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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 26 "TP2 - Calculs alg\351briq
ues." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 64 "
Premi\350re partie : introduction aux variables et aux expressions." }
}{PARA 0 "" 0 "" {TEXT 276 1 "\n" }{TEXT 258 10 "VARIABLES " }{TEXT 
-1 2 ":\n" }}{PARA 0 "" 0 "" {TEXT -1 71 "Une variable en maple est d
\351crite par un nom commen\347ant par une lettre." }}{PARA 0 "" 0 "" 
{TEXT -1 53 "L'assignation des variable se faire grace au symbole " }
{TEXT 262 2 ":=" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x:=3;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"xG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "x+2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "On peut m
ettre n'importe quelle expression math\351matique dans une variable :
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expr:=u^2+sin(v);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%exprG,&*$%\"uG\"\"#\"\"\"-%$sinG6#%\"vGF)" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "expr-u^2;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%$sinG6#%\"vG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
69 "Pour lib\351rer une variable, il suffit de lui assigner son propre
 nom :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x:='x';  expr:='expr';" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%exprGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "On p
eut aussi lib\351rer toutes les variables gr\342ce \340 la commande " 
}{TEXT 259 7 "restart" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Certains
 noms de variables sont r\351serv\351es pour des constantes et ne peuv
ent \352tre modifi\351es" }}{PARA 0 "" 0 "" {TEXT -1 37 "par exemple P
i est egal a 3.14159...." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
cos(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT 277 173 "Exercice 1.1\nMettre l'expression sin(u)+si
n(v) dans la variable 's',\npuis l'expression (sin(u)+sin(v))^2+(sin(u
)+sin(v))^3 dans la variable 't' en utilisant la variable 's'." }}
{PARA 256 "" 0 "" {TEXT 279 60 "Assigner 'u' \340 0 et 'v' \340 Pi/2 e
t r\351\351valuez la variable 't'\n" }{TEXT -1 2 "\n\n" }}{PARA 256 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 119 "--------------------
----------------------------------------------------------------------
-----------------------------" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 25 "EXPRESSIONS MATHEMATIQUES" }
{MPLTEXT 1 0 1 "\n" }{TEXT -1 122 "Les expressions math\351matiques so
nt des formules contenant\n--- des nombres : 3.4, 34,\n--- les op\351r
ations usuelles +,-,*,/,^" }}{PARA 0 "" 0 "" {TEXT -1 94 "--- des fonc
tions usuelles : sqrt, sin, cos, tan, exp,ln\n--- des noms de variable
s : x,y, expr" }}{PARA 0 "" 0 "" {TEXT -1 43 "--- des noms de constant
es comme Pi (~3.14)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x^2+sin(Pi/3
)-sqrt(7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$%\"xG\"\"#\"\"\"*$\"
\"$#F'F&F**$\"\"(F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Lorqu'une expression ne contient a
ucune variable non assign\351e, on peut " }}{PARA 0 "" 0 "" {TEXT -1 
66 "calculer une valeur approch\351e de l'expression gr\342ce \340 la \+
commande " }{TEXT 261 5 "evalf" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e
valf(exp(2)+sin(Pi/7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Q)RH#y!
\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "On peut modifier les expre
ssions :" }}{PARA 0 "" 0 "" {TEXT -1 3 "-- " }{TEXT 263 11 "simplify  \+
 " }{TEXT -1 41 "essaie de rendre l'expression plus simple" }}{PARA 0 
"" 0 "" {TEXT -1 3 "-- " }{TEXT 264 7 "expand " }{TEXT -1 22 "d\351vel
oppe l'expression" }}{PARA 0 "" 0 "" {TEXT -1 3 "-- " }{TEXT 265 7 "fa
ctor " }{TEXT -1 27 " factorise l'expression\n-- " }{TEXT 266 6 "diff \+
 " }{TEXT -1 50 "d\351rive l'expression par rapport \340 une variable
\n-- " }{TEXT 267 5 "subs " }{TEXT -1 54 "permet de remplacer une sous
-expression par une autre." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "simplify(tan(x)*cos(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sin
G6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "s:=expand((x+a)
*(x^2-1)*(x^36-2*x^18+1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG,:
*$%\"xG\"#R\"\"\"*$F'\"#@!\"#*$F'\"\"$F)*$F'\"#P!\"\"*$F'\"#>\"\"#F'F1
*&%\"aGF)F'\"#QF)*&F6F)F'\"#?F,*&F6F)F'F4F)*&F6F)F'\"#OF1*&F6F)F'\"#=F
4F6F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(s);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*0,(*$%\"xG\"\"#\"\"\"F&F(F(F(F',(F%F(
F&!\"\"F(F(F',(*$F&\"\"'F(*$F&\"\"$F(F(F(F',(F,F(F.F*F(F(F',&F&F(F*F(F
/,&F&F(F(F(F/,&F&F(%\"aGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "diff(ln(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$%\"xG!\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "subs(x=a+b,sin(x)+cos(x))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#,&%\"aG\"\"\"%\"bGF)F)
-%$cosGF&F)" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 268 12 "Exercice 1.2" }
}{PARA 260 "" 0 "" {TEXT 269 60 "Mettre l'expression (x^2+1)/(x^2+x+1)
 dans la variable fonc," }}{PARA 261 "" 0 "" {TEXT 270 74 "d\351river \+
fonc par rapport \340 x et mettez le r\351sultat dans la variable dfon
c." }}{PARA 262 "" 0 "" {TEXT 271 75 "Factoriser dfonc, en d\351duire \+
que la fonction x->fonc admet un minimum en 1." }}{PARA 257 "" 0 "" 
{TEXT -1 38 "Calculer ce minimum en utilisant subs;" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 97 "---------------------------------------------------------
----------------------------------------" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 272 43 "R\351solution formelle de syst\350mes d'\351quations" }
{TEXT -1 73 "\nelle se fait gr\342ce \340 la commande solve\nSYNTAXE :
solve(\351quation,inconnue)" }}{PARA 0 "" 0 "" {TEXT -1 64 " par exemp
le pour r\351soudre l'equation x^2+3x+1=0 il faut faire :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(x^2+3*x+1=0,x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$,&#!\"$\"\"#\"\"\"*$\"\"&#F'F&F*,&F$F'F(#!\"\"F&
" }}}{EXCHG {PARA 258 "" 1 "" {TEXT -1 79 "On peut aussi r\351soudre u
n systeme de plusieurs \351quations a plusieurs inconnues " }}{PARA 0 
"" 0 "" {TEXT -1 23 "en utilisant la syntaxe" }{TEXT 273 32 " solve(\{
equations\},\{variables\});" }}{PARA 0 "" 0 "" {TEXT -1 48 "(les \{...
\} sont les delimitateurs des ensembles)" }}{PARA 0 "" 0 "" {TEXT -1 
9 "exemple :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(\{2*x
+3*y=7,5*x-3*y=3\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"
yG#\"#H\"#@/%\"xG#\"#5\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 12 "E
xercice 1.3" }}{PARA 263 "" 0 "" {TEXT -1 103 "1.3.1)    R\351soudre l
'\351quation x^3+x+1=2,\n             donner une valeur approch\351e d
e la racine r\351elle, " }}{PARA 264 "" 0 "" {TEXT -1 104 "           \+
   (vous pouvez utiliser le copier-coller, m\352me si  il y a des %1,%
2,%3 dans les expressions." }}{PARA 265 "" 0 "" {TEXT -1 30 "1.3.2)   \+
 R\351soudre le syst\350me " }}{PARA 283 "" 0 "" {TEXT -1 1 " " }
{TEXT 284 26 "                 x^3+y^3=1" }}{PARA 284 "" 0 "" {TEXT 
-1 72 "                  x^2+y^2=1\n            Combien y-a-t-il de so
lutions  ?" }}{PARA 285 "" 0 "" {TEXT -1 37 "            (Pour \351vit
er d'avoir des " }{TEXT 280 6 "RootOf" }{TEXT -1 30 " il faut ex\351cu
ter la commande " }{TEXT 281 20 "_EnvExplicit:=true; " }{TEXT 283 1 ")
" }}{PARA 0 "" 0 "" {TEXT 282 1 "-" }{TEXT -1 95 "--------------------
----------------------------------------------------------------------
-----" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 47 "Deuxi\350me partie : pr
euve de la formule de H\351ron" }}{PARA 266 "" 0 "" {TEXT -1 11 "Quest
ions :" }}{PARA 267 "" 0 "" {TEXT -1 84 "Le but de cette partie est de
 trouver l'aire d'un triangle en fonction des longueurs" }}{PARA 268 "
" 0 "" {TEXT -1 22 "des c\364t\351s du triangle." }}{PARA 269 "" 0 "" 
{TEXT -1 58 "On consid\350re donc un triangle ABC tel que BC=a, AC=b,A
B=c." }}{PARA 270 "" 0 "" {TEXT -1 102 "On se place dans un rep\350re \+
orthonorm\351 d'origine A, et dont l'axe des abscisses est la demi-dro
ite [AB)" }}{PARA 271 "" 0 "" {TEXT -1 35 "Le point A a donc pour coor
donn\351es " }{TEXT 278 5 "(0,0)" }{TEXT -1 40 " et le point B a pour \+
coordonn\351es (c,0);" }}{PARA 272 "" 0 "" {TEXT -1 40 "Soit (xC,yC) l
es coordonn\351es du point C." }}{PARA 273 "" 0 "" {TEXT -1 89 "2.1  D
eterminer les longueurs AC et BC en fonction de a,b,c,xC,yC. Mettez ce
s expressions" }}{PARA 274 "" 0 "" {TEXT -1 50 "       respectivement \+
dans les variables AC et BC." }}{PARA 275 "" 0 "" {TEXT -1 102 "2.2 Ce
s longueurs v\351rifient AC=b, AB=c. En d\351duire les valeurs possibl
e pour (xC,yC). (utiliser solve)" }}{PARA 276 "" 0 "" {TEXT -1 110 "  \+
    Combien y-a-t-il de solutions ? interpretez.\n      Mettez dans le
s variables xC,yC une des solutions...." }}{PARA 277 "" 0 "" {TEXT -1 
156 "2.3 Quelle est l'aire du triangle ABC en fonction de a,b,c,xC,yC.
 En d\351duire l'aire en fonction de a,b,c\n      Cette expression ser
a dans la variable AIRE. " }}{PARA 278 "" 0 "" {TEXT -1 41 "2.4 Factor
iser cette derni\350re expression." }}{PARA 279 "" 0 "" {TEXT -1 81 "2
.5 (facultatif) en consid\351rant p=1/2*(a+b+c) simplifier encore cett
e expression." }}{PARA 280 "" 0 "" {TEXT -1 79 "2.6 d\351river AIRE pa
r rapport \340 la variable c, et factoriser le resultat obtenu." }}
{PARA 281 "" 0 "" {TEXT -1 93 "      En d\351duire parmi les triangle \+
qui deux longueurs de cot\351 fix\351es \340 a et b quel est celui" }}
{PARA 282 "" 0 "" {TEXT -1 22 "      d'aire maximale." }}}}{MARK "26 1
5 0" 6 }{VIEWOPTS 1 1 0 1 1 1803 }