb#F*Fa\\l-%&C OLORG6&Ff[lF`[l$\"\"(!\"\"F`[l-F$6$7$7$$\"33++++++]8F*$\"33+++()ov0;F- 7$$\"3'*************\\>F*$\"3'******p`&)HO\"F--Fd[l6&Ff[lFj[lFj[lFg[l- F$6$7$7$$\"33++++++]BF*$\"3/+++RmS$Q\"F-7$$\"3=++++++]HF*$\"3#******HK F0`\"F--Fd[l6&Ff[lFg[lFj[lFg[l-%%TEXTG6%7$$\"$_\"!\"#$\"$Y\"F[]lQ*f'(x )~<~06\"Fi]l-F[_l6%7$$\"$:#F`_l$\"$P\"F[]lQ*f'(x)~=~0Fd_lFf\\l-F[_l6%7 $$\"$&GF`_lFa_lQ*f'(x)~>~0Fd_lFh^l-%+AXESLABELSG6%Q\"xFd_lQ!Fd_l-%%FON TG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$\"#9F[]lF^[lF[al" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(x)=A" "6#/-%\"f G6#%\"xG%\"AG" }{XPPEDIT 18 0 "`` = x^2+20*x^(-1);" "6#/%!G,&*$%\"xG\" \"#\"\"\"*&\"#?F))F',$F)!\"\"F)F)" }{TEXT -1 19 ", it follows that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*&%#dAG\"\"\"%#dxG!\"\"" }{XPPEDIT 18 0 "`` = 2*x-20 *x^(-2);" "6#/%!G,&*&\"\"#\"\"\"%\"xGF(F(*&\"#?F()F),$F'!\"\"F(F." } {XPPEDIT 18 0 "`` = 2*x-20/(x^2);" "6#/%!G,&*&\"\"#\"\"\"%\"xGF(F(*&\" #?F(*$F)F'!\"\"F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`f '`(x) = 0;" "6#/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 6 " w hen " }{XPPEDIT 18 0 "2*x=20/(x^2)" "6#/*&\"\"#\"\"\"%\"xGF&*&\"#?F&*$ F'F%!\"\"" }{TEXT -1 16 ", that is, when " }{XPPEDIT 18 0 "2*x^3=20" " 6#/*&\"\"#\"\"\"*$%\"xG\"\"$F&\"#?" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x^3=10" "6#/*$%\"xG\"\"$\"#5" }{TEXT -1 14 ", which gives " } {XPPEDIT 18 0 "x=10^(1/3)" "6#/%\"xG)\"#5*&\"\"\"F(\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "We can verify that the turnin g point on the graph " }{XPPEDIT 18 0 "A = f(x);" "6#/%\"AG-%\"fG6#%\" xG" }{TEXT -1 10 " given by " }{TEXT 348 1 "x" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "x[min] = 10^(1/3);" "6#/&%\"xG6#%$minG)\"#5*&\"\"\"F+\" \"$!\"\"" }{TEXT -1 1 " " }{TEXT 347 1 "~" }{TEXT -1 83 " 2.154434690 \+ is a minimum point by investigating the sign of the second derivative \+ " }{XPPEDIT 18 0 "`f ''`(x) = d^2*A/(d*x^2);" "6#/-%%f~''G6#%\"xG*(%\" dG\"\"#%\"AG\"\"\"*&F)F,*$)F'F*F,F,!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(x) = 2+40*x^(-3);" "6#/-% %f~''G6#%\"xG,&\"\"#\"\"\"*&\"#SF*)F',$\"\"$!\"\"F*F*" }{XPPEDIT 18 0 "``=2+40/x^3" "6#/%!G,&\"\"#\"\"\"*&\"#SF'*$%\"xG\"\"$!\"\"F'" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "`f ''`(x[min]) = ``;" "6#/-%%f~''G6 #&%\"xG6#%$minG%!G" }{XPPEDIT 18 0 "`f ''`(10^(1/3)) = 2+40/10;" "6#/- %%f~''G6#)\"#5*&\"\"\"F*\"\"$!\"\",&\"\"#F**&\"#SF*F(F,F*" }{XPPEDIT 18 0 "``=6" "6#/%!G\"\"'" }{TEXT -1 53 ". Since this value is positive , we can conclude that " }{XPPEDIT 18 0 "x=x[min]" "6#/%\"xG&F$6#%$min G" }{TEXT -1 35 " does indeed give a minimum point. " }}{PARA 0 "" 0 " " {TEXT -1 50 "Alternatively, writing the derivative in the form " } {XPPEDIT 18 0 "`f '`(x) = 2*(x^3-10)/(x^2);" "6#/-%$f~'G6#%\"xG*(\"\"# \"\"\",&*$)F'\"\"$F*F*\"#5!\"\"F**$)F'F)F*F0" }{TEXT -1 74 " helps in \+ determinimg the sign of the derivative in each of the intervals " } {XPPEDIT 18 0 "0 " } {XPPEDIT 18 0 "x[min]" "6#&%\"xG6#%$minG" }{TEXT -1 35 ", as shown in \+ the following table. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 284 104 104 {PLOTDATA 2 "68-%'CURVESG6$7$7$$!3++++++++N!#<$!3+++++++]7 F*7$$\"3++++++++bF*F+-%'COLOURG6&%$RGBG\"\"!F4F4-F$6$7$7$F($\"3+++++++ +]!#=7$F.F9F0-F$6$7$7$F($\"\"\"F47$F.FAF0-F$6$7$F'F@F0-F$6$7$7$$!3++++ ++++DF*F+7$FKFAF0-F$6$7$7$$!3++++++++]F;F+7$FRFAF0-F$6$7$7$F9F+7$F9FAF 0-F$6$7$7$$\"3++++++++DF*F+7$FhnFAF0-F$6'7$7$$!\"#F4$!\"$!\"\"7$$FcoF4 $!#5Fco7%7$$!+C.*p5\"!\"*$!+ggdN$)FgoFdo7$$!+w'4I>\"F\\p$!+SRUk&*Fgo-% &STYLEG6#%,PATCHNOGRIDGF0-%*THICKNESSG6#\"\"#-F$6'7$7$FAFeo7$$F[qF4Fao 7%7$$\"+C.*p!=F\\p$!*1wbV$F\\pF`q7$$\"+w'4I*=F\\p$!*%RUkYF\\pFdpF0Fhp- %%TEXTG6&7$$FboF4$\"#vF`oQ\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-F^r6&7$$! #:FcoFbrQ.~0~<~x~<~xminFerF0Ffr-F^r6&7$$F4F4FbrQ%xminFerF0Ffr-F^r6&7$$ \"#:FcoFbrQ)x~>~xminFerF0Ffr-F^r6&7$Far$F[qFcoQ'f~'(x)FerF0Ffr-F^r6&7$ FdsF_tQ\"0FerF0Ffr-F^r6&7$$\"\"$F4F_tFdtF0Ffr-F^r6&7$F^s$\"\"%FcoQ\"-F erF0-Fgr6$Fir\"#9-F^r6&7$Fis$FitFcoQ\"+FerF0F`u-%+AXESLABELSG6$Q!FerF[ v-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!#NFco$\"#DFco;$!#8Fco$\"#6Fco" 1 2 0 1 10 0 2 9 1 1 2 1.000000 47.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" }}{TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 109 "This provides another way to see that we have found a minimum point, independently of the graph drawn above. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The c orresponding value of " }{TEXT 341 1 "h" }{TEXT -1 3 " is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "h[min]=5/x[min]^2" "6#/&% \"hG6#%$minG*&\"\"&\"\"\"*$&%\"xG6#F'\"\"#!\"\"" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "5/(10^(2/3)) = 1/2;" "6#/*&\"\"&\"\"\")\"#5*&\"\"#F&\" \"$!\"\"F,*&F&F&F*F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "`.`*10^(1/3)" "6 #*&%\".G\"\"\")\"#5*&F%F%\"\"$!\"\"F%" }{TEXT -1 2 " " }{TEXT 342 1 " ~" }{TEXT -1 15 " 1.077217345, " }}{PARA 0 "" 0 "" {TEXT -1 31 "and t he corresponding value of " }{TEXT 346 1 "A" }{TEXT -1 4 " is " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[min] = ``(x[min])^2 +4*``(x[min])*``(h[min]);" "6#/&%\"AG6#%$minG,&*$-%!G6#&%\"xG6#F'\"\"# \"\"\"*(\"\"%F1-F+6#&F.6#F'F1-F+6#&%\"hG6#F'F1F1" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "10^(2/3)+4*`.`*10^(1/3)*`.`;" "6#,&)\"#5*&\"\"#\"\"\"\" \"$!\"\"F(**\"\"%F(%\".GF()F%*&F(F(F)F*F(F-F(F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`.`*10^(1/3) = 3*`.`*10^(2/3)" "6#/*&%\".G\"\"\")\"#5*& F&F&\"\"$!\"\"F&*(F*F&F%F&)F(*&\"\"#F&F*F+F&" }{TEXT -1 2 " " }{TEXT 343 1 "~" }{TEXT -1 14 " 13.92476650. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 57 "The dimensions of the most economical box are as follows." }}{PARA 0 "" 0 "" {TEXT -1 24 "Length of side of base: " }{XPPEDIT 18 0 "10^(1/3)" "6#)\"#5*&\"\"\"F&\"\"$!\"\"" } {TEXT -1 1 " " }{TEXT 344 1 "~" }{TEXT -1 24 " 2.154434690 metres. \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "Height: " }{XPPEDIT 18 0 "1/2" "6#*&\" \"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`.`*10^(1/3);" "6#*& %\".G\"\"\")\"#5*&F%F%\"\"$!\"\"F%" }{TEXT -1 1 " " }{TEXT 345 1 "~" } {TEXT -1 22 " 1.077217345 metres. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "f := x -> x^2+20/x;\nDiff(f(x),x);\nvalue(%);\nsimplify(%);\ndf \+ := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\" 6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"F1*&\"#?F1F/!\"\"F1F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,&*$)%\"xG\"\"#\"\"\"F+*&\"# ?F+F)!\"\"F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"%\"xG F&F&*&\"#?F&F'!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\" \"\",&*$)%\"xG\"\"$F&F&\"#5!\"\"F&F*!\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*(\"\"#\"\" \",&*$)9$\"\"$F/F/\"#5!\"\"F/F3!\"#F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The derivative is zero when " }{XPPEDIT 18 0 "x^3=10" "6#/*$%\"xG\"\" $\"#5" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 "x=10^(1/3)" "6#/% \"xG)\"#5*&\"\"\"F(\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "[solve(df(x) =0)]:\nop(remove(has,%, Complex(1)));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)\"#5#\"\"\"\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+!pMW:#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can also draw the box." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "x[min] := evalf(10^(1/3 ));\nh[min] := 5/x[min]^2; \ndrawbox(x[min],x[min],h[min],\n op en=true,color=[COLOR(RGB,1,.6,.6),coral]);\nA[min]=x[min]^2+4*x[min]*h [min];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#%$minG$\"+!pMW:#!\" *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"hG6#%$minG$\"+Xt@x5!\"*" }} {PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%)POLYGONSG6'7& 7%$\"\"!F)F(F(7%F($\"+!pMW:#!\"*F(7%F(F+$\"+Xt@x5F-7%F(F(F/7&F'7%F+F(F (7%F+F(F/F17&F37%F+F+F(7%F+F+F/F47&F6F*F.F7-%'COLOURG6&%$RGBG$\"*++++ \"!\")$\")AR!)\\F?F(-F$6$7&F'F3F6F*-%&COLORG6&F<\"\"\"$\"\"'!\"\"FI-%( SCALINGG6#%,CONSTRAINEDG-%&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G " 1 2 0 1 10 0 2 1 4 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"AG6#%$minG$\"+]mZ#R\"! \")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 160 "A farmer with 750 metres of fencing wants to enclose a r ectangular area and then sudivide it into four plots with fencing para llel to one side of the rectangle." }}{PARA 0 "" 0 "" {TEXT -1 142 "Wh at is the largest possible total area for the four plots, and what are the dimensions of the fenced enclosure when this maximum is achieved? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "The maximum area is 14062.5 square metres, obtained when the dime nsions of the fenced enclosure are 75 metres by 187.5 metres. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f := x -> (750*x-5*x^2)/2;\nDiff(f(x),x);\nvalue(%);\nsolve(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,&*&\"$v$\"\"\"9$F/F/*&#\"\"&\"\"#F/*$)F0F4F/F/!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,&*&\"$v$\"\"\"%\"xGF)F)*&# \"\"&\"\"#F)*$)F*F.F)F)!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\" $v$\"\"\"*&\"\"&F%%\"xGF%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#v " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(75);\nevalf(%);\n%/75;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"&D\"G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++ D19!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++v=!\"(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 249 "A farmer wants to fen ce an area of 15000 square metres in the form of a rectangular field a nd then divide it in half with a fence parallel to one side of the rec tangle. How can this be achieved so that the total length of fencing u sed is a minimum? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 135 "The dimensions of the field should be 100 metres \+ by 150 metres where the length of the fence which subdivides the field is 100 metres. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "f := x -> 3*x+30000/x;\nDiff(f(x),x);\nvalue(% );\nsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&*&\"\"$\"\"\"9$F/F/*&\"&++$F/F0!\"\"F/F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,&*&\"\"$\"\"\"%\"xGF)F)*&\" &++$F)F*!\"\"F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"$\"\"\"*&\" &++$F%%\"xG!\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$+\"!$+\"" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }} {PARA 0 "" 0 "" {TEXT -1 75 "Find the area of the largest rectangle wh ich has two vertices on the curve " }{XPPEDIT 18 0 "y=exp(-x^2)" "6#/% \"yG-%$expG6#,$*$%\"xG\"\"#!\"\"" }{TEXT -1 25 " and two vertices on t he " }{TEXT 302 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 120 "The maximum area is 554062.5 square metres, obtained whe n the dimensions of the field are 2500 metres by 7387.5 metres. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f := x -> 2*x*exp(-x^2);\nDiff(f(x),x);\nvalue(%);\nsolve(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,$*(\"\"#\"\"\"9$F/-%$expG6#,$*$)F0F.F/!\"\"F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,$*(\"\"#\"\"\"%\"xGF)-%$expG6#,$*$)F *F(F)!\"\"F)F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"#\"\"\"-%$ expG6#,$*$)%\"xGF%F&!\"\"F&F&*(\"\"%F&F,F&F'F&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&\"\"#!\"\"F%#\"\"\"F%F(,$*&F%F&F%F'F&" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "f(sqr t(2)/2);\nevalf(%);\nsqrt(2/exp(1));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"##\"\"\"F$-%$expG6##!\"\"F$F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+Z)Qwd)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\" \"##\"\"\"F$-%$expG6#F&#!\"\"F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +Y)Qwd)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 58 "Answer the question in example 3 with the assumption that " }{TEXT 303 28 "both vertical and horizontal" } {TEXT -1 56 " margins are 5 cm and that the total area is 225 sq. cm. " }}{PARA 0 "" 0 "" {TEXT -1 31 "Thus, in the relation between " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "y" " 6#%\"yG" }{TEXT -1 23 ", you should replace " }{XPPEDIT 18 0 "``(y-6 );" "6#-%!G6#,&%\"yG\"\"\"\"\"'!\"\"" }{TEXT -1 5 " by " }{XPPEDIT 18 0 " `` (y - 10)" "6#-%!G6#,&%\"yG\"\"\"\"#5!\"\"" }{TEXT -1 44 ", a nd the \"60\" should be replaced by \"225\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The outer dimensions of the printed page which give the \+ minimum printed area are 25 cm. horizontally and 25 cm. vertically." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "isolate((x-10)*(y-10)=225,y);\nsubs(%,x*y);\nnormal(%);\nf := u napply(%,x);\nDiff(f(x),x);\nvalue(%);\nnormal(%);\nsolve(%,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*&\"$D#\"\"\",&%\"xGF(\"#5!\" \"F,F(F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"xG\"\"\",&*&\"$D#F% ,&F$F%\"#5!\"\"F+F%F*F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"& \"\"\"%\"xGF&,&\"#DF&*&\"\"#F&F'F&F&F&,&F'F&\"#5!\"\"F.F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$**\" \"&\"\"\"9$F/,&\"#DF/*&\"\"#F/F0F/F/F/,&F0F/\"#5!\"\"F7F/F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,$**\"\"&\"\"\"%\"xGF),&\"#D F)*&\"\"#F)F*F)F)F),&F*F)\"#5!\"\"F1F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"&\"\"\",&\"#DF&*&\"\"#F&%\"xGF&F&F&,&F+F&\"#5!\"\"F.F&*( F-F&F+F&F,F.F&**F%F&F+F&F'F&F,!\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*(\"#5\"\"\",(*&\"#?F&%\"xGF&!\"\"\"$D\"F+*$)F*\"\"#F&F&F&,&F*F&F% F+!