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Vinet" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 34 "Universit\351 Paris 7 - Denis Diderot" }}{PARA 0 "" 0 "" {TEXT -1 15 "Ann\351e 2000-2001" }}{PARA 257 "" 0 "" {TEXT 259 16 "MI 101 - Maple \+ V" }}{PARA 258 "" 0 "" {TEXT 259 20 "Groupes A4 - A5 - D5" }}}{EXCHG {PARA 18 "" 0 "" {TEXT 260 42 "4. Fonctions, limites d\351riv\351s, in t\351grales." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Sommaire" }}{PARA 14 "" 0 "" {TEXT -1 99 "1. Fonctions.\n 1.1 Fonctions \340 plusieur s variables.\n 1.2 Fonctions compos\351es.\n 1.3 Suites." }} {PARA 14 "" 0 "" {TEXT -1 11 "2. Limites." }}{PARA 14 "" 0 "" {TEXT -1 64 "3. D\351riv\351s.\n 3.1 D\351riv\351es partielles.\n 3.2. La commande D." }}{PARA 14 "" 0 "" {TEXT -1 12 "4.Integrales" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Note:" }}{PARA 0 "" 0 "" {TEXT -1 36 "1. Le rappel des pr\351cedents calculs." }}{PARA 0 "" 0 "" {TEXT -1 70 "Nous avons vu, precedement, que ce rappel se fait grace \340 la commande " }{TEXT 270 1 "%" }{TEXT -1 78 ". On peut, avec Maple rappe ler jusqu'\340 trois calculs pr\351cedents, en utilisant " }{TEXT 271 1 "%" }{TEXT -1 2 ", " }{TEXT 272 2 "%%" }{TEXT -1 4 " et " }{TEXT 273 3 "%%%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "exemple :" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "plots[implicitplot](y^2=4 *x, x=0..16,y=0..16, color=green):\nplot(x/2,x=0..16, color=blue):\npl ot(ln(theta),theta=0..2*Pi, coords=polar, color=red):\nplots[display]( %,%%,%%%);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "?%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "2. Affichage sans calculs" }}{PARA 0 "" 0 "" {TEXT -1 268 "On p eut, avec Maple afficher quelque chsoe sans le calculer. Il suffit de \+ mettre des apostrophes (') au d\351but et \340 la fin de ce qu'on \351 crit. \nSi l'on veut inserer une espace, il faut alors cr\351er une ch aine. Pour ce faire, il faut utiliser les guillements anglais (\")." }}{PARA 0 "" 0 "" {TEXT -1 9 "exemple :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "'sin(Pi)' = sin(Pi);\n\"\347a marche\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "?'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "?\"" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 13 "1. Fonctions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Une fonction sous Maple se d \351fini de la fa\347on suivante :" }}{PARA 260 "" 0 "" {TEXT 264 18 " nom_de_la_fonction" }{TEXT -1 1 " " }{TEXT 261 14 ":= variable(s)" } {TEXT -1 1 " " }{TEXT 263 2 "->" }{TEXT -1 1 " " }{TEXT 262 25 "defini tion_de_la_fonction" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x ->2*x^2-3*x+4;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "Voici cimment avoir la valeur de la f onction pour x=2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "De meme, on peut tracer le graphe de cette fonction entre -5 et 5" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-5..5);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 37 "Voici un autre exemple de fonctions :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=x->x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "'f(5)' = f(5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "'f(y+1)' = f(y+1);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }} {PARA 5 "" 0 "" {TEXT -1 36 "1.1 Fonctions \340 plusieurs variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "De meme que pour une fonction \340 une seule variable, on peut d \351finir une fonction \340 plusieurs variable." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "f := (x,y) -> sin(x)*cos(y);" }{TEXT -1 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 " Et donc la tracer dans la plan (n'oubliez pas que c'est un graphe en t rois dimensions) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot3 d(f(x,y),x=-5..5,y=-5..5);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Ou bien evaluer sa valeurt \340 un point donn\351 (x,y) d u plan :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "'f(15,2)' = f(1 5,2);\n'f(Pi,7)' = f(Pi,7);\n'f(8,Pi/2)' = f(8,Pi/2);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 24 "1.3 Fonct ions compos\351es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Maple sait composer des fonctions entres-elles. O n se sert de l'op\351rateur " }{TEXT 274 1 "@" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(sin@cos)(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "g := x -> x^3 +2:\nh := y -> ln(y): \n'g(h(x))' = (g@h)(x);\n'h(g(x))' = (h@g)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "\nOn peut \351galement composer une fonction par ell e-meme. On utilise alors l'operateur " }{TEXT 275 2 "@@" }{TEXT -1 1 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "'(g@@2)(x)' = (g@@2)(x );\n'(g@@3)(x)' = (g@@3)(x);\n'(g@@4)(x)' = (g@@4)(x);" }{TEXT -1 0 " " }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 10 "1.3 Suites" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Les suites sont des fonctions \"comme les autres\", \+ si ce n'est qu'elles sont d\351finies dans " }{TEXT 277 1 "N" }{TEXT -1 12 " au lieu de " }{TEXT 276 1 "R" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "u := n -> 3*n + 4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Note : Si on d\351finie la suite par " }{XPPEDIT 18 0 "u[0];" "6#&%\"uG6#\"\"!" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "u[n] = \+ f(u[n-1]);" "6#/&%\"uG6#%\"nG-%\"fG6#&F%6#,&F'\"\"\"!\"\"F." }{TEXT -1 12 " alors on a " }{XPPEDIT 18 0 "u[n] = `@@`(f,n)(u[0]);" "6#/&%\" uG6#%\"nG--%#@@G6$%\"fGF'6#&F%6#\"\"!" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 11 "Exercice 1." }}{PARA 14 "" 0 "" {TEXT -1 26 "a) D\351finissez la fonction " } {XPPEDIT 18 0 "f[1](x) = sin(x)*cos(x);" "6#/-&%\"fG6#\"\"\"6#%\"xG*&- %$sinG6#F*\"\"\"-%$cosG6#F*F/" }{TEXT -1 32 ". Tracer la fonction entr e 0 et " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 11 ". calculez " } {XPPEDIT 18 0 "f[1](0);" "6#-&%\"fG6#\"\"\"6#\"\"!" }{TEXT -1 4 " et \+ " }{XPPEDIT 18 0 "f[1](Pi);" "6#-&%\"fG6#\"\"\"6#%#PiG" }{TEXT -1 1 ". " }}{PARA 14 "" 0 "" {TEXT -1 58 " f est-elle injective ou surject ive de [0,Pi] dans R ?" }}{PARA 14 "" 0 "" {TEXT -1 26 "b) Definissez \+ la fonction " }{XPPEDIT 18 0 "f[2](x,y) = exp(x*y)*cos(x)*sin(y);" "6# /-&%\"fG6#\"\"#6$%\"xG%\"yG*(-%$expG6#*&F*\"\"\"F+F1F1-%$cosG6#F*F1-%$ sinG6#F+F1" }{TEXT -1 38 " Tracer sa courbe representative pour " } {XPPEDIT 18 0 "x = -Pi .. Pi;" "6#/%\"xG;,$%#PiG!\"\"F'" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "y = -Pi .. Pi;" "6#/%\"yG;,$%#PiG!\"\"F'" } {TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 " f[2];" "6#&%\"fG6#\"\"#" }{TEXT -1 83 "est-elle injective ? Si non, t rouvez un contre-exemple.\nc) D\351finissez la fonction " }{XPPEDIT 18 0 "g(x) = 3*x/ln(x);" "6#/-%\"gG6#%\"xG*(\"\"$\"\"\"F'F*-%#lnG6#F'! \"\"" }{TEXT -1 88 ". Que vaut exp(g) ? Et g(exp) ? (o\371 exp est la \+ fonction exponentielle)\nd) Soit la suite " }{XPPEDIT 18 0 "u[n];" "6# &%\"uG6#%\"nG" }{TEXT -1 12 " d\351fine par " }{XPPEDIT 18 0 "u[0] = 3 ;" "6#/&%\"uG6#\"\"!\"\"$" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "u[n] = l n(u[n-1])+5;" "6#/&%\"uG6#%\"nG,&-%#lnG6#&F%6#,&F'\"\"\"\"\"\"!\"\"F/ \"\"&F/" }{TEXT -1 53 ". Calculez les 5\350, 10\350 et 100\350 termes \+ en fonction de " }{XPPEDIT 18 0 "u[0];" "6#&%\"uG6#\"\"!" }{TEXT -1 30 ". \n Refaites le calcul si " }{XPPEDIT 18 0 "u[0] := 10;" "6#> &%\"uG6#\"\"!\"#5" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 " " 0 "" {TEXT -1 11 "2. Limites." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 91 " La syntaxe pour calculer la limit e de la fonction f au point a est de la forme suivante :" }}{PARA 261 "" 0 "" {TEXT 265 6 "limit(" }{TEXT -1 4 "f(x)" }{TEXT 267 1 "," } {TEXT -1 3 " x " }{TEXT 268 1 "=" }{TEXT -1 2 " a" }{TEXT 266 1 ")" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Voici quelques exemples :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(sin(x)/x,x=0);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "limit((1+a) ^(1/a),a=0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "limit((1+x^2+3*x^3)/(2-3*x+x^3),x=infinity);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Si 'lon veut juste \351crire la limite, il suffit de mettre une majuscul e \340 " }{TEXT 269 5 "Limit" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Limit(sin(x),x=0);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "De nombr euses options sont disponible concernant la directions o\371 vous voul ez calculez la limite (si vous voulez calculer la limite \340 gauche o u \340 droite par exemple)." }}{PARA 0 "" 0 "" {TEXT -1 74 "Notez que \+ Maple \340 le courage de vous dire si une limite n'est pas d\351finie. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "limit(floor(x),x=1);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(floor (x),x=1,right);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "limit(floor(x),x=1,left);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 295 11 "Exercice 2." }}{PARA 14 "" 0 "" {TEXT -1 25 "a) T rouvez la limite en +" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" } {TEXT -1 5 " et -" }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT -1 17 " de la fonction " }{XPPEDIT 18 0 "f[3] = sin(x)*cos(x)/x;" "6# /&%\"fG6#\"\"$*(-%$sinG6#%\"xG\"\"\"-%$cosG6#F,F-F,!\"\"" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 44 "b) Quelle est la limite en 0 de la fonction " }{XPPEDIT 18 0 "f[4] = 4*cos(x)/(1-x^2);" "6#/&%\"fG6#\"\" %*(\"\"%\"\"\"-%$cosG6#%\"xGF*,&\"\"\"F**$F.\"\"#!\"\"F3" }{TEXT -1 2 " ?" }}{PARA 14 "" 0 "" {TEXT -1 106 "c) Quelle est la limite en 1 \+ \340 droite de floor(x) ? Quelle est la limite en 1 \340 gauche de cet te fonction ?" }}{PARA 14 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 12 "3. D\351riv\351es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Les d\351riv\351es \"ordinaires\" ou partiels peuvent etre faite en u tilisant la commande " }{TEXT 278 4 "diff" }{TEXT -1 36 ". La syntaxe \+ de cette commande est :" }}{PARA 262 "" 0 "" {TEXT 279 5 "diff(" } {TEXT -1 8 "f(x,y,z)" }{TEXT 281 1 "," }{TEXT -1 2 " x" }{TEXT 280 1 " )" }}{PARA 0 "" 0 "" {TEXT -1 118 "o\371 f(x,y,z) est la fonction \340 d\351river et x la variable par rapport \340 laquelle vous derivez.\n Notez que si vous \351crivez " }{TEXT 282 4 "Diff" }{TEXT -1 15 " \340 la palce de " }{TEXT 283 4 "diff" }{TEXT -1 33 ", Maple n'evaluera pa s la d\351riv\351e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff( sin(x),x) = diff(sin(x),x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 70 "Diff((1+x^2+4*x^3)/(1+2*x-x^2),x) = diff((1+x^2+4*x ^3)/(1+2*x-x^2),x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "\nLes d\351riv\351es multiples peuvent etre evalu\351s en repet ant la variable plusieurs fois." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(sin(x^2),x,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Ou bien, de mani\350re equvalente en \351crivant " } {TEXT 284 3 "x$n" }{TEXT -1 44 ", o\371 n est le nombre de fois o\371 \+ l'on d\351rive." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diff(sin (x^2),x$2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }} {PARA 5 "" 0 "" {TEXT -1 23 "3.1 D\351riv\351es partielles" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "De mani\350re similaire, on peut evaluer les d \351riv\351es partielles d'une fonction \340 plusieurs variables :" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(cos(x*tan(y)),x);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 173 "Notez que l'ordre dans lequel Maple d\351rive les fonc tion est de gauche \340 droite (dans l'ordre o\371 vous les \351crivez ). Parfois (assez rarerement), cet ordre peut etre important." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "diff(cos(x+y^2),x,y);\ndiff( cos(x+y^2),y,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "diff((x^2+y^2)*(ln(x)-ln(y)),x$2,y);\ndiff((x^2+y^2)*(ln(x)-ln (y)),y,x$2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" } }{PARA 5 "" 0 "" {TEXT -1 18 "3.2 La commande D." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "Un autre moyen p our deriver les fonction est d'utiliser le symbole D, qui peut ^etre a ppliqu\351 \340 une fonction sans avoir \340 specifier ses arguments. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "D(sin);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "unassign('f','g');\nD(f@g );\nD(f*g);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "D(exp@cos@ln );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "Cela peut bien sur etre it\351r\351 en utilisant l e symbole " }{TEXT 285 3 "@@n" }{TEXT -1 47 " (o\371 n est un entier) \+ dans la composition de D." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "(D@@2)(sin);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 " Or, used with a subscript i in the form D[i], it picks out the \+ partial derivative with respect to the ith variable." }}{PARA 0 "" 0 " " {TEXT -1 56 "Ou encore en utilisant l'indexation en i, dans la forme " }{TEXT 286 4 "D[i]" }{TEXT -1 56 ", ce qui donnera une d\351riv\351 e partielle par rapport \340 la " }{TEXT 287 1 "i" }{TEXT -1 13 "\350m e variable." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "D[2]((x,y)-> x*y^3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "\n Fo r mixed partial derivatives, we use a list of subscripts in the order \+ the derivatives are to be applied (from left to right)." }}{PARA 0 "" 0 "" {TEXT -1 113 "Pour des d\351riv\351es partielles, on utilise une \+ idexation par rapport \340 l'ordre dont on veut deriver les fonctions \+ :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "D[2,1]((x,y)->x*y^3); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "D[1,2$2] ((x,y)->x*y^3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 298 11 "Exercice 3." }}{PARA 14 "" 0 "" {TEXT -1 28 "a) Quelle est la d\351riv\351e de " }{XPPEDIT 18 0 "u(x)/ v(x);" "6#*&-%\"uG6#%\"xG\"\"\"-%\"vG6#F'!\"\"" }{TEXT -1 46 " ? Calcu lez la deriv\351e seconde de ce quotient." }}{PARA 14 "" 0 "" {TEXT -1 56 "b) Donnez toutes les d\351riv\351es partielles (d'ordre 1) de \+ " }{XPPEDIT 18 0 "f[2](x,y) = exp(x*y)*cos(x)*sin(y)" "6#/-&%\"fG6#\" \"#6$%\"xG%\"yG*(-%$expG6#*&F*\"\"\"F+F1F1-%$cosG6#F*F1-%$sinG6#F+F1" }{TEXT -1 53 ". Donnez toutes les d\351riv\351es partielles d'ordre 2 \+ de " }{XPPEDIT 18 0 "f[2];" "6#&%\"fG6#\"\"#" }{TEXT -1 55 " . Ec rivez ces formules de deux fa\347ons differentes" }}{PARA 14 "" 0 "" {TEXT -1 26 "c) Definissez la fonction " }{XPPEDIT 18 0 "g(x) = x^2+ln (3*x);" "6#/-%\"gG6#%\"xG,&*$F'\"\"#\"\"\"-%#lnG6#*&\"\"$F+F'F+F+" } {TEXT -1 155 ". Puis, en une seule ligne de commande, tracez sur un me me graphe la fonction g, ainsi que ses deux premieres d\351riv\351es, \+ pour x compris entre entre 0.5 et 2" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 14 "4. Int\351grales." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 " Maple sa it calculer les integrales, grace \340 la commande " }{TEXT 288 3 "int " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 65 "La syntaxe pour les i nt\351grales indefinie (les \"primitives\") est :" }}{PARA 263 "" 0 " " {TEXT 289 4 "int(" }{TEXT -1 8 " expr, x" }{TEXT 290 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 96 "o\371 expr est l'expression que vous integrez e t x la variable par rappor \340 laquelle vosu integrez." }}{PARA 0 "" 0 "" {TEXT -1 58 "Pour les integrales d\351finies, la syntaxe est la s uivante :" }}{PARA 264 "" 0 "" {TEXT 292 5 "int( " }{TEXT -1 12 "expr, x=a..b" }{TEXT 291 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 88 "o\371 expr es t l'expression que vous integrez et a..b l'intevale sur lequel vous in tegrez.\n" }}{PARA 0 "" 0 "" {TEXT -1 45 "Note: comme pour la d\351riv \351e, si vous ecrivez " }{TEXT 293 3 "Int" }{TEXT -1 15 " \340 la pla ce de " }{TEXT 294 3 "int" }{TEXT -1 34 ", vous aurez la forme non eva lu\351e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int( tan(x), x \+ ) = int( tan(x), x );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Int( (1-x^2)/(1+x^2), x ) = int( (1-x^2)/(1+x^2), x ) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int( x^ 2*exp(-x^2), x ) = int( x^2*exp(-x^2), x );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Voici que lques integrales d\351finies et impropre :" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "Int( cos(x), x=-Pi/2..Pi/2 ) = int( cos(x), x=-Pi/2 ..Pi/2 );" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "Int( x^2*exp(-x^2), x=0..infinity ) = int( x^2*exp(-x^2), x=0..infini ty );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Int( exp(-x^2), x= 0..infinity ) = int( exp(-x^2), x=0..infinity );" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Le s integrales multiples sont faites par iterationd e l'integrale :" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "Int(Int(Int(exp(-x-y-z),x=0 ..y), y=0..z), z=0..1)\n = int(int(int(exp(-x-y-z),x=0..y), y=0..z), z =0..1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 297 11 "Exercice 4." }}{PARA 14 "" 0 "" {TEXT -1 8 "a) Soit " }{XPPEDIT 18 0 "f[5](x) = exp(-x^2);" "6#/-&%\"fG6#\" \"&6#%\"xG-%$expG6#,$*$F*\"\"#!\"\"" }{TEXT -1 25 " calculez l'integra le de " }{XPPEDIT 18 0 "f[5];" "6#&%\"fG6#\"\"&" }{TEXT -1 46 " entre \+ 0 et l'infini. Calculez l'integrale de " }{XPPEDIT 18 0 "f[5];" "6#&% \"fG6#\"\"&" }{TEXT -1 8 " entre +" }{XPPEDIT 18 0 "infinity;" "6#%)in finityG" }{TEXT -1 5 " et -" }{XPPEDIT 18 0 "infinity" "6#%)infinityG " }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 29 "b) tracez les deux \+ courbes: " }{XPPEDIT 18 0 "y = sqrt(4*x);" "6#/%\"yG-%%sqrtG6#*&\"\"% \"\"\"%\"xGF*" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "y = x/2;" "6#/%\"yG* &%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 154 " entre 0 et 16. Calculez les \+ integrales de ces deux fonctions entre 0 et 16.\n Puis, par un ca lcul d'integrale, donnez l'aire entre les deux courbes" }}{PARA 14 "" 0 "" {TEXT -1 46 "c) Cherchez une primitive en x de la fonction " } {XPPEDIT 18 0 "f[6](x,y,z) = exp(x+y)-3*sin(x*z)+ln(z)/x;" "6#/-&%\"fG 6#\"\"'6%%\"xG%\"yG%\"zG,(-%$expG6#,&F*\"\"\"F+F2F2*&\"\"$F2-%$sinG6#* &F*F2F,F2F2!\"\"*&-%#lnG6#F,F2F*F9F2" }{TEXT -1 2 ". " }}{PARA 14 "" 0 "" {TEXT -1 88 " Calcullez une primitive en y et une en z de cet te fonction. Verifiez vos resultats." }}{PARA 14 "" 0 "" {TEXT -1 0 " " }}}}{MARK "91 5 0" 18 }{VIEWOPTS 1 1 0 1 1 1803 }