diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.asy b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.asy new file mode 100644 index 0000000..6fa20b2 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.asy @@ -0,0 +1,32 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(1cm,1cm); + +limits(-5.5,5.5); +ylimits(-5.5,5.5); + +xaxis(BottomTop,xmin=-5.5,xmax=5.5,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); +yaxis(LeftRight,ymin=-5.5,ymax=5.5,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); + +real F(real x) {return 1/10*x^3+1/10*x^2-1/5*x+1;} +draw(graph(F,-6,6,n=400),linewidth(1pt)+red+solid); + +real G(real x) {return 0.3*(x-1)+1;} +draw(graph(G,-6,6,n=400),linewidth(1pt)+darkblue+solid); + +xlimits(-5.5,5.5,Crop); +ylimits(-5.5,5.5,Crop); + +xaxis(axis=YEquals(0),xmin=-5.5,xmax=5.5,Ticks(scale(.7)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=-5.5,ymax=5.5,Ticks(scale(.7)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + +label(scale(1.25)*"$\mathcal{C}_f$", (4,5)); +label(scale(1.25)*"$\mathcal{T}$", (4,2), N); + +shipout(bbox(0.1cm,0.1cm,white)); + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.pag b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.pag new file mode 100644 index 0000000..93ecf06 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.pag @@ -0,0 +1,23 @@ + + + 1 + -3.5 + 10.5 + 1 + -.75 + 8.5 + 1 + 1 + 1 + 1 + 1 + 1 + 0 + 1 + + x^2#sqrt(x) + -4#0 + 11#11 + noir#noir + ligne#ligne + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.pdf b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.pdf new file mode 100644 index 0000000..a7775d5 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.png b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.png new file mode 100644 index 0000000..449c9a0 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.svg b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.svg new file mode 100644 index 0000000..3c22725 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig1.svg @@ -0,0 +1,3619 @@ + + + + +Created by potrace 1.16, written by Peter Selinger 2001-2019 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig2.png b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig2.png new file mode 100644 index 0000000..f7412f4 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig2.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig3.png b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig3.png new file mode 100644 index 0000000..740f4b5 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig3.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.asy b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.asy new file mode 100644 index 0000000..b4c9ba3 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.asy @@ -0,0 +1,30 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(1cm,1cm); +xlimits(-5,5); +ylimits(-5,3); + +xaxis(BottomTop,xmin=-5,xmax=5,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); +yaxis(LeftRight,ymin=-5,ymax=3,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); + +real F(real x) {return log(x+1);} +draw(graph(F,-0.9999999999,6,n=400),linewidth(1.5pt)+black+solid); + +point pA=(-1,-1), pB=(-1,1); +draw(line(pA,pB),linewidth(1.5pt)+red+dashed); +label(rotate(90)*"$\boldsymbol{ x=-1 }$", (-1,-4)-.5, fontsize(12pt)+red); + +xlimits(-5,5,Crop); +ylimits(-5,5,Crop); + +xaxis(axis=YEquals(0),xmin=-5,xmax=5,Ticks(scale(.7)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=10,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=-5,ymax=3,Ticks(scale(.7)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + + +shipout(bbox(0.1cm,0.1cm,white)); + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.pdf b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.pdf new file mode 100644 index 0000000..1a444b8 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.png b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.png new file mode 100644 index 0000000..b88d31f Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.svg b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.svg new file mode 100644 index 0000000..a4cdc06 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig4.svg @@ -0,0 +1,2180 @@ + + + + +Created by potrace 1.16, written by Peter Selinger 2001-2019 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.asy b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.asy new file mode 100644 index 0000000..5947978 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.asy @@ -0,0 +1,35 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(0.5cm,0.125cm); +xlimits(-3,6); +ylimits(-20,20); + +xaxis(BottomTop,xmin=-3,xmax=6,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(.8)+dotted); +yaxis(LeftRight,ymin=-20,ymax=20,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(.8)+dotted); + +real F(real x) {return 2/(x-3)+5;} +draw(graph(F,-4,2.999999,n=400),linewidth(1.5pt)+black+solid); +real G(real x) {return 2/(x-3)+5;} +draw(graph(G,3.000001,6,n=400),linewidth(1.5pt)+black+solid); + +point pA=(3,-10), pB=(3,10); +draw(line(pA,pB),linewidth(1.5pt)+red+dashed); +label(rotate(90)*"$\boldsymbol{x=3}$", (3,-10)+.5, fontsize(8pt)+red); +point pA=(-3,5), pB=(3,5); +draw(line(pA,pB),linewidth(1.5pt)+heavygreen+dashed); +label("$\boldsymbol{y=5}$", (-2,6), fontsize(8pt)+heavygreen); + +xlimits(-3,6,Crop); +ylimits(-20,20,Crop); + +xaxis(axis=YEquals(0),xmin=-3,xmax=6,Ticks(scale(.4)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=-20,ymax=20,Ticks(scale(.4)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=10,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + + +shipout(bbox(0.1cm,0.1cm,white)); + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.pdf b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.pdf new file mode 100644 index 0000000..f1f5662 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.png b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.png new file mode 100644 index 0000000..9a75255 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.svg b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.svg new file mode 100644 index 0000000..2b4ebc2 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig5.svg @@ -0,0 +1,116 @@ + + + + +Created by potrace 1.16, written by Peter Selinger 2001-2019 + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.asy b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.asy new file mode 100644 index 0000000..c8b5ff5 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.asy @@ -0,0 +1,30 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(1cm,1cm); +xlimits(-2,7); +ylimits(-3,5); + +xaxis(BottomTop,xmin=-2,xmax=7,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); +yaxis(LeftRight,ymin=-3,ymax=5,Ticks("%",extend=true,Step=1),p=linewidth(1pt)+gray(0)+dotted); + +real F(real x) {return x-3+1/x;} +draw(graph(F,-0.9999999999,8,n=400),linewidth(1.5pt)+black+solid); +real G(real x) {return x-3;} +draw(graph(G,-0.9999999999,8,n=400),linewidth(1.5pt)+red+solid+dashed); + +label(rotate(45)*"$\boldsymbol{ y = x-3 }$", (5,2)+.5, fontsize(12pt)+red); + +xlimits(-2,7,Crop); +ylimits(-3,5,Crop); + +xaxis(axis=YEquals(0),xmin=-2,xmax=7,Ticks(scale(.7)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=10,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=-3,ymax=5,Ticks(scale(.7)*Label(),beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + + +shipout(bbox(0.1cm,0.1cm,white)); + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.pdf b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.pdf new file mode 100644 index 0000000..787167c Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.png b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.png new file mode 100644 index 0000000..520994e Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.svg b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.svg new file mode 100644 index 0000000..fa78a60 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/images/fig6.svg @@ -0,0 +1,1468 @@ + + + + +Created by potrace 1.16, written by Peter Selinger 2001-2019 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/.pdf b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/.pdf new file mode 100644 index 0000000..0f3d86e Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/images b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/images new file mode 120000 index 0000000..e4c5bd0 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/images @@ -0,0 +1 @@ +../images/ \ No newline at end of file diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieI.tex b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieI.tex new file mode 100644 index 0000000..edd330f --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieI.tex @@ -0,0 +1,93 @@ +% !TeX root = prof_-_etude_de_fonction_-_2023.1.tex + +\partie{Rappels : dérivation} + +\sspartie{Nombre dérivé, tangente} + +\noindent +\begin{multicols}{2} + \noindent + On considère une fonction $f$ définie sur un intervalle $I$, sa courbe représentative $\ronde{C}_f$ dans un repère orthogonal, et un réel $a$ dans $I$. + + Si $\ronde{C}_f$ admet une tangente $\mathsf{T}$ au point d'abscisse $a$ non parallèle à l'axe des abscisses, on dit que $f$ est dérivable en $a$. + + Le coefficient directeur de la tangente $\mathsf{T}$ au point $a$ est le nombre dérivé $f^{\prime}(a)$.\\\\ + \columnbreak + + \begin{center} + \includegraphics[scale=.6]{fig1.pdf} + \end{center} +\end{multicols} + +\medskip + +\begin{propriete}[équation de la tangente] + La tangente $\mathsf{T}$ à $\ronde{C}_f$ au point de coordonnées $\coord{a, f(a)}$ a pour équation $y = f^{\prime}(a)(x - a) + f(a)$. +\end{propriete} + +\sspartie{Fonction dérivée et dérivées usuelles} + +\noindent +On dit que la fonction \textbf{$\boldsymbol{f}$ est dérivable sur l'intervalle $I$} si elle est dérivable en tout point $a$ de $I$.\\ + +\noindent +La \textbf{fonction dérivée de $\boldsymbol{f}$} est notée $f^{\prime}$ et correspond à la fonction qui à tout réel $x$ de $I$, associe le nombre $f^{\prime}(x)$. + +\medskip + +\ssspartie{Dérivées des fonctions usuelles} + +\begin{center} + \begin{tabularx}{.7\textwidth}{| C{4.5cm} | >{\centering\arraybackslash}X | C{4.5cm} |} + \hline + \textbf{\small$ \boldsymbol{f(x)} $} & \textbf{\footnotesize est dérivable sur ...}\vspace{.2em} & \textbf{\small$ \boldsymbol{f^{\prime}(x)} $} \\ + \hline + \rule[-1ex]{0pt}{2.5ex} $ k $ (constante) & $ \mathbb{R} $ & \rule[-1ex]{0pt}{5ex}$ 0 \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ x $ & $ \mathbb{R} $ & \rule[-1ex]{0pt}{5ex}$ 1 \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ x^n $ où $ n \ge 1 $ & $ \mathbb{R} $& \rule[-1ex]{0pt}{5ex}$ nx^{n-1} $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \dfrac{1}{x} $ & $ \mathbb{R}^{*} $& \rule[-1ex]{0pt}{5ex}$ -\dfrac{1}{x^2} $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \sqrt{x} $ & $ ]0\; ;\; +\infty[ $ & \rule[-1ex]{0pt}{5ex}$ \dfrac{1}{2\sqrt{x}} $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \ln x $ & $]0\ ;\ +\infty[$ & \rule[-1ex]{0pt}{5ex}$\dfrac1{x}$\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \e^{x} $ & $\R$ & \rule[-1ex]{0pt}{5ex}$\e^{x}$\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \cos x $ & $\R$ & \rule[-1ex]{0pt}{5ex}$-\sin x$\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \sin x $ & $\R$ & \rule[-1ex]{0pt}{5ex}$\cos x$\vspace{1ex} \\ + \hline + \end{tabularx} +\end{center} + +\ssspartie{Opérations sur les fonctions dérivées} + +\noindent +On considère $u$ et $v$ deux fonctions dérivables sur un intervalle $I$ de $\R$. +\begin{center} + \begin{tabularx}{.7\textwidth}{| C{4.5cm} | >{\centering\arraybackslash}X | C{4.5cm} |} + \hline + \textbf{$ \boldsymbol{f} $} & \textbf{\footnotesize est dérivable ...}\vspace{.2em} & \textbf{$ \boldsymbol{f^{\prime}} $} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ k \times u $ ($k$ constante)& sur $I$ & \rule[-1ex]{0pt}{5ex}$ k \times u' $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ u + v $ & sur $I$ & \rule[-1ex]{0pt}{5ex}$ u' + v' $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ u \times v $ & sur $I$ & \rule[-1ex]{0pt}{5ex}$ u' \times v + u \times v' $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \dfrac{u}{v} $ & si $v(x) \neq 0$ & \rule[-1ex]{0pt}{5ex}$ \dfrac{u' \times v - u \times v'}{v^2} \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \dfrac{1}{v} $ & si $v(x) \neq 0$ & \rule[-1ex]{0pt}{5ex}$ \dfrac{-v'}{v^2} \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ u^n $ avec $n \ge 1$ & sur $I$ & \rule[-1ex]{0pt}{5ex}$ n \times u^{\prime} \times u^{n-1} \vspace{1ex}$ \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \ln(u) $ & si $u(x) > 0$ & \rule[-1ex]{0pt}{5ex}$ \dfrac{u^{\prime}}{u} $\vspace{1ex} \\ + \hline + \rule[-1ex]{0pt}{2.5ex}$ \e^{u} $ & sur $I$ & \rule[-1ex]{0pt}{5ex}$ u^{\prime} \times \e^{u} $\vspace{1ex} \\ + \hline + \end{tabularx} +\end{center} + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieII.tex b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieII.tex new file mode 100644 index 0000000..0535c88 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieII.tex @@ -0,0 +1,153 @@ +% !TeX root = prof_-_etude_de_fonction_-_2023.1.tex + +\partie{Rappels : limites et asymptotes} + +\sspartie{Limites des fonctions de références} + +\ssspartie{Limite en $\boldsymbol{\infty}$} + +\begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{B5B5B5}}c|c|c|c|c|c|c|c|} + \hline + \rule[-1ex]{0pt}{2.5ex}$f(x)$ & $x^n$ & $\dfrac1{x^n}$ & $\sqrt{x}$ & $\dfrac1{\sqrt{x}}$ & $e^{x}$ & $e^{ax}$ & $\ln(x)$ \\[2ex] \hline + \rule[-1ex]{0pt}{2.5ex}$\lim\limits_{x \to +\infty}f(x)=$ & $+\infty$ & $0$ & $+\infty$ & $0$ & $+\infty$ & + \begin{tabular}{lr} + $+\infty$ & si $a>0$ \\ + $0$ & si $a<0$ + \end{tabular} & $+\infty$ \\ \hline + \rule[-1ex]{0pt}{2.5ex}$\lim\limits_{x \to -\infty}f(x)=$ + & + \begin{tabular}{lr} + $-\infty$ & si $n$ impair \\ + $+\infty$ & si $n$ pair + \end{tabular} + & $0$ & non définie & non définie & $0$ & + \begin{tabular}{lr} + $0$ & si $a>0$ \\ + $+\infty$ & si $a<0$ + \end{tabular} & non définie \\ \hline + \end{tabular}% + % } +\end{center} + +\vspace{-.2cm} + +\ssspartie{Limite en $\boldsymbol{0}$} + +\begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{B5B5B5}} c|c|c|c|} + \hline + \rule[-1ex]{0pt}{2.5ex}$f(x)$ & $\dfrac1{x^n}$ & $\dfrac1{\sqrt{x}}$ & $\ln(x)$ \\[2ex] \hline + \rule[-1ex]{0pt}{2.5ex}$\lim\limits_{\substack{x \to 0 \\ x > 0}}f(x)=$ & $+\infty$ & $+\infty$ & $-\infty$ \\[2ex] \hline + \rule[-1ex]{0pt}{2.5ex}$\lim\limits_{\substack{x \to 0 \\ x < 0}}f(x)=$ + & + \begin{tabular}{lr} + $-\infty$ & si $n$ impair \\ + $+\infty$ & si $n$ pair + \end{tabular} + & non définie & non définie \\ \hline + \end{tabular}% + % } +\end{center} + +\vspace{-.2cm} + +\ssspartie{Opérations sur les limites} + +\begin{tasks}[style=itemize](4) + \task $ f $ et $ g $ sont deux fonctions ; + \task $A$ est soit $+\infty$, soit $-\infty$ ou un réel ; + \task \textsf{\textbf{F. I.}} : Forme indéterminée ; + \task \textup{*} : règle des signes. +\end{tasks} + +\begin{multicols}{2} + ~ + \vspace{-1.3cm} + \begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{FFFFFF}}c |c|c|c|c|c|>{\columncolor[HTML]{EFEFEF}}c|} + \hline + Si $\lim\limits_{x \to A}f(x)=$ & $\ell$ & $\ell$ & $\ell$ & $+\infty$ & $-\infty$ & {$+\infty$} \\ + \hline + Si $\lim\limits_{x \to A}g(x)=$ & $\ell'$ & $+\infty$ & $-\infty$ & $+\infty$ & $-\infty$ & {$-\infty$} \\ + \hline + alors $\lim\limits_{x \to A}f(x)+g(x)=$ & $\ell + \ell'$ & $+\infty$ & $-\infty$ & $+\infty$ & $-\infty$ & {\textsf{\textbf{F. I}}} \\ + \hline + \end{tabular}% + % } + \end{center} + + \columnbreak + + \begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{FFFFFF}}c |c|c|>{\columncolor[HTML]{EFEFEF}}c|c|} + \hline + Si $\lim\limits_{x \to A}f(x)=$ & $\ell$ & $\ell \neq0 $ & {$0$} & $\infty$ \\ + \hline + Si $\lim\limits_{x \to A}g(x)=$ & $\ell'$ & $\infty$ & {$\infty$} & $\infty$ \\ + \hline + alors $\lim\limits_{x \to A}f(x) \times g(x)=$ & $\ell \times \ell'$ & $\infty^{*}$ & {\textsf{\textbf{F. I.}}} & $\infty^{*}$ \\ + \hline + \end{tabular}% + % } + \end{center} +\end{multicols} + +\begin{center} + % \resizebox{\textwidth}{!}{% + \begin{tabular}{|>{\columncolor[HTML]{FFFFFF}}c |c|c|>{\columncolor[HTML]{EFEFEF}}c|c|c|>{\columncolor[HTML]{EFEFEF}}c|} + \hline + Si $\lim\limits_{x \to A}f(x)=$ & $\ell$ & $\ell \neq 0$ & {$0$} & $\ell$ & $\infty$ & {$\infty$} \\ + \hline + Si $\lim\limits_{x \to A}g(x)=$ & $\ell' \neq 0$ & $0$ & {$0$} & $\infty$ & $\ell'$ & {$\infty$} \\ + \hline + alors $\lim\limits_{x \to A}\dfrac{f(x)}{g(x)}=$ & $\dfrac{\ell}{\ell'}$ & $\infty^{*}$ & {\textsf{\textbf{F. I.}}} & $0$ & $\infty^{*}$ & {\textsf{\textbf{F. I.}}} \\ + \hline + \end{tabular}% + % } +\end{center} + +\sspartie{Comportement asymptotique} + +\noindent +On considère une fonction $f$, sa courbe $\ronde{C}_f$, $a$ et $\ell$ des nombres réels. + +\vspace{.4cm} + +\begin{definition}[existence d'asymptote horizontale] + \bi + \item Lorsque $ \lim\limits_{x \to +\infty} f(x) = \ell $, on dit que la courbe $\ronde{C}_f$ admet la droite d'équation $y=\ell$ comme asymptote horizontale en $+\infty$. + \item Lorsque $ \lim\limits_{x \to a} f(x) = +\infty $, on dit que la courbe $\ronde{C}_f$ admet la droite d'équation $x=a$ comme asymptote verticale. + \item Lorsque $ \lim\limits_{x \to +\infty} f(x) - (mx+p)= \ell $, on dit que la courbe $\ronde{C}_f$ admet la droite d'équation $y=mx+p$ comme asymptote oblique en $+\infty$. + \ei +\end{definition} + +\vspace{.2cm} + +\noindent +On retrouve les même situations lorsque $x$ tend vers $-\infty$. + +\vspace{.4cm} + +% exemple +\begin{exemple} + \begin{center} + \begin{tabular}{ccc} + \rule[-1ex]{0pt}{2.5ex} $f(x) = \ln(x+1)$ & \hspace{.2cm} & $g(x) = \dfrac{2}{x-3}+5$ \\ + \rule[-1ex]{0pt}{2.5ex} \includegraphics[scale=.8]{fig4.pdf} & \hspace{.2cm} & \includegraphics[scale=1.25]{fig5.pdf} \\ + \rule[-1ex]{0pt}{2.5ex} la droite $x=-1$ est asymptote verticale & \hspace{.2cm} & les droites $y=5$ et $x=3$ sont asymptotes horizontale et verticale \\ + \end{tabular} + \end{center} + ~\vspace{.4cm} + \begin{center} + \begin{tabular}{ccc} + \rule[-1ex]{0pt}{2.5ex} \hspace{.2cm} & $h(x) = \dfrac{x^2-3x+1}{x}$ & \hspace{.2cm} \\ + \rule[-1ex]{0pt}{2.5ex} \hspace{.2cm} & \includegraphics[scale=.8]{fig6.pdf} & \hspace{.2cm} \\ + \rule[-1ex]{0pt}{2.5ex} \hspace{.2cm} & la droite $y=x-3$ est asymptote oblique & \hspace{.2cm} \\ + \end{tabular} + \end{center} +\end{exemple} diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieIII.tex b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieIII.tex new file mode 100644 index 0000000..ea24704 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/partieIII.tex @@ -0,0 +1,45 @@ +% !TeX root = prof_-_etude_de_fonction_-_2023.1.tex + +\newpage + +\partie{Applications} + +\begin{application}[calculer une dérivée] + Déterminer la fonction dérivée de chacune des fonctions sur l'intervalle donné. + + \begin{tasks} (2)[] + \task\ $f_1(x) = x^2 \e^{x}$ \qquad pour $x \in \R$ + \task\ $f_2(x) = 3 \ln(x) - \dfrac5{x} + 2\sqrt{x}$ \qquad pour $ x \in ]0\ ;\ +\infty[ $ + \task\ $f_3 = \dfrac{5x+3}{x-4}$ \qquad pour $x \in ]-\infty\ ;\ 4[ \cup ]4\ ;\ +\infty[$ + \task\ $f_4(x) = \e^{x^2-x}$ \qquad pour $x \in \R$ + \task\ $f_5(x) = \ln(x^3+1)$ \qquad pour $x \in ]0\ ;\ +\infty[$ + \end{tasks} +\end{application} + +\vspace{.4cm} + +\begin{application}[calculer une limite] + Déterminer les limites suivantes: + \begin{tasks} (4)[] + \task\ $ \lim\limits_{x \to +\infty } 5-x^2 \e^{x} $ + \task\ $ \lim\limits_{\substack{ x \to 0 \\ x>0 }} \dfrac{3+x}{\ln x} $ + \task\ $ \lim\limits_{\substack{ x \to -2 \\ x>-2 }} \dfrac{x^2-6x+7}{4+2x} $ + \task\ $ \lim\limits_{x \to -\infty } \e^{x^2-x} $ + \end{tasks} +\end{application} + +\vspace{.4cm} + +\begin{application}[étudier une fonction et ses asymptotes] + Soit $f$ la fonction définie sur $]-\infty\ ;\ -2[\cup]-2\ ;\ +\infty[$ par $f(x)=x+3-\dfrac1{x+2}$ et $\ronde{C}_f$ sa courbe représentative. + \begin{questions} + \item Calculer la dérivée de la fonction $f$ et étudier ses variations. + \item Déterminer les limites de $f$ aux bornes de son ensemble de définition. + \item En déduire l'existence d'une asymptote verticale et préciser son équation. + \item Montrer que la droite $D$ d'équation $y=x+3$ est asymptote oblique à la coiurbe $\ronde{C}_f$ en $-\infty$ et en $+\infty$. + \item Déterminer la position relative de $\ronde{C}_f$ et de la droite $D$. Justifier. + \end{questions} +\end{application} + +\vspace{.4cm} + diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.log b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.log new file mode 100644 index 0000000..f1d39b1 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.log @@ -0,0 +1,1302 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Arch Linux) (preloaded format=pdflatex 2023.8.3) 9 AUG 2023 11:09 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**\scrollmode + 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+*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$1$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{$\boldsymbol{ x=-1 }$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(43.55823pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(7.73332pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.59998pt)dim + +*\fontsize{12}{14.4}\selectfont% + +* +(Please type a command or say `\end') +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$0$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(5.87494pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(0.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-5$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-4$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-3$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim + +*\setbox\ASYbox=\hbox{\vphantom{$10^4$}$-2$} + +* +(Please type a command or say `\end') +*\immediate\write16{>dim(\the\wd\ASYbox)dim} +>dim(15.2083pt)dim + +*\immediate\write16{>dim(\the\ht\ASYbox)dim} +>dim(9.51027pt)dim + +*\immediate\write16{>dim(\the\dp\ASYbox)dim} +>dim(1.0pt)dim diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.pdf b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.pdf new file mode 100644 index 0000000..7c78ef4 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.tex b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.tex new file mode 100644 index 0000000..2baa709 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 1 - etude de fonction: rappels derivation et limites/cours - etude de fonction - 2023.1/prof - etude de fonction - 2023.1/prof_-_etude_de_fonction_-_2023.tex @@ -0,0 +1,22 @@ +\documentclass[10pt]{jl-cours} +\usepackage{asymptote} + +\newcolumntype{M}[1]{>{\centering}m{#1}} + +\begin{document} +\pagenumbering{arabic} + +\titre{\Jd Étude de fonctions}{\Jd Étude de fonctions} +\lohead*{BTS2E} +\rofoot*{\anneescolaire} + +\cofoot[\thepage]{\thepage} +\cefoot[\thepage]{\thepage} + +\nskip{3} + +\import{.}{partieI.tex} +\import{.}{partieII.tex} +\import{.}{partieIII.tex} + +\end{document} diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig1.png b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig1.png new file mode 100644 index 0000000..760d21c Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig1.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig2.png b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig2.png new file mode 100644 index 0000000..754f4d8 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig2.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig3.png b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig3.png new file mode 100644 index 0000000..6b58446 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig3.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig4.asy b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig4.asy new file mode 100644 index 0000000..5ccf51a --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig4.asy @@ -0,0 +1,49 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(1cm,2cm); + +xlimits(-1.25,10); +ylimits(-0.5,3.5); + +xaxis(BottomTop,xmin=-1.25,xmax=10,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted); +yaxis(LeftRight,ymin=-0.5,ymax=3.5,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted); + +xlimits(-1.25,10,Crop); +ylimits(-0.5,3.5,Crop); + +real f(real x) {return 1/8*(x-5)^2+3/2;} + +pair O=(0,0), I=(1,0), J=(0,1), A=(1,1), D=(3,0), E=(8,0), F=(3,f(3)), G=(8,f(8)); +label("$O$",O,4.5000(-0.2121,-0.2121)); +label("$I$",I,1.5*S); +label("$J$",J,1.5*W); +label("$A$",A, .125*NE); + +filldraw(O--I--A--J--cycle, lightgray, linewidth(1pt)); + +path Cf=graph(f, 2.5, 8.5, n=400); +draw(Cf, linewidth(1pt)); +path domaineD = buildcycle(F--D--E--G, graph(f,3,8)); + +fill(domaineD, pattern("hachures1")); + +//draw(F--G, linewidth(1.5pt)); +draw((D.x,D.y-.5)--(F.x,F.y+.5), gray(0)+dashed); +draw((E.x,E.y-.5)--(G.x,G.y+.5), gray(0)+dashed); + +label("Domaine $D$", (5.5,1), fontsize(18pt), Fill(white)); +label("$\int_{a}^{b} f(t)\; dt$", (5.5,.5), fontsize(18pt), Fill(white)); +label("$a$", D, 1.75*S, fontsize(18pt), Fill(white)); +label("$b$", E, 1.25*S, fontsize(18pt), Fill(white)); + +xaxis(axis=YEquals(0),xmin=-1.25,xmax=10,Ticks("%",NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=-0.5,ymax=3.5,Ticks("%",NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + +shipout(bbox(0.1cm,0.1cm,white)); + + diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig4.pdf b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig4.pdf new file mode 100644 index 0000000..4546e87 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig4.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.asy b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.asy new file mode 100644 index 0000000..7f3f2eb --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.asy @@ -0,0 +1,56 @@ +// préambule asymptote +usepackage("amsmath,amssymb"); +usepackage("inputenc","utf8"); +usepackage("icomma"); + +import lib_jl; + +unitsize(1cm,1cm); + +real xmin=-5.5, ymin=-1.5, xmax=5.5, ymax=5.5; + +xlimits(xmin,xmax); +ylimits(ymin,ymax); + +xaxis(BottomTop,xmin=xmin,xmax=xmax,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted); +yaxis(LeftRight,ymin=ymin,ymax=ymax,Ticks("%",extend=true,Step=1),p=linewidth(0.5pt)+gray(0)+dotted); + +real F(real x) {return .5*(x+5)*(.5x^2+.25x+.5);} + +draw(graph(F,-6,4,n=400),linewidth(1pt)+deepred+solid); + +xlimits(xmin,xmax,Crop); +ylimits(ymin,ymax,Crop); + +xaxis(axis=YEquals(0),xmin=xmin,xmax=xmax,Ticks("%",beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); +yaxis(axis=XEquals(0),ymin=ymin,ymax=ymax,Ticks("%",beginlabel=true,endlabel=true,begin=true,end=true,NoZero,Step=1,Size=1mm),p=linewidth(1pt)+black,Arrow(2mm),true); + +pair O=(0,0), I=(1,0), J=(0,1), A=(-4,0), B=(1,0), C=(-1.5,0); + +labelx(Label("$O$",NoFill), 0, SW); +label("$I$",I,SE); +label("$J$",J,NW); + +draw(line(A-(0,2),A+(0,2)),deepblue+pen1); +label("$a$",A,SE,deepblue); + +path integrale1=buildcycle((A.x, F(A.x))--A--C--(C.x, F(C.x)),graph(F,A.x,C.x)); +add("integrale1",hatch(H=.5mm,dir=NE,lightred)); +fill(integrale1,pattern("integrale1")); + +label(scale(.9)*"$\int_{a}^{c} f(x)\; dx$",((A.x+C.x)/2,2),deepred); + +path integrale2=buildcycle((C.x, F(C.x))--C--B--(B.x, F(B.x)),graph(F,C.x,B.x)); +add("integrale2",hatch(H=.5mm,dir=NW,lightgreen)); +fill(integrale2,pattern("integrale2")); + +label(scale(.9)*"$\int_{c}^{b} f(x)\; dx$",((C.x+B.x)/2,.5),deepgreen); + +draw(line(B-(0,2),B+(0,2)),deepblue+pen1); +label("$b$",B,SW,deepblue); + +draw(line(C-(0,2),C+(0,2)),deepgreen+pen1); +label("$c$",C,SW,deepgreen); + +shipout(bbox(0.1cm,0.1cm,white)); + diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.pdf b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.pdf new file mode 100644 index 0000000..bf8eaa0 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.png b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.png new file mode 100644 index 0000000..7fc2981 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.svg b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.svg new file mode 100644 index 0000000..59ae281 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig5.svg @@ -0,0 +1,463 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig6.pdf b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig6.pdf new file mode 100644 index 0000000..4baffb7 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig6.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig6.png b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig6.png new file mode 100644 index 0000000..5effd05 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/images/fig6.png differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/.correction applications du cours - partie VI - 2020.1.autosave.xopp b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/.correction applications du cours - partie VI - 2020.1.autosave.xopp new file mode 100644 index 0000000..5d30863 Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/.correction applications du cours - partie VI - 2020.1.autosave.xopp differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/images b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/images new file mode 120000 index 0000000..e4c5bd0 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/images @@ -0,0 +1 @@ +../images/ \ No newline at end of file diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieI.tex b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieI.tex new file mode 100644 index 0000000..9327f27 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieI.tex @@ -0,0 +1,147 @@ +% !TeX root = prof_-_calcul_integral_-_2023_1.tex + +\partie{Primitives} + +\noindent +Soit les fonctions $f(x) = 2x^2 + 3x + 5$ et $F(x) = \dfrac23 x^3 + \dfrac32 x^2 + 5x + 3$, définies sur $ \mathbb{R} $.\\ + +\noindent +Quel lien existe entre ces deux fonctions? + +\vspace{2cm} + +\hrule + +\sspartie{Définitions} + +\noindent +Dans cette partie, on considère une fonction $f$ définie sur un intervalle $I$ de $ \mathbb{R} $. + +\vspace{.4cm} + +% Définition +\begin{definition}[primitive] + On dit que $ F $ est une \textbf{primitive} de $ f $ sur $ I $ lorsque $ F $ est dérivable sur $ I $ et $ F'=f $. +\end{definition} + +\begin{propriete}[] + Toute fonction continue sur un intervalle $I$ admet une primitive sur cet intervalle. +\end{propriete} + +\begin{application} + \vspace{-.4cm} + $ f $ est la fonction définie sur $ \R $ par $ f(x) = e^x+2 $.\\ + Déterminer une primitive de la fonction $f$ sur $\R$. + \vspace{1cm} +\end{application} + +\vspace{.4cm} + +\begin{propriete}[primitives d'une même fonction] + Deux primitives d'une même fonction continue sur un intervalle diffèrent d'une constante. + \Cad que si $F$ est une primitive de $f$ sur $I$, alors la fonction $G_k$ définie sur $I$ par $G_k(x) = F(x) + k$ avec $k \in \mathbb{R}$ est aussi une primitive de $f$. +\end{propriete} + +\vspace{.4cm} + +\begin{propriete} + Soit $x_0$ et $y_0$ deux réels de $I$.\\ + Il existe une unique primitive $F$ de $f$ telle que $F(x_0) = y_0$.\\ + Cette valeur s'appelle une condition initiale. +\end{propriete} + +\begin{application}[] + Soit la fonction $f(x) = 5x^3 - 8x^2 + \dfrac54$ définie sur $ R $. + \begin{questions} + \item Déterminer l'ensemble des primitives $F_k$, où $k \in \R$ de la fonction $f$. + \item Déterminer la primitive de la fonction $f$ prenant la valeur $25$ en $3$. + \end{questions} +\end{application} + +\sspartie{Primitives des fonctions usuelles} + +\begin{center} + \small + \begin{tabularx}{1\textwidth}{| >{\centering\arraybackslash}X | >{\centering\arraybackslash}X | >{\centering\arraybackslash}X |} + \hline + \textbf{Fonction $\boldsymbol{f}$} & \textbf{Une primitive $\boldsymbol{F}$} & \textbf{Sur l'intervalle $\boldsymbol{I}$}\\ + \hline + \rule[-1ex]{0pt}{5ex}$ f(x)=a $, $ a \in \R $ & $ F(x)=ax+C $ & $ \R $\\[2ex] + \hline + \rule[-1ex]{0pt}{6ex}$ f(x)=x^n $, $ n \in \Z $ et $ n \neq -1 $ & $ F(x)=\dfrac{x^{n+1}}{n+1}+C $ & $\begin{array}{ccr}\R&\text{ si }n\in\N\\ ]-\infty\ ;\ 0[ \text{ ou } ]0\ ;\ +\infty[&\text{ si }n<0\\\end{array}$\\[3ex] + \hline + \rule[-1ex]{0pt}{6ex}$ f(x)=\dfrac1{x^2} $ & $ F(x)=-\dfrac1{x} +C $ & $ ]-\infty\ ;\ 0[ $ ou $ ]0\ ;\ +\infty[ $\\[2ex] + \hline + \rule[-1ex]{0pt}{6ex}$ f(x)=\dfrac1{x} $ & $ F(x)=\ln x +C $ & $ ]0\ ;\ +\infty[ $\\[2ex] + \hline + \rule[-1ex]{0pt}{6ex}$ f(x)=\dfrac1{\sqrt{x}} $ & $ F(x)=2\sqrt{x} +C $ & $ ]0\ ;\ +\infty[ $\\[4ex] + \hline + \rule[-1ex]{0pt}{5ex}$ f(x)=e^{x} $ & $ F(x)=e^x +C$ & $ \R $\\[2ex] + \hline + \rule[-1ex]{0pt}{5ex}$ f(x)=\cos{x} $ & $ F(x)=\sin{x} +C$ & $ \R $\\[2ex] + \hline + \rule[-1ex]{0pt}{5ex}$ f(x)=\sin{x} $ & $ F(x)=-\cos{x} +C$ & $ \R $\\[2ex] + \hline + \end{tabularx} +\end{center} + +\vspace{.4cm} + +\begin{application}[calcul de primitives (1)] + Calculer les primitives des fonctions suivantes: + \begin{tasks}(4)[style=enumerate] + \task $f(x) = x$ + \task $f(x) = x^3$ + \task $f(x) = 3x^5+1$ + \task $f(x) = \sqrt{x}$ + \task $f(x) = \dfrac1{x^2}$ + \task $f(x) = \dfrac1{\sqrt{x}}$ + \task $f(x) = \dfrac{-2}{x^4} + \dfrac1{\sqrt{x}}$ + \task $f(x) = 7\cos x - 2\sin x$ + \end{tasks} +\end{application} + +\sspartie{Primitives et composition} + +\begin{center} + \small + \begin{tabularx}{1\textwidth}{| >{\centering\arraybackslash}X | >{\centering\arraybackslash}X | >{\centering\arraybackslash}X |} + \hline + \textbf{Fonction $\boldsymbol{f}$} & \textbf{Une primitive $\boldsymbol{F}$ sur $\boldsymbol{I}$} & \textbf{Condition sur $\boldsymbol{u}$}\\ + \hline + \rule[-1ex]{0pt}{6ex}$ a\, u' $, $ a \in \R $ & $ a\, u + C $ & \\[3ex] + \hline + \rule[-1ex]{0pt}{6ex}$ u'+v' $ & $ u + v + C $ & \\[3ex] + \hline + \rule[-1ex]{0pt}{6ex}$ u'\, u^n $, $ n \in \Z $ et $ n \notin \{-1\; ;\ 0\} $ & $ \dfrac{u^{n+1}}{n+1}+C $ & $u(x) \neq 0 \text{ pour } x\in I \text{ si } n\le -2$\\[3ex] + \hline + \rule[-1ex]{0pt}{6ex}$ \dfrac{u'}{u} $ & $ \ln(u) +C $ & $ u(x)>0,\ x \in I $\\[2ex] + \hline + \rule[-1ex]{0pt}{6ex}$ \dfrac{u'}{u^2} $ & $ -\dfrac1{u} +C $ & $ u(x)\neq 0,\ x \in I $\\[2ex] + \hline + \rule[-1ex]{0pt}{6ex}$ \dfrac{u'}{\sqrt{u}} $ & $ 2\sqrt{u} +C $ & $ u(x)>0,\ x \in I $\\[2ex] + \hline + \rule[-1ex]{0pt}{6ex}$ u'e^{u} $ & $ e^{u} +C $ & \\[2ex] + \hline + \rule[-1ex]{0pt}{6ex}$ \cos(\omega\, x + \varphi) $, $ \omega \in \R $ et $ \varphi \in \R $ & $ \dfrac1{\omega}\sin(\omega\, x + \varphi) + C $ & \\[3ex] + \hline + \rule[-1ex]{0pt}{6ex}$ \sin(\omega\, x + \varphi) $, $ \omega \in \R $ et $ \varphi \in \R $ & $ -\dfrac1{\omega}\cos(\omega\, x + \varphi) + C $ & \\[3ex] + \hline + \end{tabularx} +\end{center} + +\vspace{.4cm} + +\begin{application}[calcul de primitives (2)] + Calculer les primitives des fonctions suivantes: + \begin{tasks}(4)[style=enumerate] + \task $ f(x) = \dfrac{2x}{x^2+1} $ + \task $ f(x) = (x+1)\e^{x^2+2x+1} $ + \task $ f(x) = \dfrac{x}{\sqrt{x^2+1}} $ + \task $ f(x) = \cos(2x + 7) $ + \task $ f(x) = \dfrac{x^3}{\left( x^4+1 \right)} $ + \task $ f(x) = \sin x\, \cos^3 x $ + \task $ f(x) = \tan x $ + \task $ f(x) = \dfrac{\sin x}{\cos^2 x } $ + \end{tasks} +\end{application} diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieII.tex b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieII.tex new file mode 100644 index 0000000..6b46844 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieII.tex @@ -0,0 +1,165 @@ +% !TeX root = prof_-_calcul_integral_-_2023_1.tex + +\newpage + +\partie{Intégration} + +\sspartie{Notion d'aire} + +\begin{definition}[intégrale] + On appelle intégrale de la fonction $f$ entre $a$ et $b$ est l'aire signée sous la courbe de $f$. + \begin{center} + \includegraphics[scale=.3]{fig1.png} + \end{center} + Ceci signifie qu'on compte positivement les aires situées au-dessus de l'axe des abscisses et négativement celles situées en-dessous, puis qu'on en fait la somme.\\ + + \noindent + On note alors cette intégrale \(\int_{a}^{b} f(x)\; dx\) et on lira <>. +\end{definition} + +\noindent +$\int_{a}^{b} f(x)\; dx$ peut être interpréter comme la <> des aires $f(x) \times dx$ des rectangles infinitésimaux de hauteur $f(x)$ et de largeur $dx$. + +\vspace{-8pt} + +\begin{center} + \begin{tabular}[c]{ccc} + \includegraphics[scale=.25]{fig2.png} & \hspace{.4cm} & \includegraphics[scale=.2]{fig3.png} + \end{tabular} +\end{center} + +\begin{multicols}{2} + \noindent + Sur l'intervalle $[a\ ;\ b]$ l'aire sous la courbe est la surface du domaine hachuré $D$.\\ + Cette aire est donc obtenue par calcul de \(\int_{a}^{b} f(t)\; dt\). + + \noindent + L'unité d'aire est donnée par la surface du rectangle $OIAJ$.\\\\ + + \columnbreak + + \begin{center} + \includegraphics[scale=.6]{fig4.pdf} + \end{center} +\end{multicols} + +\sspartie{Calcul d'intégrales} + +\begin{propriete}[calcul d'une intégrale] + Soit $F$ une primitive de $f$ sur l'intervalle $I$. Alors: \qquad $\int_{a}^{b} f(x)\; dx = [F(x)]_{a}^{b} = F(b)-F(a)$. +\end{propriete} + +\begin{remarque} + Dans l'écriture \(\int_{a}^{b} f(x)\; dx\), la lettre $x$ est une variable <>. Ainsi: $\int_{a}^{b} f(x)\; dx = \int_{a}^{b} f(y)\; dy = \int_{a}^{b} f(t)\; dt = \int_{a}^{b} f$ +\end{remarque} + +\medskip + +\begin{propriete}[] + \begin{tasks}(3)[style=enumerate] + \task \(\int_{a}^{a} f(x)\; dx = 0 \) + \task \(\int_{a}^{b} f(x)\; dx = -\int_{b}^{a} f(x)\; dx \) + \task \(\int_{a}^{a} k\; dx = k\, \left(b-a\right) \) + \end{tasks} +\end{propriete} + +\medskip + +\begin{application}[calculer une intégrale depuis une primitive] + Calculer : + % Questions + \begin{tasks}(2) + \task $ A = \int_{1}^{4} \dfrac3{x^2}\; dx $ + \task $ B = \int_{2}^{5} (3t^2+4t-5)\; dt $ + \task $ C = \int_{-1}^{1} 2t^2-1\; dt $ + \task $ D = \int_{-1}^{1} 2+e^{-2x}\; dx $ + \end{tasks} +\end{application} + +\sspartie{Propriétés de l'intégrale} + +\begin{propriete}[relation de Chasles] + Soit \(f\) une fonction continue sur un intervalle $ I=[a\ ;\ b] $. Soit $ c $ un réel de $ I $. + \begin{center} + \begin{tabular}{cc} + \rule[-1ex]{0pt}{2.5ex} + \begin{minipage}{.4\textwidth} + \[ + \int_{a}^{b} f(x)\; dx = \int_{a}^{c} f(x)\; dx + \int_{c}^{b} f(x)\; dx + \] + \end{minipage} + & + \begin{minipage}{.7\textwidth} + \includegraphics[scale=.7]{fig5.pdf} + \end{minipage} + \end{tabular} + \end{center} +\end{propriete} + +\medskip + +\begin{propriete}[linéarité] + \vspace{-.4cm} + Soient \(f\) et \(g\) deux fonctions continues sur un intervalle $ I=[a\ ;\ b] $ et $\lambda$ un réel quelconque. + \[ + \int_{a}^{b} (f(x) + \lambda\, g(x))\; dx = \int_{a}^{b} f(x)\; dx + \lambda\, \int_{a}^{b} g(x)\; dx + \] +\end{propriete} + +\medskip + +\begin{propriete}[inégalité] + Soient \(f\) et \(g\) deux fonctions continues sur un intervalle $ I=[a\ ;\ b] $. + \begin{itemize} + \item Si pour tout $ x \in [a\ ;\ b], f(x) \geq 0 $, alors $ \int_{a}^{b} f(x)\; dx \geq 0 $ ; + \item Si pour tout $ x \in [a\ ;\ b], f(x) \geq g(x) $, alors $ \int_{a}^{b} f(x)\; dx \geq \int_{a}^{b} g(x)\; dx $. + \end{itemize} +\end{propriete} + +\medskip + +\begin{application}[] + \vspace{-.4cm} + \small + % Questions + \begin{questions} + \item Calculer $ \int_{1}^{e} \dfrac1{x}\; dx $. + \item + % Questions + \begin{questions} + \item Justifier que la fonction $ x \mapsto \dfrac1{x+1} $ a pour primitive sur $ ]-1\ ;\ +\infty[ $ la fonction $ x \mapsto \ln(x+1) $.\\ + On admettra que la dérivée d'une fonction sous la forme $ \ln(u) $ est $ \dfrac{u'}{u} $. + \item Calculer $ \int_{1}^{e} \dfrac1{1+x}\; dx $. + \end{questions} + \item Démontrer que pour tout $ x > 0 $, $ \dfrac1{x(x+1)} = \dfrac1{x}-\dfrac1{x+1}$ + \item En déduire la valeur de $ \int_{1}^{e} \dfrac1{x(x+1)}\; dx $. + \end{questions} +\end{application} + +\medskip + +\begin{propriete}[valeur moyenne d'une fonction sur un intervalle] + % \underline{\textbf{Graphiquement :}} + \begin{center} + \begin{tabular}{l c r} + \begin{minipage}{.4\textwidth} + \noindent + On appelle valeur moyenne de $ f $ sur $ I $ le nombre réel: + \[ + m = \dfrac1{b-a}\, \int_{a}^{b} f(x)\, dx + \]\\ + + \noindent + L'aire sous la courbe représentative de $ f $ (en rouge ci-contre) est égale à l'aire sous la droite d'équation $ y = m $ (en bleu). + \end{minipage} + & + \hspace{.5cm} + & + \begin{minipage}{.7\textwidth} + \includegraphics[scale=.5]{fig6.pdf} + \end{minipage} + \end{tabular} + \end{center} +\end{propriete} + +\medbreak diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieIII.tex b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieIII.tex new file mode 100644 index 0000000..5a9f44a --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/partieIII.tex @@ -0,0 +1,47 @@ +% !TeX root = prof_-_calcul_integral_-_2023_1.tex + +\partie{Intégration par parties} + +\begin{propriete}[] + \vspace{-.4cm} + Soit $ u $ et $ v $ deux fonctions dérivables sur $ [a\ ;\ b] $ de dérivées $ u' $ et $ v' $ continues, alors : + + \[ + \int_{a}^{b} uv'(x)\; dx = \left[ u\; v(x)\right ]_{a}^{b} - \int_{a}^{b} u'v(x)\; dx + \] +\end{propriete} + +% remarque +\begin{remarque} + Le choix de $ u $ et $ v' $ est à faire judicieusement puisqu'il faut pouvoir trouver une primitive de $ u'v $. +\end{remarque} + +\medskip + +\begin{application}[intégration par partie] + \vspace{-.4cm} + Calculer $ \int_{0}^{1} x\e^{x}\; dx $. +% +% \noindent +% On ne peut pas trouver une primitive de $ x \mapsto x\e^{x} $ car non de la forme $ u'\e^{u} $.\\ +% On intègre par partie : +% +% \[ +% \begin{array}{rcl} +% u(x) = x & & u'(x) = 1 \\ +% v'(x) = \e^{x} & & v(x) = \e^{x} +% \end{array} +% \] +% +% \[ +% \int_{0}^{1} x\e^{x}\; dx = \left[ x\e^{x} \right]_{0}^{1} - \int_{0}^{1} \e^{x} \times 1\; dx = \left[ 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Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.36 \savepicturepage + {pgfid2}{1} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.37 \pgfsyspdfmark + {pgfid2}{1305270}{23310906} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.38 \savepicturepage + {pgfid3}{1} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.39 \pgfsyspdfmark + {pgfid3}{1305270}{17174441} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.40 \savepicturepage + {pgfid4}{1} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.41 \pgfsyspdfmark + {pgfid4}{1305270}{11295568} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.42 \savepicturepage + {pgfid5}{1} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.43 \pgfsyspdfmark + {pgfid5}{1305270}{4539756} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.46 \savepicturepage + {pgfid6}{2} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.47 \pgfsyspdfmark + {pgfid6}{1305270}{44980978} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.49 \savepicturepage + {pgfid7}{2} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.50 \pgfsyspdfmark + {pgfid7}{1305270}{14660377} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.52 \savepicturepage + {pgfid8}{3} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.53 \pgfsyspdfmark + {pgfid8}{1305270}{18097414} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.58 \savepicturepage + {pgfid9}{4} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.59 \pgfsyspdfmark + {pgfid9}{1305270}{33508000} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.60 \savepicturepage + {pgfid10}{4} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.61 \pgfsyspdfmark + {pgfid10}{1436342}{17297253} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +! Undefined control sequence. +l.63 \pagesLTS@ifcounter + {pagesLTS.arabic.1.local.cnt} +The control sequence at the end of the top line +of your error message was never \def'ed. 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calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/prof_-_calcul_integral_-_2023_1.pdf b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/prof_-_calcul_integral_-_2023_1.pdf new file mode 100644 index 0000000..447159b Binary files /dev/null and b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/prof_-_calcul_integral_-_2023_1.pdf differ diff --git a/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/prof_-_calcul_integral_-_2023_1.tex b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/prof_-_calcul_integral_-_2023_1.tex new file mode 100644 index 0000000..27e17c3 --- /dev/null +++ b/bts/bts2e/fonction variable reelle/sequence 2 - calcul integral/cours - calcul integral - 2023_1/prof - calcul integral - 2023_1/prof_-_calcul_integral_-_2023_1.tex @@ -0,0 +1,29 @@ +% File : prof_-_calcul_integral_-_2023_1.tex +% Author : Jeff Lance +% Date : 18.08.2023 12:20:01 +% Last Modified Date: 18.08.2023 12:20:01 +% Last Modified By : Jeff Lance + +\documentclass[11pt]{jl-cours} +\usepackage{cancel} +\usepackage{asymptote} + +\begin{document} +\pagenumbering{arabic} + +\titre{\Jd\ Calcul intégral}{\Jd\ Calcul intégral} +\lohead*{BTS2E} +\rofoot*{\anneescolaire} + +Dans ce document, $ f $ est une fonction définie et continue sur un intervalle $ I=[a\ ;\ b] $ de $ \R $. + +% Primitives +\import{.}{partieI.tex} + +% Intégration +\import{.}{partieII.tex} + +% Intégration par parties +\import{.}{partieIII.tex} + +\end{document}