\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#D!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(x=25 ,225/(x-10)+10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#D" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 90 "The top and bottom mar gins of a poster are each 6 cm. and the side margins are each 4 cm. " }}{PARA 0 "" 0 "" {TEXT -1 118 "If the area of printed material is fix ed at 384 square cm., find the dimensions of the poster with the small est area. " }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________ ________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 275 "A rectangular box w ith an open top is to have a volume of 10 cubic metres. The length of \+ the base is twice the width. Material for the base costs $10 per squar e metre and material for the sides costs $6 per square metre. Find the cost of materials for the cheapest such box. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "f := x -> 20*x^2+180/x;;\nDiff(f(x),x);\nvalue(%);\n[solve(%)]:\nop(remove (has,%,Complex(1)));\nf(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"#?\"\"\")9$\"\"#F /F/*&\"$!=F/F1!\"\"F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Diff G6$,&*&\"#?\"\"\")%\"xG\"\"#F)F)*&\"$!=F)F+!\"\"F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#S\"\"\"%\"xGF&F&*&\"$!=F&F'!\"#!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"\"\"\"'#F%\"\"$\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#!*\"\"\")\"\"'#F&\"\"$F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M&3aj\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 17 "Code for \+ pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 27 "code for rectangular field " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1677 "A1 \+ := [0,0]: B1 := [4,0]: C1 := [4,2]: D1 := [0,2]:\nP1 := [-2.5,2]: Q1 : = [6.5,2]:\nf := x -> 2.8+0.015*sin(5*x):\np1 := plots[polygonplot]([A 1,B1,C1,D1],\n style=patchnogrid,color=COLOR(RGB,.7,.85,.7)):\np2 : = plot([D1,A1,B1,C1],color=brown,thickness=3):\ncv := op(op(1,op(1,plo t(f(x),x=-2.5..6.5)))):\np3 := plots[polygonplot]([[2,2],[-2.5,2],cv,[ 6.5,2],[2,2]],\n color=COLOR(RGB,.8,.8,1),style=patchnogri d):\np4 := plot(f(x),x=-2.5..6.5,color=navy):\np5 := plot([[-2.5,2],[6 .5,2]],color=navy):\np6 := plot([[[-2.5,2],[-2.5,f(-2.5)]],\n \+ [[6.5,2],[6.5,f(6.5)]]],color=navy,linestyle=2):\np7 := plottools[ar row]([-.4,1.2],[-.4,2],0,.15,.2,\n \+ arrow,color=brown):\np8 := plottools[arrow]([4.4,1.2],[4.4,2],0,.15,. 2,\n arrow,color=brown):\np9 := pl ottools[arrow]([-.4,.8],[-.4,0],0,.15,.2,\n \+ arrow,color=brown):\np10 := plottools[arrow]([4.4,.8],[4.4,0 ],0,.15,.2,\n arrow,color=brown): \np11 := plottools[arrow]([1.8,-.3],[0,-.3],0,.1,.1,\n \+ arrow,color=brown):\np12 := plottools[arrow]([2.2 ,-.3],[4,-.3],0,.1,.1,\n arrow,col or=brown):\np13 := plot([[[0,0],[-.8,0]],[[4,0],[4.8,0]],\n [[0,0] ,[0,-.5]],[[4,0],[4,-.5]]],color=brown,linestyle=2):\nt1 := plots[text plot]([[-.4,1.05,`y`],[4.4,1.05,`y`],\n [2,-.27,`x`]],colo r=brown):\nt2 := plots[textplot]([1.9,1.2,`A`],color=COLOR(RGB,0,.2,0) ):\nt3 := plots[textplot]([2,2.4,`river`],color=navy):\nplots[display] ([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,\n p11,p1 2,p13,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 828 "p1 := plot([[[-3.5,-1.25],[ 5.5,-1.25]],[[-3.5,.5],[5.5,.5]],\n [[-3.5,1],[5.5,1]],[[-3.5,-1.2 5],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5,-1.25],[-.5,1]],\n \+ [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]]],color=black):\np2 := plo ttools[arrow]([-2,-1],[-1,-.3],\n 0,.15,.15,arrow,color=black,t hickness=2):\np3 := plottools[arrow]([1,-.3],[2,-1],\n 0,.15,.1 5,arrow,color=black,thickness=2):\nt1 := plots[textplot]([[-3,.75,`x`] ,[-1.5,.75,` x < 150`],\n [0,.75,`150`],[1.5,.75,`x > 150`],[3,.75, `3`]],color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-3,.2 ,`f '(x)`],[0,.2,`0`],[3,.2,`0`]],\n color=black,font=[HELVETIC A,10]):\nt3 := plots[textplot]([[-1.5,.3,`+`],[1.5,.4,`_`]],\n \+ color=black,font=[HELVETICA,14]):\nplots[display]([p1,p2,p3,t1,t2,t3], axes=none,\n view=[-3.5..2.5,-1.3..1.1]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 29 "code for semi-circle picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 724 "h := evalf(sqrt(3)/2): v := 0.5:\np1 := plot([cos(t),sin(t),t=0..Pi],co lor=red):\np2 := plot([[h,0],[h,v],[-h,v],[-h,0]],\n color=COLOR (RGB,.9,.15,.85)):\np3 := plots[polygonplot]([[h,0],[h,v],[-h,v],[-h,0 ]],\n style=patchnogrid,color=COLOR(RGB,.9,.75,.85)):\np4 := plo t([[[h,v],[-h,v]]$3],color=COLOR(RGB,.6,0,1),\n style=point, symbol=[cross,circle,diamond]):\nt1 := plots[textplot]([[1.2,-.06,`x`] ,[-0.06,1.25,`y`],\n [-.04,1.08,`1`],[1,-.06,`1`],[-1,-.06,`-1`]], \n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot]([[1,. 58,`P(x,y)`],[-1,.58,`Q(-x,y)`]],\n color=COLOR(RGB,.5,0,.9)): \nplots[display]([p1,p2,p3,p4,t1,t2],tickmarks=[0,0],\n scaling=cons trained,view=[-1.1..1.2,-0.2..1.25]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "xmax := 'xmax':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 844 "p1 := plot([[[-3.5,-1.25],[ 5.5,-1.25]],[[-3.5,.5],[5.5,.5]],\n [[-3.5,1],[5.5,1]],[[-3.5,-1.2 5],[-3.5,1]],\n [[-2.5,-1.25],[-2.5,1]],[[-.5,-1.25],[-.5,1]],\n \+ [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2.5,1]]],color=black):\np2 := plo ttools[arrow]([-2,-1],[-1,-.3],\n 0,.15,.15,arrow,color=black,t hickness=2):\np3 := plottools[arrow]([1,-.3],[2,-1],\n 0,.15,.1 5,arrow,color=black,thickness=2):\nt1 := plots[textplot]([[-3,.75,`x`] ,[-1.5,.75,` x < xmax`],\n [0,.75,`xmax`],[1.5,.75,`x > xmax`],[3,. 75,`3`]],color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-3 ,.2,`f '(x)`],[0,.2,`0`],[3,.2,`0`]],\n color=black,font=[HELVE TICA,10]):\nt3 := plots[textplot]([[-1.5,.3,`+`],[1.5,.4,`_`],[4.5,.2, `+`]],\n color=black,font=[HELVETICA,14]):\nplots[display]([p1, p2,p3,t1,t2,t3],axes=none,\n view=[-3.5..2.5,-1.3..1.1]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "code for semi-circle animation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1789 " g := x -> x*sqrt(1-(x/2)^2):\np1 := plot([cos(t)-1.5,sin(t),t=0..Pi],c olor=red):\np2 := plot(g(x),x=0..2,color=red):\np3 := plot([[[-1.5,-.1 ],[-1.5,1.2]],[[0,-.1],[0,1.2]],\n [[-.1,0],[2.2,0]],[[-2.7,0],[-. 3,0]],\n [[-.02,1],[.02,1]]],color=black):\nt1 := plots[textplot]( [[2.15,-.06,`x`],[-.35,-.06,`x`],\n [-.1,.75,`A`],[-1.56,1.15,`y`], [2.,-.07,`1`],\n [-.06,1,`1`]],color=COLOR(RGB,.01,.01,.01)):\nfrm s := NULL:\nfor i from 40 to 0 by -1 do\n d := evalf(Pi/80);\n if \+ i<>0 and i<>40 then\n cs := cos(d*i); sn := sin(d*i);\n elif i= 0 then\n cs := 1; sn := 0;\n else\n cs := 0; sn := 1;\n \+ end if;\n x1 := -1.5+cs; x2 := -1.5-cs;\n xx := evalf(Pi/4+(cs-Pi /4)*0.9)*1.17-1.5;\n yy := evalf(Pi/4+(sn-Pi/4)*0.9)*1.1;\n p4 := \+ plot([[x1,0],[x1,sn],[x2,sn],[x2,0]],\n color=COLOR(RGB,.9,.15,. 85));\n p5 := plots[polygonplot]([[x1,0],[x1,sn],[x2,sn],[x2,0]],\n \+ style=patchnogrid,color=COLOR(RGB,.9,.75,.85));\n p6 := plot([ [[x1,sn]]$3],color=COLOR(RGB,.4,0,.9),\n style=point,symbol= [cross,circle,diamond]):\n p7 := plot([[[2*cs,g(2*cs)]]$3],style=poi nt,\n symbol=[cross,circle,diamond],color=brown):\n t2 := plots[textplot]([[1.2,.6,x=evalf[6](cs)],\n [1.2,.4,y=evalf[6](sn )],[1.2,.2,A=evalf[6](2*cs*sn)]],\n color=COLOR(RGB,.4,0 ,.9)):\n if i<>0 and i<>40 then\n t3 := plots[textplot]([xx,yy, `(x,y)`],\n color=COLOR(RGB,.4,0,.9)):\n frms := frms,plots[display]([p1,p2,p3,p4,p5,p6,p7,t1,t2,t3],\n \+ view=[-2.7..2.2,-.1..1.2]);\n else\n frms := frms,plots [display]([p1,p2,p3,p4,p5,p7,t1,t2],\n view=[-2.7..2.2,-.1..1.2] );\n end if; \n end do:\nplots[display]([frms],tickmarks=[0,0],\n view=[-2.7..2.2,-.1..1.2],scaling=unconstrained,\n insequence=tr ue,axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "cod e for printed page" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1957 "darkgreen := COLOR(RGB,0,.4,0):\nA1 := [0, 0]: B1 := [0,21]: C1 := [18,21]: D1 := [18,0]:\nA2 := [5,3]: B2 := [5, 18]: C2 := [13,18]: D2 := [13,3]:\np1 := plot([A1,B1,C1,D1,A1],color=b rown):\np2 := plot([A2,B2,C2,D2,A2],color=navy):\np3 := plots[polygonp lot]([A2,B2,C2,D2],color=COLOR(RGB,.8,.8,.685)):\np4 := plot([[[18,0], [21,0]],[[18,0],[18,-3]],\n [[18,21],[21,21]],[[0,0],[0,-3]]],\n \+ linestyle=2,color=darkgreen):\n\np5 := plottools[arrow]([20,11. 5],[20,21],0,.5,.08,\n arrow,color=da rkgreen):\np6 := plottools[arrow]([20,9.5],[20,0],0,.5,.08,\n \+ arrow,color=darkgreen):\np7 := plottools[arro w]([10,-2],[18,-2],0,.5,.08,\n arrow, color=darkgreen):\np8 := plottools[arrow]([8,-2],[0,-2],0,.5,.08,\n \+ arrow,color=darkgreen):\np9 := plottool s[arrow]([3,10.5],[5,10.5],0,.5,.2,\n \+ arrow,color=brown):\np10 := plottools[arrow]([2,10.5],[0,10.5],0,.5,. 2,\n arrow,color=brown):\np11 := plot tools[arrow]([16,10.5],[18,10.5],0,.5,.2,\n \+ arrow,color=brown):\np12 := plottools[arrow]([15,10.5],[13,10.5 ],0,.5,.2,\n arrow,color=brown):\np13 := plottools[arrow]([9,20.2],[9,21],0,.5,.4,\n \+ arrow,color=brown):\np14 := plottools[arrow]([9,18.8],[9,18 ],0,.5,.4,\n arrow,color=brown):\np15 := plottools[arrow]([9,2.2],[9,3],0,.5,.4,\n \+ arrow,color=brown):\np16 := plottools[arrow]([9,0.8],[9,0],0, .5,.4,\n arrow,color=brown):\nt1 := p lots[textplot]([[15.5,10.6,`5`],\n [2.5,10.6,`5`],[9,19.5,`3`],[9,1 .5,`3`]],color=brown):\nt2 := plots[textplot]([[20,10.7,'`y`'],\n [ 9,-1.8,'`x`']],color=darkgreen):\nplots[display]([p1,p2,p3,p4,p5,p6,p7 ,p8,p9,p10,p11,p12,p13,p14,\n p15,p16,t1,t2],axes=none); " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2457 "darkgreen := COLOR(RGB,0,.4,0):\nA1 := [0,0]: B1 := [0,21]: C1 : = [18,21]: D1 := [18,0]:\nA2 := [5,3]: B2 := [5,18]: C2 := [13,18]: D2 := [13,3]:\np1 := plot([A1,B1,C1,D1,A1],color=brown):\np2 := plot([A2 ,B2,C2,D2,A2],color=navy):\np3 := plots[polygonplot]([A2,B2,C2,D2],col or=COLOR(RGB,.8,.8,.685)):\np4 := plot([[[18,0],[21,0]],[[18,0],[18,-3 ]],\n [[18,21],[21,21]],[[0,0],[0,-3]]],\n linestyle=2,col or=darkgreen):\n\np5 := plottools[arrow]([20,11.5],[20,21],0,.5,.08,\n arrow,color=darkgreen):\np6 := plott ools[arrow]([20,9.5],[20,0],0,.5,.08,\n \+ arrow,color=darkgreen):\np7 := plottools[arrow]([10,-2],[18,-2],0,. 5,.08,\n arrow,color=darkgreen):\np8 \+ := plottools[arrow]([8,-2],[0,-2],0,.5,.08,\n \+ arrow,color=darkgreen):\np9 := plottools[arrow]([3,10.5],[5,1 0.5],0,.5,.2,\n arrow,color=brown):\n p10 := plottools[arrow]([2,10.5],[0,10.5],0,.5,.2,\n \+ arrow,color=brown):\np11 := plottools[arrow]([16,10.5] ,[18,10.5],0,.5,.2,\n arrow,color=bro wn):\np12 := plottools[arrow]([15,10.5],[13,10.5],0,.5,.2,\n \+ arrow,color=brown):\np13 := plottools[arrow]([ 9,20.2],[9,21],0,.5,.4,\n arrow,color =brown):\np14 := plottools[arrow]([9,18.8],[9,18],0,.5,.4,\n \+ arrow,color=brown):\np15 := plottools[arrow]([ 9,2.2],[9,3],0,.5,.4,\n arrow,color=b rown):\np16 := plottools[arrow]([9,0.8],[9,0],0,.5,.4,\n \+ arrow,color=brown):\np17 := plottools[arrow]([7.5, 13],[5,13],0,.5,.2,\n arrow,color=red ):\np18 := plottools[arrow]([10.5,13],[13,13],0,.5,.2,\n \+ arrow,color=red):\np19 := plottools[arrow]([11,11] ,[11,18],0,.5,.1,\n arrow,color=red): \np20 := plottools[arrow]([11,10],[11,3],0,.5,.1,\n \+ arrow,color=red):\nt1 := plots[textplot]([[15.5,10.6,`5 `],\n [2.5,10.6,`5`],[9,19.5,`3`],[9,1.5,`3`]],color=brown):\nt2 := plots[textplot]([[20,10.7,`y`],\n [9,-1.8,`x`]],color=darkgreen): \nt3 := plots[textplot]([[9,13,`x - 10`],\n [11,10.5,`y - 6`]] ,color=red):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p 13,p14,\n p15,p16,p17,p18,p19,p20,t1,t2,t3],axes=none); " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 819 "p1 := plot([[[-3.5,-1.25],[5.5,-1.25]],[[-3.5,.5],[5.5,.5]],\n \+ [[-3.5,1],[5.5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[-2. 5,1]],[[-.5,-1.25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2 .5,1]]],color=black):\np2 := plottools[arrow]([-2,-.3],[-1,-1],\n \+ 0,.15,.15,arrow,color=black,thickness=2):\np3 := plottools[arrow]([ 1,-1],[2,-.3],\n 0,.15,.15,arrow,color=black,thickness=2):\nt1 \+ := plots[textplot]([[-3,.75,`x`],[-1.5,.75,` 10 < x < 20`],\n [0,.7 5,`20`],[1.5,.75,`x > 20`],[3,.75,`3`]],color=black,font=[HELVETICA,10 ]):\nt2 := plots[textplot]([[-3,.2,`f '(x)`],[0,.2,`0`]],\n col or=black,font=[HELVETICA,10]):\nt3 := plots[textplot]([[-1.5,.4,`_`],[ 1.5,.3,`+`]],\n color=black,font=[HELVETICA,14]):\nplots[displa y]([p1,p2,p3,t1,t2,t3],axes=none,\n view=[-3.5..2.5,-1.3..1.1]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 33 "code for c ardboard sheet pictures" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 881 "p1 := plots[polygonplot]([[2,2],[8 ,2],[8,6],[2,6]],color=COLOR(RGB,.5,.8,.9)):\np2 := plots[polygonplot] ([[[0,2],[2,2],[2,6],[0,6]],\n [[8,2],[10,2],[10,6],[8,6]],[[2,0],[8, 0],[8,2],[2,2]],\n [[2,6],[8,6],[8,8],[2,8]]],color=COLOR(RGB,.5,.9, .8)):\np3 := plot([[[0,6],[0,8],[2,8]],[[8,8],[10,8],[10,6]],\n [[ 10,2],[10,0],[8,0]],[[0,2],[0,0],[2,0]]],linestyle=3,color=navy):\np4 \+ := plot([[[10,8],[11.5,8]],[[10,0],[11.5,0]],[[0,8],[0,9.5]],\n [[1 0,8],[10,9.5]]],linestyle=2,color=brown):\np5 := plottools[arrow]([11, 4.5],[11,8],0,.25,.1,arrow,color=brown):\np6 := plottools[arrow]([11,3 .5],[11,0],0,.25,.1,arrow,color=brown):\np7 := plottools[arrow]([4.2,9 ],[0,9],0,.25,.07,arrow,color=brown):\np8 := plottools[arrow]([5.8,9], [10,9],0,.25,.07,arrow,color=brown):\nt1 := plots[textplot]([[5,9,`10 \+ cm`],[11,4,`8 cm`]],color=brown):\nplots[display]([p1,p2,p3,p4,p5,p6,p 7,p8,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1361 "p1 := p lots[polygonplot]([[2,2],[8,2],[8,6],[2,6]],color=COLOR(RGB,.5,.8,.9)) :\np2 := plots[polygonplot]([[[0,2],[2,2],[2,6],[0,6]],\n [[8,2],[10, 2],[10,6],[8,6]],[[2,0],[8,0],[8,2],[2,2]],\n [[2,6],[8,6],[8,8],[2, 8]]],color=COLOR(RGB,.5,.9,.8)):\np3 := plot([[[0,6],[0,8],[2,8]],[[8, 8],[10,8],[10,6]],\n [[10,2],[10,0],[8,0]],[[0,2],[0,0],[2,0]]],li nestyle=3,color=navy):\np4 := plot([[[10,8],[11.5,8]],[[10,0],[11.5,0] ],[[0,8],[0,9.5]],\n [[10,8],[10,9.5]]],linestyle=2,color=brown):\n p5 := plottools[arrow]([11,4.5],[11,8],0,.25,.1,arrow,color=brown):\np 6 := plottools[arrow]([11,3.5],[11,0],0,.25,.1,arrow,color=brown):\np7 := plottools[arrow]([4.5,9],[0,9],0,.25,.07,arrow,color=brown):\np8 : = plottools[arrow]([5.5,9],[10,9],0,.25,.07,arrow,color=brown):\np9 := plottools[arrow]([7,4.5],[7,6],0,.25,.1,arrow,color=red):\np10 := plo ttools[arrow]([7,3.5],[7,2],0,.25,.1,arrow,color=red):\np11 := plottoo ls[arrow]([4.1,5],[2,5],0,.25,.1,arrow,color=red):\np12 := plottools[a rrow]([5.9,5],[8,5],0,.25,.1,arrow,color=red):\nt1 := plots[textplot]( [[-.3,7,`x`],[1,8.3,`x`],[9,8.3,`x`],[10.3,7,`x`],\n [10.3,1,`x`],[9,- .3,`x`],[1,-.3,`x`],[-.3,1,`x`]],color=navy):\nt2 := plots[textplot]([ [5,9,`10`],[11,4,`8`]],color=brown):\nt3 := plots[textplot]([[5,5,`10 \+ - 2 x`],[7,4,`8 - 2 x`]],color=red):\nplots[display]([p1,p2,p3,p4,p5,p 6,p7,p8,p9,p10,p11,p12,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 "code for bending flaps aimation " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 647 "A1 := [0,0,0]: B1 := [6,0,0]: \nC1 := [6,4,0]: D1 := [0,4,0]:\nd \+ := evalf(Pi/40):i:=10:\np1:=plots[polygonplot3d]([A1,B1,C1,D1],color=C OLOR(RGB,.5,.8,.9)):\nfrms := NULL:\nfor i from 0 to 20 do\n h := 2* cos(d*i);\n v := 2*sin(d*i); \n A2 := [-h,0,v]; B2 := [6+h,0,v]; \+ \n C2 := [6+h,4,v]; D2 := [-h,4,v];\n A3 := [0,-h,v]; B3 := [6,-h, v];\n C3 := [6,4+h,v]; D3 := [0,4+h,v];\n plt := plots[polygonplot 3d]([[A1,D1,D2,A2],[B1,C1,C2,B2],\n [A1,B1,B3,A3],[C1,D1,D3,C3]] ,color=COLOR(RGB,.5,.9,.8)); \n frms := frms,display([p1,plt]):\nend do:\nplots[display]([frms],insequence=true,\n scaling=constrained,ori entation=[-135,50],axes=none);\ni := 'i':" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 31 "code for e conomical box picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1269 "A1 := [0,0]: B1 := [0,3]: C1 := [2,3]: \+ D1 := [2,0]:\ncs := evalf(cos(Pi/5)): sn := evalf(sin(Pi/5)):\nh := 1. 5*cs: v := 1.5*sn:\nA2 := [h,v]: B2 := [h,3+v]: C2 := [2+h,3+v]: D2 := [2+h,v]:\nA3 := [-.6,0]: B3 := [-.6,3]: A4 := [0,-.6]: D3 := [2,-.6]: \nD4:= [2.6,0]: D5 := [2.6+h,v]:\np1 := plot([[D1,A1,B1,B2,C2,D2,D1,C 1,B1],[C1,C2]],color=brown):\np2 := plot([[A1,A2,D2],[A2,B2]],color=ta n,linestyle=3):\np3 := plots[polygonplot]([A1,B1,C1,D1],color=COLOR(RG B,1,.8,.65)):\np4 := plots[polygonplot]([D2,C2,C1,D1],color=COLOR(RGB, 1,.77,.62)):\np5 := plots[polygonplot]([B1,B2,C2,C1],color=COLOR(RGB,1 ,.85,.7)):\np6 := plot([[A3,A1,A4],[B3,B1],[D3,D1,D4],[D2,D5]],color=b rown,linestyle=2):\np7 := plottools[arrow]([-.3,1.3],[-.3,0],0,.1,.1,a rrow,color=tan):\np8 := plottools[arrow]([-.3,1.7],[-.3,3],0,.1,.1,arr ow,color=tan):\np9 := plottools[arrow]([.8,-.3],[0,-.3],0,.1,.15,arrow ,color=tan):\np10 := plottools[arrow]([1.2,-.3],[2,-.3],0,.1,.15,arrow ,color=tan):\np11 := plottools[arrow]([2.3+h*.42,v*.42],[2.3,0],0,.1,. 18,arrow,color=tan):\np12 := plottools[arrow]([2.3+h*.58,v*.58],[2.3+h ,v],0,.1,.18,arrow,color=tan):\nt1 := plots[textplot]([[-.3,1.5,'`h`'] ,[1,-.3,'`x`'],\n [2.3+h*.5,v*.5,'`x`']],color=brown):\nplots[displa y]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,t1],axes=none); " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 823 "p1 := plot([[[-3.5,-1.25],[5.5,-1.25]],[[-3.5,.5],[5.5,.5]],\n \+ [[-3.5,1],[5.5,1]],[[-3.5,-1.25],[-3.5,1]],\n [[-2.5,-1.25],[-2. 5,1]],[[-.5,-1.25],[-.5,1]],\n [[.5,-1.25],[.5,1]],[[2.5,-1.25],[2 .5,1]]],color=black):\np2 := plottools[arrow]([-2,-.3],[-1,-1],\n \+ 0,.15,.15,arrow,color=black,thickness=2):\np3 := plottools[arrow]([ 1,-1],[2,-.3],\n 0,.15,.15,arrow,color=black,thickness=2):\nt1 \+ := plots[textplot]([[-3,.75,`x`],[-1.5,.75,` 0 < x < xmin`],\n [0,. 75,`xmin`],[1.5,.75,`x > xmin`]],color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-3,.2,`f '(x)`],[0,.2,`0`],[3,.2,`0`]],\n \+ color=black,font=[HELVETICA,10]):\nt3 := plots[textplot]([[-1.5,.4,`- `],[1.5,.3,`+`]],\n color=black,font=[HELVETICA,14]):\nplots[di splay]([p1,p2,p3,t1,t2,t3],axes=none,\n view=[-3.5..2.5,-1.3..1.1 ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 355 144 144 {PLOTDATA 2 "68-%'CURVES G6$7$7$$!3++++++++N!#<$!3+++++++]7F*7$$\"3++++++++bF*F+-%'COLOURG6&%$R GBG\"\"!F4F4-F$6$7$7$F($\"3++++++++]!#=7$F.F9F0-F$6$7$7$F($\"\"\"F47$F .FAF0-F$6$7$F'F@F0-F$6$7$7$$!3++++++++DF*F+7$FKFAF0-F$6$7$7$$!3+++++++ +]F;F+7$FRFAF0-F$6$7$7$F9F+7$F9FAF0-F$6$7$7$$\"3++++++++DF*F+7$FhnFAF0 -F$6'7$7$$!\"#F4$!\"$!\"\"7$$FcoF4$!#5Fco7%7$$!+C.*p5\"!\"*$!+ggdN$)Fg oFdo7$$!+w'4I>\"F\\p$!+SRUk&*Fgo-%&STYLEG6#%,PATCHNOGRIDGF0-%*THICKNES SG6#\"\"#-F$6'7$7$FAFeo7$$F[qF4Fao7%7$$\"+C.*p!=F\\p$!*1wbV$F\\pF`q7$$ \"+w'4I*=F\\p$!*%RUkYF\\pFdpF0Fhp-%%TEXTG6&7$$FboF4$\"#vF`oQ\"x6\"F0-% %FONTG6$%*HELVETICAG\"#5-F^r6&7$$!#:FcoFbrQ.~0~<~x~<~xminFerF0Ffr-F^r6 &7$$F4F4FbrQ%xminFerF0Ffr-F^r6&7$$\"#:FcoFbrQ)x~>~xminFerF0Ffr-F^r6&7$ Far$F[qFcoQ'f~'(x)FerF0Ffr-F^r6&7$FdsF_tQ\"0FerF0Ffr-F^r6&7$$\"\"$F4F_ tFdtF0Ffr-F^r6&7$F^s$\"\"%FcoQ\"-FerF0-Fgr6$Fir\"#9-F^r6&7$Fis$FitFcoQ \"+FerF0F`u-%+AXESLABELSG6$Q!FerF[v-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$! #NFco$\"#DFco;$!#8Fco$\"#6Fco" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 27 "code \+ for gradient pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 608 "g := x->150*x-x^2/2:\np1 := plot(g (x),x=110..190,color=red):\ndg := x->150-x:\na := 150: b := g(a): d := 20:\np2:=plot([[a-d,b],[a+d,b]],color=COLOR(RGB,0,.7,0)):\na := 125: \+ b := g(a): m := dg(a): d := 10:\np3:=plot([[a-d,b-m*d],[a+d,b+m*d]],co lor=magenta,thickness=2):\na := 175: b := g(a): m := dg(a): d := 10:\n p4:=plot([[a-d,b-m*d],[a+d,b+m*d]],color=blue,thickness=2):\nt1 := plo ts[textplot]([116,11000,`f'(x) > 0`],color=magenta):\nt2 := plots[text plot]([150,11350,`f'(x) = 0`],color=COLOR(RGB,0,.7,0)):\nt3 := plots[t extplot]([184,11000,`f'(x) < 0`],color=blue):\nplots[display]([p1,p2,p 3,p4,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 647 "g := x->2*x*sqrt(1-x^2):\np1 := pl ot(g(x),x=0.62..0.78,color=red):\ndg := x->2*(1-2*x^2)/sqrt(1-x^2):\na := evalf(1/sqrt(2)): b := g(a): d := .03:\np2:=plot([[a-d,b],[a+d,b]] ,color=COLOR(RGB,0,.7,0)):\na := .65: b := g(a): m := dg(a): d := .02: \np3:=plot([[a-d,b-m*d],[a+d,b+m*d]],color=magenta,thickness=2):\na := .76: b := g(a): m := dg(a): d := .02:\np4:=plot([[a-d,b-m*d],[a+d,b+m *d]],color=blue,thickness=2):\nt1 := plots[textplot]([.636,.99,`f'(x) \+ > 0`],color=magenta):\nt2 := plots[textplot]([.71,1.0035,`f'(x) = 0`], color=COLOR(RGB,0,.7,0)):\nt3 := plots[textplot]([.776,.99,`f'(x) < 0` ],color=blue):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],axes=none);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 596 "g := x->6*x^2/(x-10):\np1 := plot(g(x),x=13.3..35,color=red):\ndg := x->6*x*(x-20)/(x-10)^2:\na := 20: b := g(a): d := 4:\np2:=plot([[a -d,b],[a+d,b]],color=COLOR(RGB,0,.7,0)):\na := 15: b := g(a): m := dg( a): d := 2:\np3:=plot([[a-d,b-m*d],[a+d,b+m*d]],color=blue):\na := 30: b := g(a): m := dg(a): d := 5:\np4:=plot([[a-d,b-m*d],[a+d,b+m*d]],co lor=magenta,thickness=2):\nt1 := plots[textplot]([14,251,`f'(x) > 0`], color=blue):\nt2 := plots[textplot]([20,234,`f'(x) = 0`],color=COLOR(R GB,0,.7,0)):\nt3 := plots[textplot]([30,254,`f'(x) < 0`],color=magenta ):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 611 "g := x ->x*(10-2*x)*(8-2*x):\np1 := plot(g(x),x=.9..2.2,color=red):\ndg := x- >80-72*x+12*x^2:\na := evalf(3-sqrt(21)/3): b := g(a): d := .4:\np2:=p lot([[a-d,b],[a+d,b]],color=COLOR(RGB,0,.7,0)):\na := 1.2: b := g(a): \+ m := dg(a): d := .3:\np3:=plot([[a-d,b-m*d],[a+d,b+m*d]],color=magenta ):\na := 1.8: b := g(a): m := dg(a): d := .3:\np4:=plot([[a-d,b-m*d],[ a+d,b+m*d]],color=blue):\nt1 := plots[textplot]([1.05,51,`f'(x) > 0`], color=magenta):\nt2 := plots[textplot]([1.75,53.5,`f'(x) = 0`],color=C OLOR(RGB,0,.7,0)):\nt3 := plots[textplot]([1.95,51,`f'(x) < 0`],color= blue):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 604 "g : = x -> x^2+20/x:\np1 := plot(g(x),x=1.4..3,color=red):\ndg := x->2*(x^ 3-10)/x^2:\na := evalf(10^(1/3)): b := g(a): d := .4:\np2:=plot([[a-d, b],[a+d,b]],color=COLOR(RGB,0,.7,0)):\na := 1.65: b := g(a): m := dg(a ): d := .3:\np3:=plot([[a-d,b-m*d],[a+d,b+m*d]],color=blue):\na := 2.6 5: b := g(a): m := dg(a): d := .3:\np4:=plot([[a-d,b-m*d],[a+d,b+m*d]] ,color=magenta):\nt1 := plots[textplot]([1.52,14.6,`f'(x) < 0`],color= blue):\nt2 := plots[textplot]([2.15,13.7,`f'(x) = 0`],color=COLOR(RGB, 0,.7,0)):\nt3 := plots[textplot]([2.85,14.6,`f'(x) > 0`],color=magenta ):\nplots[display]([p1,p2,p3,p4,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "6 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }