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1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {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 "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 "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 "Times" 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 16 "by Peter Stone. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Versi on: 17.11.2006" }}{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 "Suggestions specific to the solut ion of max/min problems " }}{PARA 15 "" 0 "" {TEXT -1 99 "Define any s ymbols you wish to use which are not already specified in the statemen t of the problem." }}{PARA 15 "" 0 "" {TEXT -1 29 "Make a sketch if ap propriate." }}{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 var iable (say n variables) find n-1 equations relating these variables (c onstraints).\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 variables and hence express Q as \+ a function of one variable, say " }{TEXT 283 1 "x" }{TEXT -1 2 ". " }} {PARA 15 "" 0 "" {TEXT -1 47 "Determine the interval I in which the va riable " }{TEXT 285 1 "x" }{TEXT -1 40 " must lie for the problem to m ake 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 (w here the derivative " }{XPPEDIT 18 0 "dQ/dx" "6#*&%#dQG\"\"\"%#dxG!\" \"" }{TEXT -1 46 " is zero) or at an endpoint of the interval I." }} {PARA 15 "" 0 "" {TEXT -1 183 "Give some justification that one partic ular value is the required extreme value. Probably the safest and most instructive way to do this is to make a sketch of the graph of Q agai nst " }{TEXT 284 1 "x" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 58 "Make a concluding statement answering the given question. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 " General advice for problem solving " }}{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 Geomet ry\", 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 and other courses is devoted to solving problems. It is worth your while to develop sound techniques. Here ar e a few suggestions." }}{PARA 15 "" 0 "" {TEXT 259 5 "Think" }{TEXT -1 350 ": Before plunging into a problem, take a moment to think. Read the problem again. Think about it. What are its essential features? H ave you seen a problem like it before? What techniques are needed?\nTr y to make a rough estimate of the answer. It will help you understand \+ the problem and will serve as a check against unreasonable answers. A \+ car will " }{TEXT 260 3 "not" }{TEXT -1 61 " go 1000 km in 3 hr; a wei ght dropped from 3,000 metres will " }{TEXT 260 3 "not" }{TEXT -1 65 " hit the earth at 5 km. per hour; 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 understan d what is given. Translate the data into mathematical language. Whenev er 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 typical 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 mean s \"equals\" or \"is equal to\". The equal sign is not to be used in p lace of \"and\", nor as a punctuation mark. " }{TEXT 287 1 "*" }{TEXT -1 60 "Quantities on opposite sides of an equal sign must be equal." } {TEXT 286 1 "*" }{TEXT -1 282 "\nUse short simple sentences. Avoid pro nouns such as \"it\" and \"which\". Give names and use them. Otherwise you may write gibberish like the following: \"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 derivative " }{XPPEDIT 18 0 "`f ' `(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 20 " equal to zero. Let " } {XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 49 " be the solution \+ of the resulting equation. Then " }{XPPEDIT 18 0 "f(x[0])" "6#-%\"fG6# &%\"xG6#\"\"!" }{TEXT -1 25 " is the minimum value of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{XPPEDIT 18 0 ")" "6#%#%?G" }{TEXT -1 7 " .\"\n(b) " }{TEXT 261 31 "Avoid sloppiness in computation" }{TEXT -1 274 ". \nDo calculations in a sequence of neat, orderly steps. Include all steps except utterly trivial ones. This will help eliminate error s, or at least make errors easier to find. Check any numbers used; be \+ sure that you have not dropped a minus sign or transposed digits.\n(c) " }{TEXT 261 25 "Avoid sloppiness in units" }{TEXT -1 199 ". \nIf you start out measuring lengths in metres, all lengths must be in metres, all areas in square metres, and all volumes in cubic metres. Do not m ix metres and millimetres, seconds and years.\n(d) " }{TEXT 261 30 "Av oid sloppiness in the answer" }{TEXT -1 77 ". \nBe sure to answer the \+ question that is asked. If the problem asks for the " }{TEXT 260 13 "m aximum value" }{TEXT -1 6 " of f(" }{TEXT 288 1 "x" }{TEXT -1 47 "), t he answer is not the point on the graph of " }{XPPEDIT 18 0 "f(x)" "6# -%\"fG6#%\"xG" }{TEXT -1 53 " where the maximum occurs. If the problem asks for a " }{TEXT 260 7 "formula" }{TEXT -1 30 ", the answer is not a number.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "Example 1 - rectangular enclosure fenced on 3 sides " }}{PARA 0 "" 0 "" {TEXT 281 8 "Question" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 278 "A farmer wants to construct a rectangular enclosure. Fencing is needed for just three sides of th e enclosure because one side is formed by the bank of a long straight \+ river. If 300 metres of fencing is available for the job, what is the \+ largest possible area for the enclosure? 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P4'4+'zbFgcq$\"3!GFgcq7$$\"3Y**\\P*)>OacFgcq$\"3v,8**e y.+GFgcq7$$\"3)*[(o/i#GzcFgcq$\"39w.ycL<)z#Fgcq7$$\"3T*\\i:D.Uq&Fgcq$ \"3[5'f#yrL'z#Fgcq7$$\"3$)\\il#)Q7HdFgcq$\"3]$Hs\\zdXz#Fgcq7$$\"3Q*** \\P^WSv&Fgcq$\"3UuA!yziGz#Fgcq7$$\"3u*\\7G8B')z&Fgcq$\"3/-\"3_(H7!z#Fg cq7$$\"37+](=v,K%eFgcq$\"3-!ea\"G=(yy#Fgcq7$$\"3]+v$4P!y()eFgcq$\"3c') =OM2A'y#Fgcq7$$\"3*3+++**eB$fFgcq$\"3C[MK$Q^_y#Fgcq7$$\"328`W;:CWfFgcq $\"3#zT6)G]6&y#Fgcq7$$\"3CD1*G/Ch&fFgcq$\"3)z1B2?J]y#Fgcq7$$\"3WPfLpl+ ofFgcq$\"3KbG([>+]y#Fgcq7$$\"3]]7y&4*))zfFgcq$\"3%z#\\n?@-&y#Fgcq7$$\" 3vv=n[Tl.gFgcq$\"3QxMBrUA&y#Fgcq7$$\"36+Dc,#>u-'Fgcq$\"3sB@],[j&y#Fgcq 7$$\"3s\\PM2$\\\\2'Fgcq$\"3W#=2b)\\0(y#Fgcq7$$\"3L**\\78%zC7'Fgcq$\"3G t.lSG?*y#Fgcq7$$\"3%*[P4,cAXhFgcq$\"3e'[*>tUX!z#Fgcq7$$\"3U*\\i!*yrz;' Fgcq$\"3_US3W!H=z#Fgcq7$$\"3#*\\7.xzr!>'Fgcq$\"3_evF7F]jwQ\"AFg`x-F26&F4F)$F/F7F)-Fa`x6%7$F.$\"#CF7Q&riverFg `xF]hw-%+AXESLABELSG6%Q!Fg`xFcbx-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%N ONEG-%%VIEWG6$;FTFjbqFgbx" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 282 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 67 "Let the length of the enclosure (in the directi on of the river) be " }{TEXT 289 1 "x" }{TEXT -1 22 " metres, the widt h be " }{TEXT 290 1 "y" }{TEXT -1 24 " metres and the area be " } {TEXT 291 1 "A" }{TEXT -1 15 " square metres." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " A=x*y" "6#/%\"AG*&%\"xG\"\"\"%\"yGF'" }{TEXT -1 15 " ------- (i). " } }{PARA 0 "" 0 "" {TEXT -1 101 "Since 300 metres of fencing is availabl e, the area will be a maximum when all of the fencing is used." }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x+2*y = 300;" "6#/,&%\"xG\"\"\"*&\"\"#F&%\"yGF&F&\"$ +$" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 28 "Thi s equation constitutes a " }{TEXT 259 10 "constraint" }{TEXT -1 17 " f or the problem." }}{PARA 0 "" 0 "" {TEXT -1 45 "We can use equation (i i) to express the area " }{TEXT 292 1 "A" }{TEXT -1 41 " as a function of a single variable, say " }{TEXT 295 1 "x" }{TEXT -1 27 ", since we can substitute " }{XPPEDIT 18 0 "y = (300-x)/2;" "6#/%\"yG*&,&\"$+$ \"\"\"%\"xG!\"\"F(\"\"#F*" }{TEXT -1 19 " in equation (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 346 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;\n plot(f(x),x=0..300,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,&*&\"$]\"\"\"\"9$F/F/*&#F/\"\"#F/*$)F0F 3F/F/!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3%*******\\i9Rl!#<$\"39abg1s\"\\f* !#:7$$\"3*****\\PC#)GA\"!#;$\"3?[5_;8bffS*\\F4$\"3Rb54Hu0WiF77$$\"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$$\"3-+]P>:mk:F0$\"3Reg&RW4H7\"Fhp7$$\"3,+]iv&QAi\"F0$ \"3`&e9`')Gv6\"Fhp7$$\"3%****\\PPBWo\"F0$\"3$QL2'4S*z5\"Fhp7$$\"3-+++b jm[_#3%4\"Fhp7$$\"3!)***\\(yb^6=F0$\"3KE!)4A!zk2\"Fhp7$$\" 3)***\\PMaKs=F0$\"3uL(ea)oob5Fhp7$$\"3*)***\\7TW)R>F0$\"3iqg%pWo#G5Fhp 7$$\"3s*****\\@80+#F0$\"3=A=03EV(***F77$$\"3'*****\\7,Hl?F0$\"3+&o[NWN Al*F77$$\"3')**\\P4w)R7#F0$\"36jBt;t>.$*F77$$\"3?++]x%f\")=#F0$\"3!=jM wm#=#)))F77$$\"3q**\\P/-a[AF0$\"3-n6<7yV[%)F77$$\"38+](=Yb;J#F0$\"3#*) RUg0xg&zF77$$\"3y****\\i@OtBF0$\"3:@mclE>OuF77$$\"3v**\\PfL'zV#F0$\"3q D)\\BoB6&oF77$$\"35+++!*>=+DF0$\"3--#)RM*z\"[iF77$$\"3?++DE&4Qc#F0$\"3 C[()Hfka\"f&F77$$\"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(-%+AXESLABE LSG6$Q\"x6\"Q\"AFb[l-%%VIEWG6$;F(Fez%(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 40 "It looks as though the maximum value of " }{TEXT 293 1 "A" }{TEXT -1 14 " occurs where " }{XPPEDIT 18 0 "x =150" "6#/% \"xG\"$]\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 167 "To check \+ this, we note that the maximum point occurs where the tangent line to \+ the graph is horizontal, that is, where the slope, as given by the der ivative, is zero. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 278 126 126 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$\"$5\"\"\"!$\"&]/\"F*7$$\"3RLL LBxV<6!#:$\"3i3eB_I#=0\"!#87$$\"3imm;$>5E8\"F0$\"39$[<\\O7v0\"F37$$\"3 MLLLAKn\\6F0$\"3!\\%[Uubjj5F37$$\"3QLLLc$\\o;\"F0$\"35VAJC`]p5F37$$\"3 gmm;bQ%R=\"F0$\"3[XllcU0v5F37$$\"3ILL$Qk#z*>\"F0$\"395;6F37$$\"3cmmmqFc#Q\"F0$\"39bKe[U5=6F37$$\"3(*****\\9!H.S\"F0$\"3IHZK ZG.?6F37$$\"3pmmm]^0;9F0$\"37-_8JmZ@6F37$$\"30++]9#4LV\"F0$\"3Z'pk&phx A6F37$$\"3'*****\\j\"R(\\9F0$\"39gr:CptB6F37$$\"3mmmm\"4#)oY\"F0$\"3K7 Q2-;XC6F37$$\"3wmm;^Yi#[\"F0$\"3\")fki[!\\[7\"F37$$\"3CLLLG^g*\\\"F0$ \"3H$=Q?#***\\7\"F37$$\"3PLL$=2Vs^\"F0$\"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$$\"3h mmm!>qMj\"F0$\"3'>q,T&G4;6F37$$\"3=+++ISu];F0$\"3[z'4(=\"QO6\"F37$$\"3 ILL$ep'Rm;F0$\"3_uy!)pg:66F37$$\"35+++%>4No\"F0$\"3;N(ey=i\"36F37$$\"3 cmm;@2h*p\"F0$\"3'=m(**zx206F37$$\"38++]c9W;P4\"F3 7$$\"3aLLL(>:nw\"F0$\"3Y?dv,:V*3\"F37$$\"3eLLLSDo$y\"F0$\"3E8]:3@w%3\" F37$$\"3!pmm^Q40!=F0$\"3-wLraq%)z5F37$$\"34+++3:(f\"=F0$\"3A.h1.53v5F3 7$$\"3$pmmc%GpL=F0$\"3!3WvBaC$p5F37$$\"3GLLL@Ia\\=F0$\"3U#e=\"Q)4R1\"F 37$$\"3))****\\9EWm=F0$\"3C-#R^!*fy0\"F37$$\"3A++]\"oy,\\& =!=0\"F37$$\"$!>F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*F^[l-F$6$7$7$ $\"$I\"F*$\"&]7\"F*7$$\"$q\"F*Fe[l-%&COLORG6&FjzF*$\"\"(!\"\"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$$\"$%=F*Fd^lQ*f'(x)~ <~0Fg^lF\\^l-%+AXESLABELSG6%Q\"xFg^lQ!Fg^l-%%FONTG6#%(DEFAULTG-%*AXESS TYLEG6#%%NONEG-%%VIEWG6$;F(FezF^`l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 47.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dA/dx=150-x " "6#/*&%#dAG\"\"\"%#dxG!\"\",&\"$]\"F&%\"xGF(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 15 "The derivative " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*&%#dAG\"\"\"%#dxG!\"\"" }{TEXT -1 11 " i s 0 when " }{XPPEDIT 18 0 "x=150" "6#/%\"xG\"$]\"" }{TEXT -1 15 ", as \+ suggested." }}{PARA 0 "" 0 "" {TEXT -1 21 "The maximum value of " } {TEXT 329 1 "A" }{TEXT -1 14 " is therefore " }{XPPEDIT 18 0 "f(150) = 150*``((300-150)/2);" "6#/-%\"fG6#\"$]\"*&F'\"\"\"-%!G6#*&,&\"$+$F)F' !\"\"F)\"\"#F0F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(150)*`.`*``(75) = 11250;" "6#/*(-%!G6#\"$]\"\"\"\"%\".GF)-F&6#\"#vF)\"&]7\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "We can check that " }{XPPEDIT 18 0 "x=150" "6#/%\"xG\"$]\"" }{TEXT -1 39 " gives a maximum point on \+ the graph of " }{XPPEDIT 18 0 "A = 150*x-x^2/2" "6#/%\"AG,&*&\"$]\"\" \"\"%\"xGF(F(*&F)\"\"#F+!\"\"F," }{TEXT -1 35 " from the fact that the derivative " }{XPPEDIT 18 0 "dA/dx = 150-x;" "6#/*&%#dAG\"\"\"%#dxG! \"\",&\"$]\"F&%\"xGF(" }{TEXT -1 18 " is positive for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` < 150;" "6#2%!G\"$]\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "dA/dx" "6#*&%#dAG\"\"\"%#dxG!\"\"" }{TEXT -1 18 " is negative for " }{XPPEDIT 18 0 "150\"!\"*$!*1wbV$Fjo Fco7$$!+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$$!#:FboFbrQ)~x~<~150FerF0Ffr-F^r6&7$$F4 F4FbrQ$150FerF0Ffr-F^r6&7$$\"#:FboFbrQ(x~>~150FerF0Ffr-F^r6&7$$\"\"$F4 FbrQ\"3FerF0Ffr-F^r6&7$Far$FipFboQ'f~'(x)FerF0Ffr-F^r6&7$FdsFetQ\"0Fer F0Ffr-F^r6&7$F_tFetFjtF0Ffr-F^r6&7$F^s$F`tFboQ\"+FerF0-Fgr6$Fir\"#9-F^ r6&7$Fis$\"\"%FboQ\"_FerF0Fcu-%+AXESLABELSG6$Q!FerF_v-%*AXESSTYLEG6#%% NONEG-%%VIEWG6$;$!#NFbo$\"#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" "Curv e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Cur ve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 1 7" "Curve 18" "Curve 19" "Curve 20" }}{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 21 "Alternatively, since " }{XPPEDIT 18 0 "d^2*A/(d*x^2)=-1 " "6#/*(%\"dG\"\"#%\"AG\"\"\"*&F%F(*$%\"xGF&F(!\"\",$F(F," }{TEXT -1 31 " is negative for all values of " }{TEXT 350 1 "x" }{TEXT -1 36 ", \+ any turning point on the graph of " }{XPPEDIT 18 0 "A = 150*x-x^2/2" " 6#/%\"AG,&*&\"$]\"\"\"\"%\"xGF(F(*&F)\"\"#F+!\"\"F," }{TEXT -1 22 " i s a maximum point. " }}{PARA 0 "" 0 "" {TEXT -1 147 "The maximum area \+ is 11250 square metres, and it is obtained when the length of the rect angular enclosure is 150 metres and its width is 75 metres. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 22 ": Since the graph of " } {XPPEDIT 18 0 "A = 150*x-x^2/2" "6#/%\"AG,&*&\"$]\"\"\"\"%\"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 us e the derivative to see that the maximum value of A occurs where " } {XPPEDIT 18 0 "x=150" "6#/%\"xG\"$]\"" }{TEXT -1 31 ", which is mid-wa y between the " }{TEXT 294 1 "x" }{TEXT -1 12 " intercepts " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=300" "6#/%\"xG\"$+$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f := x -> 150*x-x^2/2 ;\nDiff(f(x),x);\nvalue(%);\nxmax := solve(%);\nAmax = f(xmax);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&*&\"$]\"\"\"\"9$F/F/*&#F/\"\"#F/*$)F0F3F/F/!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,&*&\"$]\"\"\"\"%\"xGF)F)*&#F)\"\" #F)*$)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 "Example 2 - rectangle under a semi-circ le" }}{PARA 0 "" 0 "" {TEXT 273 8 "Question" }{TEXT -1 3 ": " }} {PARA 0 "" 0 "" {TEXT -1 90 "Find the dimensions of the rectangle of l argest area which has two of its vertices on the " }{TEXT 272 1 "x" } {TEXT -1 151 " axis, and two of its vertices on the semi-circle of rad ius 1 lying in the 1st and 2nd quadrants, with its centre at the origi n, 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 following animation to illus trate this problem." }}{PARA 0 "" 0 "" {TEXT -1 76 "(Click on the grap hic 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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m6%F`hmQ,y~=~.195090FcemFjgmFf\\o-F[em6%7$$!*::Mv$F[[n$\"+o@L&z#F_[nFg ]nFjgmFghmF_im7=F'Fe[lFc^mF`_mFe_mF\\`mFe`m-F(6$7&7$$!33+++%f;J7&F.F/7 $Fd^v$\"3/+++^YMk:F.7$$!39+++T$)o([#FjpFg^v7$Fj^vF/Feam-Fabm6%7&7$$!+% f;J7&F_[nF/7$Fa_v$\"+^YMk:F_[n7$$!+T$)o([#F[[nFd_v7$Fg_vF/FhbmF\\cm-F( 6&7#Ff^vFccmFjgmF]dm-F(6&F\\`vFbdmFjgmF]dm-F(6&F\\`vFgdmFjgmF]dm-F(6&7 #7$$\"3)******4ow`(>Fjp$\"3')*****H&*p,4$F.FccmFgcmF]dm-F(6&Fc`vFbdmFg cmF]dm-F(6&Fc`vFgdmFgcmF]dmFjdmFgemF\\fmFafmFifmF_gm-F[em6%FdgmQ,x~=~. 987688FcemFjgm-F[em6%F`hmQ,y~=~.156434FcemFjgmFhhn-F[em6%7$$!*#fs!o$F[ [n$\"+&=RET#F_[nFg]nFjgmFghmF_im7=F'Fe[lFc^mF`_mFe_mF\\`mFe`m-F(6$7&7$ $!3%)******HaJp]F.F/7$F_bv$\"3%******R(RPv6F.7$$!37+++d%oI\\#FjpFbbv7$ FebvF/Feam-Fabm6%7&7$$!+IaJp]F_[nF/7$F\\cv$\"+uRPv6F_[n7$$!+d%oI\\#F[[ nF_cv7$FbcvF/FhbmF\\cm-F(6&7#FabvFccmFjgmF]dm-F(6&FgcvFbdmFjgmF]dm-F(6 &FgcvFgdmFjgmF]dm-F(6&7#7$$\"3/+++9p8')>Fjp$\"3*)******ROXMBF.FccmFgcm F]dm-F(6&F^dvFbdmFgcmF]dm-F(6&F^dvFgdmFgcmF]dmFjdmFgemF\\fmFafmFifmF_g m-F[em6%FdgmQ,x~=~.993068FcemFjgm-F[em6%F`hmQ,y~=~.117537FcemFjgmFjdn- F[em6%7$$!*ItSi$F[[n$\"+9#ev-#F_[nFg]nFjgmFghmF_im7=F'Fe[lFc^mF`_mFe_m F\\`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~=~.996917FcemFjgm-F[em6%F`h mQ/y~=~.784591e-1FcemFjgmF\\an-F[em6%7$$!*jWNe$F[[n$\"+GIoS;F_[nFg]nFj gmFghmF_im7=F'Fe[lFc^mF`_mFe_mF\\`mFe`m-F(6$7&7$$!3p*****zj4x+&F.F/7$F fiv$\"3y*****pd\")f#RF67$$!3-+++O!H#*\\#FjpFiiv7$F\\jvF/Feam-Fabm6%7&7 $$!+Q'4x+&F_[nF/7$Fcjv$\"+x:)f#RF\\gv7$$!+O!H#*\\#F[[nFfjv7$FijvF/Fhbm F\\cm-F(6&7#FhivFccmFjgmF]dm-F(6&F^[wFbdmFjgmF]dm-F(6&F^[wFgdmFjgmF]dm -F(6&7#7$$\"3-+++s!e%)*>Fjp$\"3Y+++/0\"f%yF6FccmFgcmF]dm-F(6&Fe[wFbdmF gcmF]dm-F(6&Fe[wFgdmFgcmF]dmFjdmFgemF\\fmFafmFifmF_gm-F[em6%FdgmQ,x~=~ .999229FcemFjgm-F[em6%F`hmQ/y~=~.392598e-1FcemFjgmF]]n-F[em6%7$$!*S-#f NF[[n$\"+c,h_7F_[nFg]nFjgmFghmF_im79F'Fe[lFc^mF`_mFe_mF\\`mFe`m-F(6$7& F+F+7$FizF/F`]wFeam-Fabm6%7&7$$!\"&FjamF/Fd]w7$$!#DFjamF/Fg]wFhbmF\\cm -F(6&7#F`^mFccmFgcmF]dm-F(6&F\\^wFbdmFgcmF]dm-F(6&F\\^wFgdmFgcmF]dmFjd mFgemF\\fmFafmFifmF_gm-F[em6%FdgmQ'x~=~1.FcemFjgm-F[em6%F`hmQ'y~=~0.Fc emFjgmFbhmFghmF_im-%(SCALINGG6#%.UNCONSTRAINEDG-%*AXESSTYLEG6#%%NONEG- %*AXESTICKSG6$F0F0F_im" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT 274 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 " " {TEXT -1 30 "The semi-circle has equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1-x^2)" "6#/%\"yG-%%sqrtG6#,&\" \"\"F)*$%\"xG\"\"#!\"\"" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "If we take a general po int " }{XPPEDIT 18 0 "P(x,y);" "6#-%\"PG6$%\"xG%\"yG" }{TEXT -1 72 " o n the section of the semi-circle in the 1st quadrant then the numbers \+ " }{TEXT 276 1 "x" }{TEXT -1 5 " and " }{TEXT 277 1 "y" }{TEXT -1 23 " satisfy equation (i). " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 400 300 300 {PLOTDATA 2 "63-%'CURVESG6$7S7$$\"\"\"\"\"!$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)R3>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;I j+\"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?vEKje w7F/$\"3()pAta;==**F/7$$!3)>@&fjQH>>F/$\"3Ew$4\"yt39)*F/7$$!3Sr=G7zpuD F/$\"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$$!38IyW%*Qc7vF/$\"3cErLF[5+mF/7$$!3YEdsL13BzF/$\"3y jGuUC@,hF/7$$!3V/esK4R<$)F/$\"3S\"Q*\\Vrm^bF/7$$!3')Qlozn?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)f2v k%)RJ\"F/7$$!3w_8!pw6n(**F/$\"34(R(G%y?2#oF27$$!\"\"F*$!30BmIq&o?5%!#F -%'COLOURG6&%$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#%,P ATCHNOGRIDG-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#%(DI AMONDGFg^lF[_l-%%TEXTG6%7$$\"#7Fgz$!\"'Ff\\lQ\"x6\"-F`\\l6&F^[l$F)Ff\\ lFd`lFd`l-Fi_l6%7$F^`l$\"$D\"Ff\\lQ\"yFa`lFb`l-Fi_l6%7$$!\"%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$F*F*-%+AXESLABELSG6%Q!Fa`lF\\cl-%%FO NTG6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!#6FgzF\\`l;$Ff \\lFgzFh`l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 46.000000 45.000000 0 0 "C urve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "C urve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 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 that the area \+ of the rectangle with vertices " }{XPPEDIT 18 0 "``(x,0),``(x,y),``(-x ,y);" "6%-%!G6$%\"xG\"\"!-F$6$F&%\"yG-F$6$,$F&!\"\"F*" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "``(-x,0);" "6#-%!G6$,$%\"xG!\"\"\"\"!" }{TEXT -1 19 " has maximum area. " }}{PARA 257 "" 0 "" {TEXT -1 9 "The area " } {TEXT 275 1 "A" }{TEXT -1 30 " of the rectangle is given by " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A = 2*x*y;" "6#/%\"AG*(\"\" #\"\"\"%\"xGF'%\"yGF'" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The equation (i) provi des a constraint for the problem which can be used with equation (ii) \+ to express the area " }{TEXT 280 1 "A" }{TEXT -1 13 " in terms of " } {TEXT 330 1 "x" }{TEXT -1 15 " only, namely: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A=2* x*sqrt(1-x^2)" "6#/%\"AG*(\"\"#\"\"\"%\"xGF'-%%sqrtG6#,&F'F'*$F(F&!\" \"F'" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "This equation is valid for " }{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**$F'F)!\"\"F*" }{TEXT -1 33 ". \+ The following graph shows how " }{TEXT 331 1 "A" }{TEXT -1 13 " varie s for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 1 ;" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f := x -> 2*x*sqrt(1-x^2);\n plot(f(x),x=0..1,labels=[`x`,`A`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"fGf*6#%\"xG6\"6$%)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-$\"3d 7$)*fRM%R7!#=7$$\"3[mmmT&phN)F-$\"3-ozHW*)Ql;F:7$$\"3CLLe*=)H\\5F:$\"3 YLZ?\\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$$ \"31++vV&RY2aF:$\"3K_#*z(Rgt 4*F:7$$\"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$$\"3 GKLe9S8&\\(F:$\"3vl3'))3&RB**F:7$$\"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'\\DGY3 Xb)zF:7$$\"3CLL$3dg6<*F:$\"3g*)fIC9i6tF:7$$\"3y***\\(oTAq#*F:$\"3o9(>3 &yx_pF:7$$\"3ImmmmxGp$*F:$\"3w%e6\\F8&\\lF:7$$\"3sK$eRA5\\Z*F:$\"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$$\"\"\"F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABEL SG6$%\"xG%\"AG-%%VIEWG6$;F(F^^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "The maximum value of " }{TEXT 296 1 "A" }{TEXT -1 14 " \+ occurs where " }{TEXT 298 1 "x" }{TEXT -1 1 " " }{TEXT 297 1 "~" } {TEXT -1 6 " 0.7. " }}{PARA 0 "" 0 "" {TEXT -1 24 "To locate this valu e of " }{TEXT 299 1 "x" }{TEXT -1 168 " more precisely we note that th e maximum point occurs where the tangent line to the graph is horizont al, that is, where the slope, as given by the derivative, is zero. " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 308 133 133 {PLOTDATA 2 " 6,-%'CURVESG6$7S7$$\"3%**************>'!#=$\"39$y#)>Wi!H(*F*7$$\"3rmmm Ya([B'F*$\"3iK!eIFm#\\(*F*7$$\"3MLLL'Q?_E'F*$\"3Ef9Sh>Fm(*F*7$$\"3#pmm YWY$*H'F*$\"3W.,w#[^Zy*F*7$$\"3immm7()pLjF*$\"3i#zbBabE!)*F*7$$\"37LLL 5x)yO'F*$\"3M0\"[zjq(>)*F*7$$\"3-nmm(G&e*R'F*$\"31-vm5O+N)*F*7$$\"3L++ +$H1CV'F*$\"35RC8s\\7])*F*7$$\"35mmmB)\\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*$\"3 u6BPB`&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$$\"39+++U;9mrF*$\"3Ipu1Q\\L'***F*7$$\"3Y+++lNd )>(F*$\"3%fR'QFpP$***F*7$$\"38+++'o$eMsF*$\"3!pm>:G]!*)**F*7$$\"3ILLL \"QSpE(F*$\"3I84pkI@%)**F*7$$\"3%)******f!)[,tF*$\"3Ua=X13/y**F*7$$\"3 bmmm\"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$$ \"3!QLL8p&QnwF*$\"3Wsw%[x,V%)*F*7$$\"3SnmmUg3*p(F*$\"3??G0$o2k#)*F*7$$ \"3`+++H_)Gt(F*$\"3M<)>FKQh!)*F*7$$\"3Q+++j`BlxF*$\"3/V.9dqc&y*F*7$$\" 3E+++++++yF*$\"3i(4HmTg@w*F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F`[l F_[l-F$6$7$7$$\"3!*******4y1rnF*$\"\"\"F`[l7$$\"3W+++5y1rtF*Fg[l-%&COL ORG6&F[[lF`[l$\"\"(!\"\"F`[l-F$6%7$7$$\"3-+++++++jF*$\"3-+++!pevz*F*7$ $\"3S+++++++nF*$\"3w******42tg**F*-Fiz6&F[[lF\\[lF_[lF\\[l-%*THICKNESS G6#\"\"#-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#%%NONEG-%%VIEWG6$;$\"#iF[_l$\"#yF[_l Fe`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 the derivative " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*&%#dAG\"\"\"%#dxG!\"\"" } {TEXT -1 46 " by using the product rule and the 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(-%%sqr tG6#,&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 " when " }{XPPEDIT 18 0 "2-4*x^2=0" " 6#/,&\"\"#\"\"\"*&\"\"%F&*$%\"xGF%F&!\"\"\"\"!" }{TEXT -1 16 ", that i s, 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 347 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/sqrt(2) = ``;" "6#/* &\"\"\"F%-%%sqrtG6#\"\"#!\"\"%!G" }{TEXT 349 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 348 1 "~" }{TEXT -1 46 " 0.7071 1 which is applicable to this problem. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Although it is obvious from the grap h of " }{XPPEDIT 18 0 "A=f(x)" "6#/%\"AG-%\"fG6#%\"xG" }{TEXT -1 20 " \+ that this value of " }{TEXT 365 1 "x" }{TEXT -1 114 " gives a maximum \+ point on the graph, we can verify 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#-%%sqrtG6#,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 24 " is always positive for " }{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+-%%sqrtG6#,&F+F+*$F'F *F/F/" }{TEXT -1 45 " is determined 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~'G 6#%\"xG" }{TEXT -1 14 " positive for " }{XPPEDIT 18 0 "0\"!\"*$!*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$$FeoF4$\"#vF`oQ\"x6\"F0-%%FONTG 6$%*HELVETICAG\"#5-F^r6&7$$!#:FboFbrQ*~x~<~xmaxFerF0Ffr-F^r6&7$$F4F4Fb rQ%xmaxFerF0Ffr-F^r6&7$$\"#:FboFbrQ)x~>~xmaxFerF0Ffr-F^r6&7$$\"\"$F4Fb rQ\"3FerF0Ffr-F^r6&7$Far$FipFboQ'f~'(x)FerF0Ffr-F^r6&7$FdsFetQ\"0FerF0 Ffr-F^r6&7$F_tFetFjtF0Ffr-F^r6&7$F^s$F`tFboQ\"+FerF0-Fgr6$Fir\"#9-F^r6 &7$Fis$\"\"%FboQ\"_FerF0Fcu-F^r6&7$$\"#XFboFetFbuF0Fcu-%+AXESLABELSG6$ Q!FerFdv-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!#NFbo$\"#DFbo;$!#8Fbo$\"#6F bo" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curv e 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21 " }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The maximum area " }{XPPEDIT 18 0 "A[max]" "6#&%\"AG6#%$m axG" }{TEXT -1 31 " is 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'-%%sqrtG6#,&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 366 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 the refore " }{TEXT 300 13 "1 square unit" }{TEXT -1 36 ", and it is obtai ned when the point " }{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 join ining " }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 33 " t o the origin makes an angle of " }{XPPEDIT 18 0 "45^o" "6#)\"#X%\"oG" }{TEXT -1 36 " with the positive direction of the " }{TEXT 332 1 "x" } {TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f := x -> 2* x*sqrt(1-x^2);\nDiff(f(x),x);\nvalue(%);\nnormal(%);\ndf := unapply(%, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,$*(\"\"#\"\"\"9$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/sqrt(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 345 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$$\"#]\"\"!$\"$v$Fjgl-%'COLOURG6&%$RGBG$\"#5!\"\"$Fjg lFjglFdhl-%+AXESLABELSG6$Q\"x6\"Q\"AFihl-%%VIEWG6$;$FbhlFjglFhgl;Fdhl$ \"$+&Fjgl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "To find this value \+ more precisely we note that the minimum point occurs where the tangent line to the graph is horizontal, that is, where the slope, as given b y the derivative, is zero. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 305 153 153 {PLOTDATA 2 "6,-%'CURVESG6$7Z7$$\"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 -1 176 "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 slope, as give n 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 find \+ " }{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)%\".G F),&F*F)\"#5!\"\"F)F)**\"\"'F)*$F*\"\"#F)F+F)F)F)F.F)*$,&F*F)F-F.F2F. " }{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 " } {TEXT 367 1 "x" }{TEXT -1 22 " > 10, the derivative " }{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-" }{TEXT -1 15 " is zero when \+ " }{XPPEDIT 18 0 "x=20" "6#/%\"xG\"#?" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 114 "Although it is already obvious from the graph drawn above that we do indeed have a minimum point on the graph of " } {XPPEDIT 18 0 "A = 6*x^2/(x-10);" "6#/%\"AG*(\"\"'\"\"\"*$)%\"xG\"\"#F '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 po int 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$$\"3++++++++bF*F+-%'COLOURG6&%$RGBG \"\"!F4F4-F$6$7$7$F($\"3++++++++]!#=7$F.F9F0-F$6$7$7$F($\"\"\"F47$F.FA F0-F$6$7$F'F@F0-F$6$7$7$$!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$$!+C.*p5\"!\"*$!+ggdN$)FgoFd o7$$!+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$\"#vF`oQ\"x6\"F0-%%FO NTG6$%*HELVETICAG\"#5-F^r6&7$$!#:FcoFbrQ-~10~<~x~<~20FerF0Ffr-F^r6&7$$ F4F4FbrQ#20FerF0Ffr-F^r6&7$$\"#:FcoFbrQ'x~>~20FerF0Ffr-F^r6&7$$\"\"$F4 FbrQ\"3FerF0Ffr-F^r6&7$Far$F[qFcoQ'f~'(x)FerF0Ffr-F^r6&7$FdsFetQ\"0Fer F0Ffr-F^r6&7$F^s$\"\"%FcoQ\"_FerF0-Fgr6$Fir\"#9-F^r6&7$Fis$F`tFcoQ\"+F erF0Fau-%+AXESLABELSG6$Q!FerF\\v-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!#NF co$\"#DFco;$!#8Fco$\"#6Fco" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 52 "It is now clear th at the minimum point occurs where " }{TEXT 351 1 "x" }{TEXT -1 43 " is exactly 20. The corresponding value of " }{TEXT 362 1 "A" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "60/(20-10)+6=12" "6#/,&*&\"#g\"\"\",&\"#?F' \"#5!\"\"F+F'\"\"'F'\"#7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 133 "The outer dimensions of the printed page which give the minimum p rinted area are therefore 20 cm. horizontally and 12 cm. vertically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "f := x -> 6*x^2/(x-10);\nDif f(f(x),x);\nvalue(%);\nnormal(%);\nsolve(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*(\"\"'\"\" \"9$\"\"#,&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 p rocedure 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 "drawbox: usage" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 303 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 304 2 " " } {TEXT -1 36 " drawbox( length, width, height ) " }{TEXT 305 1 "\n" } {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 27 " length, wi dth, height - " }{TEXT -1 37 "the dimensions of the box to be drawn" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 12 "Descri ption:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "dra wbox" }{TEXT -1 53 " draws an open or closed box of the given dimensio ns." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 8 "Op tions:" }{TEXT -1 1 "\n" }}{PARA 15 "" 0 "" {TEXT 262 16 "open=true/fa lse\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 i s 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 ca n be chose for the base, sides and top respectively." }}{PARA 0 "" 0 " " {TEXT -1 2 " " }}{PARA 15 "" 0 "" {TEXT 262 44 "lightmodel=none/lig ht1/light2/light3/light4\n" }{TEXT -1 68 "This option chooses a predef ined 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 subsection, place the cursor any where after the prompt [ > and press [Enter].\nYou can then close up \+ the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "drawbox: imple mentation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "dr awbox" {MPLTEXT 1 0 2074 "drawbox := proc(length,width,height)\n loc al 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 basec lr := blue;\n topclr := red;\n opn := false;\n lightmdl := light 2;\n if nargs>=4 then\n Options:=[args[4..nargs]];\n if no t type(Options,list(equation)) then\n error \"each optional ar gument must be an equation\"\n end if;\n if hasoption(Option s,open,'opn','Options') then\n if opn<>true then opn := false \+ end if;\n end if;\n hasoption(Options,lightmodel,'lightmdl', 'Options');\n if hasoption(Options,color,'clr','Options') or\n \+ hasoption(Options,colour,'clr','Options') then\n if type (clr,list) then\n if nops(clr)>=1 then baseclr := clr[1] en d if;\n if nops(clr)>=2 then sideclr := 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 nop s(Options)>0 then\n error \"%1 is not 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(width);\n c := evalf(height) ;\n if not type([a,b,c],list(numeric)) then\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 negative\"\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],[C 1,D1,D2,C2]]: \n p1:=plots[polygonplot3d](sides,color=sideclr):\n \+ p2:=plots[polygonplot3d](base,color=baseclr):\n if opn then\n \+ plots[display]([p1,p2],style=patch,scaling=constrained,\n lig htmodel=lightmdl)\n else\n p3:=plots[polygonplot3d]([A2,B2,C2,D 2],color=topclr):\n plots[display]([p1,p2,p3],style=patch,scaling =constrained,\n lightmodel=lightmdl);\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 307 8 "Question" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 284 "An open rectangular box is to be made from a rectangular sheet of cardboard 10 cm. by 8 cm. by cutting equal squares from the four corn ers and bending the resulting four flaps to form the sides 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$F,F@F+F'7&F1F.7$F,F,7$F(F,-F36&F5F 6F:F9-%'CURVESG6%7%FA7$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$$\"3+++++++]6!#;F,-FU6&F5$\")#)eqkFY$\"))e qk\"FYFap-Fgn6#F)-FP6%7$Fao7$FjoF@F]pFcp-FP6%7$FS7$F@$\"3++++++++&*!#< F]pFcp-FP6%7$F]o7$FDF]qF]pFcp-FP6&7$7$$\"#6F*$\"#XF87$Fhq$\"#!)F87%7$$ \"+++](3\"FY$\"++++]w!\"*F\\r7$$\"+++]76FYFcr-%&STYLEG6#%,PATCHNOGRIDG F]p-FP6&7$7$Fhq$\"#NF87$FhqF@7%7$Fgr$\"*+++]$FerFcs7$FarFfsFirF]p-FP6& 7$7$$\"#UF8$F;F*7$F@F_t7%7$$\"*+++%HFer$\"++++v))FerF`t7$Fct$\"++++D\" *FerFirF]p-FP6&7$7$$\"#eF8F_t7$$\"$+\"F8F_t7%7$$\"++++1(*FerFhtF`u7$Fe uFetFirF]p-%%TEXTG6%7$$F7F*F_tQ&10~cm6\"F]p-Fiu6%7$Fhq$\"\"%F*Q%8~cmF^ vF]p-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F^vF\\w-%%FONTG6#%(DEFAULT G-%%VIEWG6$F`wF`w" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 87 "You can play t he following animation to see how the box is formed by bending the fla ps." }}{PARA 0 "" 0 "" {TEXT -1 91 "(Click on the graphic and use the \+ controls in the context bar, or use the animation menu.) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT3D 597 496 496 {PLOTDATA 3 "6&-%(AN IMATEG677$-%)POLYGONSG6$7&7%$\"\"!F-F,F,7%$\"\"'F-F,F,7%F/$\"\"%F-F,7% F,F2F,-%&COLORG6&%$RGBG$\"\"&!\"\"$\"\")F;$\"\"*F;-F(6'7&F+F47%$!\"#F- F2F,7%FDF,F,7&F.F17%$F=F-F2F,7%FIF,F,7&F+F.7%F/FDF,7%F,FDF,7&F1F47%F,F /F,7%F/F/F,-F66&F8F9F>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&F1 F47%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)R Z%>FZF2$\"+ws!*oYFgn7%FbqF,Fdq7&F.F17%$\"+T)RZ%zFZF2Fdq7%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%$\"+LI6-zFZF2F[s7%F`s F,F[s7&F+F.7%F/FirF[s7%F,FirF[s7&F1F47%F,$\"+LI6-fFZF[s7%F/FhsF[sFQ7$F '-F(6'7&F+F47%$!+l!fx%=FZF2$\"+]'oOl(Fgn7%F`tF,Fbt7&F.F17%$\"+l!fx%yFZ F2Fbt7%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.F1 7%$\"+[I,#y(FZF2Fiu7%F^vF,Fiu7&F+F.7%F/FguFiu7%F,FguFiu7&F1F47%F,$\"+[ I,#y&FZFiu7%F/FfvFiuFQ7$F'-F(6'7&F+F47%$!+H.G0\"3_\"F ZF2$\"+(4'*))H\"FZ7%F\\zF,F^z7&F.F17%$\"+J>\"3_(FZF2F^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.F17%$\"+iN@9uFZF2Fe[l7 %Fj[lF,Fe[l7&F+F.7%F/Fc[lFe[l7%F,Fc[lFe[l7&F1F47%F,$\"+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,$\"+(4'*))H&FZF\\]l7%F/Fi]lF\\]lFQ7$F'-F(6'7&F+F47%$ !+/0dv6FZF2$\"+*)R.=;FZ7%Fa^lF,Fc^l7&F.F17%$\"+/0dvrFZF2Fc^l7%Fh^lF,Fc ^l7&F+F.7%F/Fa^lFc^l7%F,Fa^lFc^l7&F1F47%F,$\"+/0dv^FZFc^l7%F/F`_lFc^lF Q7$F'-F(6'7&F+F47%$!+Gr*\\/\"FZF2$\"+H.G0FZ7%FdelF,Ffel7&F.F17%$\"+F2*oY'FZF2Ffel7%F[flF,Ffel7&F+F.7%F/FdelFf el7%F,FdelFfel7&F1F47%F,$\"+F2*oY%FZFfel7%F/FcflFfelFQ7$F'-F(6'7&F+F47 %$!+/$*oGJFgnF2$\"+\"ow`(>FZ7%F[glF,F]gl7&F.F17%$\"+I*oGJ'FZF2F]gl7%Fb glF,F]gl7&F+F.7%F/F[glF]gl7%F,F[glF]gl7&F1F47%F,$\"+I*oGJ%FZF]gl7%F/Fj glF]glFQ7$F'-F(6'7&F+F47%$!+/>=p:FgnF2$\"+oY$Q*>FZ7%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%$\"+++++gFZF2F\\jl7%FajlF,F\\jl7&F+F.7%F/FiilF\\jl 7%F,FiilF\\jl7&F1F47%F,$\"+++++SFZF\\jl7%F/FijlF\\jlFQ-%(SCALINGG6#%,C ONSTRAINEDG-%*AXESSTYLEG6#%%NONEG-%+PROJECTIONG6%$!$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 308 8 "Solution" } {TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Let the length of a side of each square to be cut from th e corners be " }{TEXT 309 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 311 1 "x" }{TEXT -1 5 " cm. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 300 300 {PLOTDATA 2 "6C-%)PO LYGONSG6$7&7$$\"\"#\"\"!F(7$$\"\")F*F(7$F,$\"\"'F*7$F(F/-%&COLORG6&%$R GBG$\"\"&!\"\"$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%F A7$F@F,FL-%'COLOURG6&F5$\")!\\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,-FU6&F5$\")#)eqkFY$\"))eqk\"FYFap-Fgn6#F)-FP 6%7$Fao7$FjoF@F]pFcp-FP6%7$FS7$F@$\"3++++++++&*!# " 0 "" {MPLTEXT 1 0 132 "xx := 1.5;\nLength := 10-2*xx;\nWi dth := 8-2*xx;\nHeight := xx; \ndrawbox(Length,Width,Height,open=true );\n'Volume'=Length*Width*Height;" }}{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.F9-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-F$6$7&F'F3 F8F*-F<6&F>F(F(F?-%&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G-%(SCALI NGG6#%,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$\"&+D&!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Taking a smaller value of " }{TEXT 317 1 "x" }{TEXT -1 123 " results in a shallower box, but this is counterbalanced, to s ome extent, by the fact that 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,open=true);\n'Volume'=Length*Width*Heigh t;" }}{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" "C urve 2" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'VolumeG$\"&+:$!\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "For a val ue of " }{TEXT 318 1 "x" }{TEXT -1 51 " close to zero the volume is si gnificantly 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(Length,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#>%&WidthG$\"$!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%F4 F+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$\"(+ 0\"R!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "On the other hand, when " }{TEXT 312 1 "x" }{TEXT -1 102 " is clos e to 4, one of the dimensions of the base is close to zero, so that th e 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;\nW idth := 8-2*xx;\nHeight := xx; \ndrawbox(Length,Width,Height,open=tru e);\n'Volume'=Length*Width*Height;" }}{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%F4 F+F(7%F4F+F/F67&F8F*F.F9-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-F$6$7&F'F 3F8F*-F<6&F>F(F(F?-%&STYLEG6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_2G-%(SCAL INGG6#%,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$\"%[O!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 25 "We can set up a function " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 25 " to calculate the volume " }{TEXT 334 1 "V " }{TEXT -1 26 " for values of the length " }{TEXT 313 1 "x" }{TEXT -1 86 " of the side of the squares cut from each corner of the origina l cardboard sheet when " }{TEXT 315 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,label s=[`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$$\"3-++]7z>^7!#=$\"3!y>VXk RQX*F07$$\"3RLLLe'40j\"F9$\"3[1_0GMV57!#;7$$\"3/++](Q&3d?F9$\"3M\\tv/E \"o\\\"FA7$$\"3mmmm;6m$[#F9$\"3mCIJR&))4x\"FA7$$\"3jmmmmW18HF9$\"3Dk)f '4c%[.#FA7$$\"3fmmm;yYULF9$\"3Y#)H_:fr'G#FA7$$\"3/++](GI)pPF9$\"3^mi_a gnDDFA7$$\"3%HLL$eF>(>%F9$\"3KiE')['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$$\"3'QLLLV!pu$)F 9$\"3*>X!z]:#)4WFA7$$\"3xmmm;c0T\"*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$$\"3G LLLj$[kL\"F0$\"3)[\"\\TpnX;_FA7$$\"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&=&FA7$$\"3/++]^FA7$$\"3 QLLLe/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!4XFA7$$\"3YmmmJy*eC#F0$\"33R'*3N>&*RVFA7 $$\"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)>$GF0$\"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$$\"3ILLLGUYoOF0$\"3Y)Hk/m*zxkF07$$\"3_mmm1^r ZPF0$\"3RDfDA45OZF07$$\"34++]sI@KQF0$\"3Cw+4Ae_.IF07$$\"34++]2%)38RF0$ \"3#)>+l))Hgy9F07$$\"\"%F)F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLAB ELSG6$%\"xG%\"VG-%%VIEWG6$;F(F]\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 30 "A maximum value of the volume " }{TEXT 344 1 "V" } {TEXT -1 13 " occurs when " }{TEXT 314 1 "x" }{TEXT -1 16 " is about 1 .45. " }}{PARA 0 "" 0 "" {TEXT -1 176 "To find this value more precise ly we note that the maximum point occurs where the tangent line to the graph is horizontal, that is, where the slope, as given by the deriva tive " }{XPPEDIT 18 0 "`f '`(x) = dA/dx;" "6#/-%$f~'G6#%\"xG*&%#dAG\" \"\"%#dxG!\"\"" }{TEXT -1 10 ", is zero." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 310 157 157 {PLOTDATA 2 "6,-%'CURVESG6$7S7$$\"3A++ +++++!*!#=$\"3-+++++gvX!#;7$$\"3/nm;/IO$G*F*$\"3\"oQBzt*>WYF-7$$\"3%RL e*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$$\"3JLekk(3k.\"F@$\"3%=q/\"*Rb'p[F-7$ $\"3om\"Hi/j@1\"F@$\"3s$*zP\"*z9:\\F-7$$\"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$$\"37++v3zF`7F@$\"3K lTS?c4f^F-7$$\"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$$\"3`LL$)>'*e89F@$\"3Q t=;*zU\\C&F-7$$\"34+DctuiT9F@$\"3/*oK5YC'\\_F-7$$\"37+voShKo9F@$\"3;Nv Ob)[8D&F-7$$\"3_L$e*)R$='\\\"F@$\"3ME\"o#=aN]_F-7$$\"3_Le9e]w@:F@$\"3q bAUE\\(pC&F-7$$\"3um;aL$e$\\:F@$\"3t5H3DqsS_F-7$$\"3im\"H<**>!y:F@$\"3 4dJA`6VJ_F-7$$\"35+vV\\+(Hg\"F@$\"3P,ks3Y0@_F-7$$\"3*om\"H&z;*H;F@$\"3 Fn6]hV]2_F-7$$\"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$$\"3A++]+PXj?F@$\"3q U`&ziWPp%F-7$$\"3mL$3U(3D#4#F@$\"3')R)\\HO+Dk%F-7$$\"3zmmm4u+=@F@$\"3G iUm#eM^f%F-7$$\"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$$\"$0\"!\"#$\"#^Fa[lQ*f'(x)~>~0 6\"F^]l-F^^l6%7$$\"$v\"Fc^l$\"$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#%%NONEG-%%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" "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 "V=x*(10-2*x)*(8-2*x)" "6#/%\"VG*(%\"xG\"\"\",&\"#5 F'*&\"\"#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 derivative " }{XPPEDIT 18 0 "`f '`(x) = dV/dx;" "6#/-%$f~'G6#% \"xG*&%#dVG\"\"\"%#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 352 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 eq uation " }{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 353 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 354 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 355 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(84)/6" "6#*&-%%sq rtG6#\"#%)\"\"\"\"\"'!\"\"" }{XPPEDIT 18 0 " ``= 3" "6#/%!G\"\"$" } {TEXT -1 1 " " }{TEXT 356 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 359 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 357 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,!\"\"" }{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 358 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 360 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 361 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 316 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 319 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 320 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 " " 0 "" {TEXT -1 49 "Let the length of one side of the square base be \+ " }{TEXT 321 1 "x" }{TEXT -1 27 " metres. and the height be " }{TEXT 322 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$F`[lF`[l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 46.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "C urve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "The volume of the box is " }{XPPEDIT 18 0 "x^2*h" "6#*&%\"xG\"\"#% \"hG\"\"\"" }{TEXT -1 60 " cubic metres, and we are given that this is 4 cubic metres." }}{PARA 0 "" 0 "" {TEXT -1 25 "This gives the equati on " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*h = 5;" " 6#/*&%\"xG\"\"#%\"hG\"\"\"\"\"&" }{TEXT -1 14 " ------- (i), " }} {PARA 0 "" 0 "" {TEXT -1 20 "which constitutes a " }{TEXT 259 10 "cons traint" }{TEXT -1 17 " for the problem." }}{PARA 0 "" 0 "" {TEXT -1 17 "The surface area " }{TEXT 324 1 "A" }{TEXT -1 48 " is made up from the area of the base, which is " }{XPPEDIT 18 0 "x^2" "6#*$%\"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,&*$%\"xG\"\"#\"\"\"*(\"\"%F)F'F) %\"hGF)F)" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "We can eliminate the variable " } {TEXT 323 1 "h" }{TEXT -1 36 " from equation (ii) by substituting " } {XPPEDIT 18 0 "h = 5/(x^2);" "6#/%\"hG*&\"\"&\"\"\"*$%\"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,&*$%\"xG\"\" #\"\"\"*&\"#?F)F'!\"\"F)" }{TEXT -1 16 " ------- (iii). " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We can experiment \+ by calculating the surface area for different choices of " }{TEXT 327 1 "x" }{TEXT -1 51 ", and draw the resulting boxes using the procedure " }{TEXT 0 7 "drawbox" }{TEXT -1 1 " " }{HYPERLNK 17 "drawbox" 1 "" " drawbox" }{TEXT -1 27 " from an earlier section. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "xx := 3.;\n hh := 5/xx^2; \ndrawbox(xx,xx,hh,\n open=true,color=[COLOR(RGB,1 ,.6,.6),coral]);\n'Area'=xx^2+4*xx*hh;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#hhG$\"+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$\"#<\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "To determine the value of " }{TEXT 328 1 "x" }{TEXT -1 28 " for which the surface area " }{TEXT 363 1 "A " }{TEXT -1 35 " is a minimum we set up a function " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 56 " to calculate the surface area in terms of the variable " }{TEXT 325 1 "x" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f := \+ x -> x^2+20/x;\nplot(f(x),x=0..5,A=0..45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)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 335 1 "A" }{TEXT -1 13 " occurs when " }{TEXT 326 1 "x" }{TEXT -1 15 " is about 2.1. " }}{PARA 0 "" 0 "" {TEXT -1 186 "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 slope, 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(\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "We can verify that the turnin g point on the graph " }{XPPEDIT 18 0 "A = f(x);" "6#/%\"AG-%\"fG6#%\" xG" }{TEXT -1 10 " given by " }{TEXT 343 1 "x" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "x[min] = 10^(1/3);" "6#/&%\"xG6#%$minG)\"#5*&\"\"\"F+\" \"$!\"\"" }{TEXT -1 1 " " }{TEXT 342 1 "~" }{TEXT -1 83 " 2.154434690 \+ is a minimum point by investigating the sign of the second derivative \+ " }{XPPEDIT 18 0 "`f ''`(x) = d^2*A/(d*x^2);" "6#/-%%f~''G6#%\"xG*(%\" dG\"\"#%\"AG\"\"\"*&F)F,*$F'F*F,!\"\"" }{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'\"\"$F*\"#5!\"\"F**$F'F)F/" }{TEXT -1 74 " helps in determ inimg 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$$!3++++++++N!#<$!3+++++++]7F*7$$\"3++++ ++++bF*F+-%'COLOURG6&%$RGBG\"\"!F4F4-F$6$7$7$F($\"3++++++++]!#=7$F.F9F 0-F$6$7$7$F($\"\"\"F47$F.FAF0-F$6$7$F'F@F0-F$6$7$7$$!3++++++++DF*F+7$F KFAF0-F$6$7$7$$!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$$\"+w'4I*=F\\p$!*%RUkYF\\pFdpF0Fhp-%%TEXTG6&7$ $FboF4$\"#vF`oQ\"x6\"F0-%%FONTG6$%*HELVETICAG\"#5-F^r6&7$$!#: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\"0F erF0Ffr-F^r6&7$$\"\"$F4F_tFdtF0Ffr-F^r6&7$F^s$\"\"%FcoQ\"-FerF0-Fgr6$F ir\"#9-F^r6&7$Fis$FitFcoQ\"+FerF0F`u-%+AXESLABELSG6$Q!FerF[v-%*AXESSTY LEG6#%%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 mi nimum point, independently of the graph drawn above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The corresponding valu e of " }{TEXT 336 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 337 1 "~" }{TEXT -1 15 " 1.0 77217345, " }}{PARA 0 "" 0 "" {TEXT -1 31 "and the corresponding value of " }{TEXT 341 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*`.`*1 0^(2/3)" "6#/*&%\".G\"\"\")\"#5*&F&F&\"\"$!\"\"F&*(F*F&F%F&)F(*&\"\"#F &F*F+F&" }{TEXT -1 2 " " }{TEXT 338 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 339 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 340 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#,&*&\"\"#\"\"\"%\"xGF&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$%)ope ratorG%&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)));\nev alf(%);" }}{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 open=true,color=[COLOR(RGB,1,.6,.6),cora l]);\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" "Curve 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 rectangular area and then sudivid e it into four plots with fencing parallel to one side of the rectangl e." }}{PARA 0 "" 0 "" {TEXT -1 142 "What is the largest possible total area for the four plots, and what are the dimensions of the fenced en closure 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 sq uare metres, obtained when the dimensions 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#>%\"fG f*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&\"$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 f ence an area of 15000 square metres in the form of a rectangular field and then divide it in half with a fence parallel to one side of the r ectangle. How can this be achieved so that the total length of fencing used 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 fiel d 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 301 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 302 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 24 "code \+ for slope 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=11 0..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]],color=mage nta,thickness=2):\na := 175: b := g(a): m := dg(a): d := 10:\np4:=plot ([[a-d,b-m*d],[a+d,b+m*d]],color=blue,thickness=2):\nt1 := plots[textp lot]([116,11000,`f'(x) > 0`],color=magenta):\nt2 := plots[textplot]([1 50,11350,`f'(x) = 0`],color=COLOR(RGB,0,.7,0)):\nt3 := plots[textplot] ([184,11000,`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 0 647 "g := x->2*x*sqrt(1-x^2):\np1 := plot(g(x), x=0.62..0.78,color=red):\ndg := x->2*(1-2*x^2)/sqrt(1-x^2):\na := eval f(1/sqrt(2)): b := g(a): d := .03:\np2:=plot([[a-d,b],[a+d,b]],color=C OLOR(RGB,0,.7,0)):\na := .65: b := g(a): m := dg(a): d := .02:\np3:=pl ot([[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]],col or=blue,thickness=2):\nt1 := plots[textplot]([.636,.99,`f'(x) > 0`],co lor=magenta):\nt2 := plots[textplot]([.71,1.0035,`f'(x) = 0`],color=CO LOR(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]],color=mage nta,thickness=2):\nt1 := plots[textplot]([14,251,`f'(x) > 0`],color=bl ue):\nt2 := plots[textplot]([20,234,`f'(x) = 0`],color=COLOR(RGB,0,.7, 0)):\nt3 := plots[textplot]([30,254,`f'(x) < 0`],color=magenta):\nplot s[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->80-7 2*x+12*x^2:\na := evalf(3-sqrt(21)/3): b := g(a): d := .4:\np2:=plot([ [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=COLOR( 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 := 2.65: b \+ := g(a): m := dg(a): d := .3:\np4:=plot([[a-d,b-m*d],[a+d,b+m*d]],colo r=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):\np lots[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 "5 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }