{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 14 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 1 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 18 0 0 0 0 1 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 18 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 14 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 14 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 1 12 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 12 0 0 0 0 2 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 14 0 0 0 0 2 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Hea ding 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helveti ca" 1 12 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 5 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 4 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 3 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 260 "" 0 "" {TEXT -1 63 " RELATIONS O F A FUNCTION WITH ITS DERIATIVES" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {PARA 4 "" 0 "" {TEXT 256 87 "Calculus I Project \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 " " } }{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 256 11 "Objectives:" } {TEXT 258 2 " " }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT 257 82 "To \+ study the relation of the first and second derivative functions (f ' a nd f '') " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 31 "with the ori ginal function, f. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 86 "Use a computer algebra system to calculate f ' and f ' ', and graph f, f ' and f '' . " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 259 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 6 "Let f( " }{TEXT 261 1 "x" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "x/(1+x^2);" "6#* &%\"xG\"\"\",&F%F%*$F$\"\"#F%!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 42 "a) Use Maple to calculate f ' and f ''. " }}{PARA 0 " " 0 "" {TEXT -1 59 "b) Use the definition of derivative to find f ' \+ and f ''." }}{PARA 0 "" 0 "" {TEXT -1 43 "c) Plot the graphs of f, f ' and f ''. " }}{PARA 0 "" 0 "" {TEXT -1 96 "d) Using the graphs \+ in c), describe the relationship between f and f ' as well as f and f ''." }}{PARA 0 "" 0 "" {TEXT -1 76 "e) Using the graphs in c), desc ribe the relationship between f ' and f ''." }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 10 " Solution:" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 265 51 "a) Define the function and calculate the drivatives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 266 15 "Define f(x ) := " }{XPPEDIT 18 0 "x/(1+x^2);" "6#*&%\"xG\"\"\",&F%F%*$F$\"\"#F%! \"\"" }{TEXT 267 49 ". Use diff command to calculate the derivatives. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := x -> x/(1 + x^2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F.F.*$)F-\"\"#F.F.! \"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "fprime(x) := \+ diff(f(x), x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%'fprimeG6#%\"x G,&*&\"\"\"F*,&F*F**$)F'\"\"#F*F*!\"\"F**(F.F*F'F.F+!\"#F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(fprime(x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"\"!\"\"*$)%\"xG\"\"#F&F&F&,&F&F&F(F&! \"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "fdoubleprime(x) := diff(fprime(x), x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%-fdoublep rimeG6#%\"xG,&*(\"\"'\"\"\",&F+F+*$)F'\"\"#F+F+!\"#F'F+!\"\"*(\"\")F+F '\"\"$F,!\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "fdoublepr ime(x) := simplify(fdoubleprime(x));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%-fdoubleprimeG6#%\"xG,$**\"\"#\"\"\"F'F+,&\"\"$!\"\"*$)F'F*F+ F+F+,&F+F+F/F+!\"$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 269 2 "b)" }{TEXT -1 1 " " }{TEXT 268 57 "Calculate the firs t and second derivatives by definition." }}{PARA 0 "" 0 "" {TEXT -1 70 "We rename the first derivative as G and the second derivative H. \+ Use " }}{PARA 0 "" 0 "" {TEXT -1 59 "upper case letters to avoid confu sion with the increment h." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "g(x) := limit((f(x + h) - f(x))/h, \+ h = 0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"gG6#%\"xG,$*&,&\"\" \"!\"\"*$)F'\"\"#F+F+F+,&F+F+F-F+!\"#F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "G := x -> - (-1 + x^2)/(1 + x^2)^2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&,&\" \"\"!\"\"*$)9$\"\"#F/F/F/,&F/F/F1F/!\"#F0F(F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 43 "H(x) := limit((G(x + k) - G(x))/k, k = 0);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"HG6#%\"xG,$**\"\"#\"\"\"F'F+,& \"\"$!\"\"*$)F'F*F+F+F+,&F+F+F/F+!\"$F+" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 270 9 "c ) Plot \+ " }{TEXT 271 80 "graphs of f, f ' and f \" on the interval [-4, 4] usi ng the same coordinate axes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots);\n" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7Z%(animateG%*animate3dG%-animatecurveG%&arrowG% -changecoordsG%,complexplotG%.complexplot3dG%*conformalG%,conformal3dG %,contourplotG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinderplot G%,densityplotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradp lotG%+gradplot3dG%,graphplot3dG%-implicitplotG%/implicitplot3dG%(inequ alG%,interactiveG%-listcontplotG%/listcontplot3dG%0listdensityplotG%)l istplotG%+listplot3dG%+loglogplotG%(logplotG%+matrixplotG%(odeplotG%'p aretoG%,plotcompareG%*pointplotG%,pointplot3dG%*polarplotG%,polygonplo tG%.polygonplot3dG%4polyhedra_supportedG%.polyhedraplotG%'replotG%*roo tlocusG%,semilogplotG%+setoptionsG%-setoptions3dG%+spacecurveG%1sparse matrixplotG%+sphereplotG%)surfdataG%)textplotG%+textplot3dG%)tubeplotG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot(\{f(x), fprime(x), fdoubleprime(x)\}, x = -4..4, title = `Graph of f, f' and f''`);\n" } }{PARA 13 "" 1 "" {GLPLOT2D 403 185 185 {PLOTDATA 2 "6(-%'CURVESG6$7ao 7$$!\"%\"\"!$!3?)eqkM)*HF-7$$!3!******\\`oz$GF1$!3Bf?V!yoW8$F-7$$!3!omm;)3DoEF1$!3O&=?K?+ iG$F-7$$!3?+++:v2*\\#F1$!3c3NiJu>\\MF-7$$!3BLLL8>1DBF1$!3&fWF1$!3(f)*HWS74+%F-7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 61 " In the interval where f '' is positive, f is concave up and " }}{PARA 0 "" 0 "" {TEXT -1 45 " where f '' is negative, f is concave down. " }}{PARA 0 "" 0 "" {TEXT -1 54 "Thus the S-shaped curve is the original function, f. " }}{PARA 0 "" 0 "" {TEXT -1 60 "The curve with one peak at 0 is \+ the first derivative, f '. " }}{PARA 0 "" 0 "" {TEXT -1 60 "The curve with a peak and a valley is the second derivative " }}{PARA 0 "" 0 " " {TEXT -1 15 "function, f ''." }}{PARA 0 "" 0 "" {TEXT -1 50 " At \+ x = -1, f has a minimum, and f ' is zero. " }}{PARA 0 "" 0 "" {TEXT -1 50 " At x = 1, f has a maximum, and f ' is zero. " }}{PARA 0 " " 0 "" {TEXT -1 74 "In the intervals x < -1 and x > 1, f is decreasin g, and f ' is negative, " }}{PARA 0 "" 0 "" {TEXT -1 33 "i.e. The grap h is below x-axis. " }}{PARA 0 "" 0 "" {TEXT -1 68 "In the interval [ -1, 1], f is increasing and f ' is greater than 0, " }}{PARA 0 "" 0 " " {TEXT -1 33 "i.e. The graph is above x-axis. " }}{PARA 0 "" 0 "" {TEXT -1 68 " At x = -1.8, approx., f changes from concave down to \+ concave up " }}{PARA 0 "" 0 "" {TEXT -1 18 "and f \" is zero. " }} {PARA 0 "" 0 "" {TEXT -1 71 " At x = 0, f changes from concave up t o concave down, f \" is zero. " }}{PARA 0 "" 0 "" {TEXT -1 74 " At x = 1.8, approximately, f changes from concave down to concave up, " }}{PARA 0 "" 0 "" {TEXT -1 75 "f \" is zero. f has a point of inflect ion at x = -1.8, x = 0, and x = 1.8. " }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 274 14 "e) Conclusio n" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 275 53 "Now we analyze the \+ relationship between f ' and f ''." }}{PARA 0 "" 0 "" {TEXT -1 65 "At \+ the points where f ' has a maximum or a minimum, f '' is zero." }} {PARA 0 "" 0 "" {TEXT -1 60 " In the intervals where f ' is decreasing , f \" is negative. " }}{PARA 0 "" 0 "" {TEXT -1 59 " In the intervals where f ' is increasing, f \" is positive." }}{PARA 0 "" 0 "" {TEXT -1 48 " At x = 0 f ' has a maximum, and f \" is zero." }}{PARA 0 " " 0 "" {TEXT -1 61 " At x = -1.8, and x = 1.8, f ' has a minimum; f \" is zero." }}{PARA 0 "" 0 "" {TEXT -1 64 " In the intervals x < \+ -1.8, and [0, 1.8], f ' is decreasing, " }}{PARA 0 "" 0 "" {TEXT -1 59 " and f \" is negative. i.e. The graph is below x-axis." }} {PARA 0 "" 0 "" {TEXT -1 63 " In the intervals [-1.8, 0] and x > 1. 8, f ' is increasing, " }}{PARA 0 "" 0 "" {TEXT -1 59 " and f \" is positive. i.e. The graph is above x-axis." }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 67 "________________________ ___________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 10 "ASSIGNMENT" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 1 " " }{TEXT 262 8 "Problem:" }}{PARA 0 "" 0 "" {TEXT -1 77 " Formulate appropriate functions in your notebook before \+ you input for Maple." }}{PARA 0 "" 0 "" {TEXT -1 50 " Let f(x) = sin \+ (x) in the interval [-Pi, Pi]. " }}{PARA 0 "" 0 "" {TEXT -1 50 " \+ (a) Use Maple to calculate f ' and f ''. " }}{PARA 0 "" 0 "" {TEXT -1 67 " (b) Use the definition of derivative to find f ' and f ''." }}{PARA 0 "" 0 "" {TEXT -1 51 " (c) Plot the graph s of f, f ' and f ''. " }}{PARA 0 "" 0 "" {TEXT -1 83 " (d) \+ Using the graphs in (c), describe the relationship between f and f ' \+ " }}{PARA 0 "" 0 "" {TEXT -1 37 " as well as f and f ''. " }}{PARA 0 "" 0 "" {TEXT -1 85 " (e) Using the graphs in (c), describe the relationship between f ' and f ''." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 260 0 "" }{TEXT 263 0 "" } {TEXT 264 66 "________________________________________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "MSIP Grant #P120A80089-98: \"Three Urban Calculus Reform prog rams: Adopting the Best\" 1998-2001 " }}{PARA 0 "" 0 "" {TEXT -1 8 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " " }}}{MARK "10 6 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }