add two first valid sequences
This commit is contained in:
parent
69bf3b6387
commit
2db6555139
@ -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));
|
||||
|
||||
@ -0,0 +1,23 @@
|
||||
<?xml version="1.0" encoding="UTF-8"?>
|
||||
<Courbes>
|
||||
<UnitX>1</UnitX>
|
||||
<Xmin>-3.5</Xmin>
|
||||
<Xmax>10.5</Xmax>
|
||||
<UnitY>1</UnitY>
|
||||
<Ymin>-.75</Ymin>
|
||||
<Ymax>8.5</Ymax>
|
||||
<GradX>1</GradX>
|
||||
<GradY>1</GradY>
|
||||
<GrilleX>1</GrilleX>
|
||||
<GrilleY>1</GrilleY>
|
||||
<Axes>1</Axes>
|
||||
<Labels>1</Labels>
|
||||
<Repere>0</Repere>
|
||||
<Grille>1</Grille>
|
||||
<Supplement></Supplement>
|
||||
<Fonctions>x^2#sqrt(x)</Fonctions>
|
||||
<Varmins>-4#0</Varmins>
|
||||
<Varmaxs>11#11</Varmaxs>
|
||||
<Couleurs>noir#noir</Couleurs>
|
||||
<Styles>ligne#ligne</Styles>
|
||||
</Courbes>
|
||||
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 85 KiB |
File diff suppressed because it is too large
Load Diff
|
After Width: | Height: | Size: 168 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 43 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 53 KiB |
@ -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));
|
||||
|
||||
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 41 KiB |
File diff suppressed because it is too large
Load Diff
|
After Width: | Height: | Size: 100 KiB |
@ -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));
|
||||
|
||||
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 39 KiB |
@ -0,0 +1,116 @@
|
||||
<?xml version="1.0" standalone="no"?>
|
||||
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 20010904//EN"
|
||||
"http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd">
|
||||
<svg version="1.0" xmlns="http://www.w3.org/2000/svg"
|
||||
width="582.000000pt" height="640.000000pt" viewBox="0 0 582.000000 640.000000"
|
||||
preserveAspectRatio="xMidYMid meet">
|
||||
<metadata>
|
||||
Created by potrace 1.16, written by Peter Selinger 2001-2019
|
||||
</metadata>
|
||||
<g transform="translate(0.000000,640.000000) scale(0.100000,-0.100000)"
|
||||
fill="#000000" stroke="none">
|
||||
<path d="M1628 6209 c-19 -11 -25 -49 -8 -49 6 0 10 9 10 20 0 24 25 27 34 4
|
||||
7 -18 -4 -42 -33 -68 -30 -28 -26 -36 19 -36 35 0 40 3 39 23 0 12 -3 16 -6
|
||||
10 -2 -7 -14 -13 -26 -13 -20 0 -19 1 6 28 32 33 35 67 8 82 -23 12 -22 12
|
||||
-43 -1z"/>
|
||||
<path d="M1728 6208 c-9 -7 -19 -29 -23 -48 -10 -61 33 -110 70 -80 21 17 20
|
||||
104 -1 124 -19 19 -25 20 -46 4z m37 -64 c0 -38 -4 -49 -17 -52 -15 -3 -18 4
|
||||
-18 46 0 27 3 52 7 56 16 16 28 -4 28 -50z"/>
|
||||
<path d="M1922 6158 c3 -20 10 -23 56 -26 28 -2 52 -7 52 -12 0 -25 -32 -154
|
||||
-55 -220 -39 -110 -40 -119 -19 -127 13 -5 26 4 51 33 l33 39 0 -567 0 -567
|
||||
-57 -3 c-54 -3 -58 -5 -58 -28 0 -23 4 -25 58 -28 l57 -3 0 -709 0 -710 -270
|
||||
0 -270 0 0 48 c0 50 -13 74 -36 65 -10 -3 -14 -22 -14 -59 l0 -54 -270 0 -270
|
||||
0 0 49 c0 56 -19 82 -45 61 -10 -8 -15 -30 -15 -61 l0 -49 -265 0 -265 0 0 49
|
||||
c0 56 -19 82 -45 61 -12 -10 -15 -38 -15 -134 0 -66 5 -128 10 -136 14 -22 44
|
||||
0 48 35 l4 28 165 -6 c92 -4 210 -7 263 -7 l97 0 7 -31 c5 -25 10 -30 28 -27
|
||||
17 2 24 11 26 32 l4 29 111 -6 c62 -4 181 -7 265 -7 l152 0 10 -25 c12 -31 40
|
||||
-33 47 -4 6 22 7 22 202 15 108 -3 229 -9 269 -13 l72 -6 0 -663 0 -664 -61 0
|
||||
c-57 0 -60 -1 -57 -23 3 -19 10 -22 61 -25 l57 -3 0 -709 0 -709 -57 -3 c-51
|
||||
-3 -58 -6 -61 -25 -3 -23 -3 -23 137 -23 86 0 142 4 146 10 13 22 -17 40 -66
|
||||
40 l-49 0 0 710 0 710 49 0 c49 0 79 18 66 40 -3 5 -31 10 -61 10 l-54 0 0
|
||||
666 0 665 53 -5 c28 -3 93 -8 142 -11 367 -24 696 -79 861 -143 34 -14 50 -26
|
||||
52 -41 4 -23 22 -29 22 -7 0 9 7 12 20 9 27 -7 25 -16 -10 -56 -16 -19 -30
|
||||
-37 -30 -41 0 -3 18 -6 40 -6 35 0 40 3 39 23 0 12 -3 16 -6 10 -2 -7 -14 -13
|
||||
-26 -13 -20 0 -20 0 3 25 13 14 28 25 34 25 25 0 129 -95 175 -160 50 -72 103
|
||||
-195 140 -325 51 -182 97 -613 121 -1155 8 -179 29 -862 30 -948 0 -8 9 -12
|
||||
23 -10 21 3 22 6 22 118 -1 163 -15 617 -30 975 -21 469 -80 962 -131 1084 -8
|
||||
19 -17 48 -20 65 -8 50 -78 187 -127 252 -120 157 -336 259 -657 310 -163 25
|
||||
-150 19 -150 75 l0 49 270 0 269 0 3 -57 c3 -50 6 -58 23 -58 17 0 20 8 23 58
|
||||
l3 57 269 0 269 0 3 -57 c3 -50 6 -58 23 -58 17 0 20 8 23 58 l3 57 269 0 270
|
||||
0 0 -55 c0 -42 4 -57 16 -62 23 -8 34 14 34 70 l0 47 270 0 269 0 3 -57 c3
|
||||
-50 6 -58 23 -58 17 0 20 8 23 58 l3 57 126 0 126 0 -36 -27 c-35 -27 -47 -56
|
||||
-27 -68 5 -3 49 8 97 25 81 29 205 60 238 60 10 0 15 -16 17 -52 3 -45 6 -53
|
||||
23 -53 19 0 20 8 23 134 1 74 -1 138 -6 143 -5 5 -19 4 -34 -2 -18 -9 -86 -7
|
||||
-284 9 -490 39 -770 104 -942 218 -224 148 -322 419 -377 1037 -29 327 -38
|
||||
484 -50 896 -3 107 -8 238 -9 290 -2 52 -4 157 -5 233 l-1 137 -25 0 -25 0 0
|
||||
-127 c0 -153 16 -632 30 -938 27 -566 79 -975 151 -1182 72 -210 189 -352 359
|
||||
-436 130 -64 316 -113 533 -142 l47 -6 0 -50 0 -49 -270 0 -270 0 0 48 c0 50
|
||||
-13 74 -36 65 -10 -3 -14 -22 -14 -59 l0 -54 -270 0 -270 0 0 54 c0 40 -4 55
|
||||
-16 60 -27 10 -34 -3 -34 -60 l0 -54 -270 0 -270 0 0 49 c0 49 -18 79 -40 66
|
||||
-5 -3 -10 -31 -10 -61 l0 -54 -270 0 -270 0 0 54 c0 40 -4 55 -16 60 -27 10
|
||||
-34 -3 -34 -60 l0 -54 -270 0 -270 0 0 710 0 710 49 0 c56 0 82 19 61 45 -8
|
||||
10 -30 15 -61 15 l-49 0 0 563 1 562 27 -33 c31 -37 62 -43 62 -11 0 12 -15
|
||||
65 -34 118 -19 53 -37 118 -41 145 -4 27 -9 55 -12 62 -4 11 7 14 50 14 41 0
|
||||
56 4 61 16 3 9 4 20 0 25 -3 5 -68 9 -145 9 -140 0 -140 0 -137 -22z m3243
|
||||
-2848 c23 0 44 -11 76 -40 l44 -40 -122 0 -123 0 0 46 0 47 46 -7 c26 -3 61
|
||||
-6 79 -6z m382 -30 c28 0 33 -3 33 -24 l0 -23 -87 18 c-133 27 -156 41 -59 34
|
||||
43 -3 94 -5 113 -5z m-4744 -107 c-29 -2 -78 -2 -110 0 -32 2 -8 3 52 3 61 0
|
||||
87 -1 58 -3z m647 -3 c0 -11 -199 -12 -410 -1 -119 7 -103 8 138 9 176 1 272
|
||||
-2 272 -8z m590 -16 c0 -25 -2 -26 -37 -21 -21 4 -143 9 -270 13 -185 6 -233
|
||||
10 -233 21 0 10 50 13 270 13 l270 0 0 -26z m590 -19 c0 -53 20 -50 -202 -30
|
||||
-90 8 -202 17 -250 20 -85 7 -88 8 -88 31 l0 24 270 0 270 0 0 -45z"/>
|
||||
<path d="M1630 4725 c-19 -11 -21 -14 -7 -15 14 0 17 -9 17 -49 0 -30 -5 -51
|
||||
-12 -54 -7 -3 5 -5 27 -5 22 0 32 2 23 5 -13 3 -18 18 -20 68 l-3 63 -25 -13z"/>
|
||||
<path d="M1728 4728 c-9 -7 -19 -29 -23 -48 -10 -61 33 -110 70 -80 21 17 20
|
||||
104 -1 124 -19 19 -25 20 -46 4z m37 -64 c0 -38 -4 -49 -17 -52 -15 -3 -18 4
|
||||
-18 46 0 27 3 52 7 56 16 16 28 -4 28 -50z"/>
|
||||
<path d="M348 2909 c-20 -11 -25 -43 -5 -36 6 2 11 10 10 16 -2 6 4 11 12 11
|
||||
8 0 15 -9 15 -20 0 -11 -7 -20 -15 -20 -22 0 -18 -17 6 -23 28 -7 21 -41 -9
|
||||
-45 -16 -3 -20 0 -16 12 4 11 1 16 -10 16 -20 0 -20 -14 0 -34 19 -19 48 -21
|
||||
62 -3 16 19 10 116 -8 127 -18 12 -21 12 -42 -1z"/>
|
||||
<path d="M1530 2905 c-19 -11 -21 -14 -7 -15 14 0 17 -9 17 -49 0 -30 -5 -51
|
||||
-12 -54 -7 -3 5 -5 27 -5 22 0 32 2 23 5 -13 3 -18 18 -20 68 l-3 63 -25 -13z"/>
|
||||
<path d="M2636 2904 c-11 -8 -14 -14 -8 -14 17 0 16 -97 0 -103 -7 -3 5 -5 27
|
||||
-5 22 0 32 2 23 5 -13 3 -18 18 -20 68 -3 61 -4 63 -22 49z"/>
|
||||
<path d="M3808 2909 c-19 -11 -25 -39 -8 -39 6 0 10 7 10 15 0 8 7 15 15 15 8
|
||||
0 15 -9 15 -20 0 -11 -7 -20 -15 -20 -22 0 -18 -17 6 -23 28 -7 21 -41 -9 -45
|
||||
-16 -3 -20 0 -16 12 4 11 1 16 -10 16 -20 0 -20 -14 0 -34 19 -19 48 -21 62
|
||||
-3 16 19 10 116 -8 127 -18 12 -21 12 -42 -1z"/>
|
||||
<path d="M4990 2876 c0 -34 2 -37 16 -25 13 10 17 10 25 -2 5 -8 8 -23 7 -34
|
||||
-3 -23 -38 -34 -38 -11 0 8 -4 18 -10 21 -6 3 -10 -5 -10 -19 0 -29 29 -42 60
|
||||
-26 27 15 27 65 0 80 -11 6 -25 8 -30 5 -6 -3 -10 1 -10 9 0 8 10 16 22 18 33
|
||||
5 32 23 -2 23 -28 0 -30 -2 -30 -39z"/>
|
||||
<path d="M5594 2902 c-27 -22 -35 -45 -27 -83 7 -37 44 -59 68 -39 30 25 13
|
||||
90 -24 90 -11 0 -11 -4 3 -19 20 -22 18 -56 -4 -56 -11 0 -15 13 -17 47 -1 44
|
||||
15 78 25 50 5 -17 32 -15 32 2 0 8 -7 17 -16 20 -19 7 -15 9 -40 -12z"/>
|
||||
<path d="M927 2904 c-11 -11 -8 -44 3 -44 6 0 10 9 10 20 0 24 25 27 34 4 7
|
||||
-18 -4 -42 -33 -68 -30 -28 -26 -36 19 -36 35 0 40 3 39 23 0 12 -3 16 -6 10
|
||||
-2 -7 -14 -13 -26 -13 -20 0 -19 2 7 29 37 39 33 75 -9 79 -17 2 -34 0 -38 -4z"/>
|
||||
<path d="M4411 2866 c-17 -25 -31 -47 -31 -50 0 -3 12 -6 26 -6 19 0 25 -4 20
|
||||
-15 -4 -11 2 -15 20 -15 16 0 23 4 18 11 -3 6 0 15 7 20 10 6 11 9 2 9 -8 0
|
||||
-13 15 -13 39 0 22 -4 42 -9 45 -4 3 -22 -14 -40 -38z m29 -16 c0 -23 -4 -30
|
||||
-20 -30 -20 0 -20 1 -5 30 9 17 18 30 20 30 3 0 5 -13 5 -30z"/>
|
||||
<path d="M170 2830 c0 -6 28 -10 65 -10 37 0 65 4 65 10 0 6 -28 10 -65 10
|
||||
-37 0 -65 -4 -65 -10z"/>
|
||||
<path d="M760 2830 c0 -6 28 -10 65 -10 37 0 65 4 65 10 0 6 -28 10 -65 10
|
||||
-37 0 -65 -4 -65 -10z"/>
|
||||
<path d="M1360 2830 c0 -6 28 -10 65 -10 37 0 65 4 65 10 0 6 -28 10 -65 10
|
||||
-37 0 -65 -4 -65 -10z"/>
|
||||
<path d="M1630 1785 c-19 -11 -21 -14 -7 -15 14 0 17 -9 17 -49 0 -30 -5 -51
|
||||
-12 -54 -7 -3 5 -5 27 -5 22 0 32 2 23 5 -13 3 -18 18 -20 68 l-3 63 -25 -13z"/>
|
||||
<path d="M1728 1788 c-9 -7 -19 -29 -23 -48 -10 -61 33 -110 70 -80 21 17 20
|
||||
104 -1 124 -19 19 -25 20 -46 4z m37 -64 c0 -38 -4 -49 -17 -52 -15 -3 -18 4
|
||||
-18 46 0 27 3 52 7 56 16 16 28 -4 28 -50z"/>
|
||||
<path d="M1465 1710 c-4 -6 17 -10 54 -10 34 0 61 4 61 10 0 6 -24 10 -54 10
|
||||
-30 0 -58 -4 -61 -10z"/>
|
||||
<path d="M1628 309 c-19 -11 -25 -39 -8 -39 6 0 10 7 10 15 0 20 26 19 34 -1
|
||||
7 -18 -4 -42 -33 -68 -30 -28 -26 -36 19 -36 35 0 40 3 39 23 0 12 -3 16 -6
|
||||
10 -2 -7 -14 -13 -26 -13 -20 0 -19 1 6 28 32 33 35 67 8 82 -23 12 -22 12
|
||||
-43 -1z"/>
|
||||
<path d="M1728 308 c-9 -7 -19 -29 -23 -48 -10 -61 33 -110 70 -80 21 17 20
|
||||
104 -1 124 -19 19 -25 20 -46 4z m37 -64 c0 -38 -4 -49 -17 -52 -15 -3 -18 4
|
||||
-18 46 0 27 3 52 7 56 16 16 28 -4 28 -50z"/>
|
||||
<path d="M1450 230 c0 -6 28 -10 65 -10 37 0 65 4 65 10 0 6 -28 10 -65 10
|
||||
-37 0 -65 -4 -65 -10z"/>
|
||||
</g>
|
||||
</svg>
|
||||
|
After Width: | Height: | Size: 7.4 KiB |
@ -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));
|
||||
|
||||
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 5.2 KiB |
File diff suppressed because it is too large
Load Diff
|
After Width: | Height: | Size: 68 KiB |
Binary file not shown.
@ -0,0 +1 @@
|
||||
../images/
|
||||
@ -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}
|
||||
|
||||
@ -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}
|
||||
@ -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}
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
Binary file not shown.
@ -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}
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 33 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 31 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 32 KiB |
@ -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));
|
||||
|
||||
|
||||
Binary file not shown.
@ -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));
|
||||
|
||||
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 81 KiB |
File diff suppressed because one or more lines are too long
|
After Width: | Height: | Size: 61 KiB |
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 1.4 MiB |
Binary file not shown.
@ -0,0 +1 @@
|
||||
../images/
|
||||
@ -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}
|
||||
@ -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 <<intégrale de \(a\) à \(b\) de \(f(x)\)>>.
|
||||
\end{definition}
|
||||
|
||||
\noindent
|
||||
$\int_{a}^{b} f(x)\; dx$ peut être interpréter comme la <<somme infinie>> 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 <<muette>>. 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
|
||||
@ -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 :
|
||||
%
|
||||
% \[
|
||||
% \begin{array}{rcl}
|
||||
% u(x) = x & & u'(x) = 1 \\
|
||||
% v'(x) = \e^{x} & & v(x) = \e^{x}
|
||||
% \end{array}
|
||||
% \]
|
||||
%
|
||||
% \[
|
||||
% \int_{0}^{1} x\e^{x}\; dx = \left[ x\e^{x} \right]_{0}^{1} - \int_{0}^{1} \e^{x} \times 1\; dx = \left[ x\e^{x} \right]_{0}^{1} - \int_{0}^{1} \e^{x} \; dx = \left[ x\e^{x} \right]_{0}^{1} - \left[\e^{x} \right]_{0}^{1} = (\e - 0) - (\e - 1) = 1.
|
||||
% \]
|
||||
\end{application}
|
||||
|
||||
\medskip
|
||||
|
||||
% application
|
||||
\begin{application}[déterminer une primitive]
|
||||
\vspace{-.4cm}
|
||||
Déterminer une primitive de $ \ln $ sur $ ]0\ ;\ +\infty[ $.
|
||||
\end{application}
|
||||
@ -0,0 +1,623 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Arch Linux) (preloaded format=pdflatex 2023.8.3) 15 AUG 2023 20:51
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
**\scrollmode
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\documentclass[12pt]{article}
|
||||
|
||||
*\let\paperwidthsave\paperwidth\let\paperwidth\undefined
|
||||
(/usr/share/texmf-dist/tex/latex/base/article.cls
|
||||
Document Class: article 2022/07/02 v1.4n Standard LaTeX document class
|
||||
(/usr/share/texmf-dist/tex/latex/base/size12.clo
|
||||
File: size12.clo 2022/07/02 v1.4n Standard LaTeX file (size option)
|
||||
)
|
||||
\c@part=\count185
|
||||
\c@section=\count186
|
||||
\c@subsection=\count187
|
||||
\c@subsubsection=\count188
|
||||
\c@paragraph=\count189
|
||||
\c@subparagraph=\count190
|
||||
\c@figure=\count191
|
||||
\c@table=\count192
|
||||
\abovecaptionskip=\skip48
|
||||
\belowcaptionskip=\skip49
|
||||
\bibindent=\dimen140
|
||||
)
|
||||
*\usepackage{graphicx}
|
||||
|
||||
*\let\paperwidth\paperwidthsave
|
||||
(/usr/share/texmf-dist/tex/latex/graphics/graphicx.sty
|
||||
Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR)
|
||||
(/usr/share/texmf-dist/tex/latex/graphics/keyval.sty
|
||||
Package: keyval 2022/05/29 v1.15 key=value parser (DPC)
|
||||
\KV@toks@=\toks16
|
||||
) (/usr/share/texmf-dist/tex/latex/graphics/graphics.sty
|
||||
Package: graphics 2022/03/10 v1.4e Standard LaTeX Graphics (DPC,SPQR)
|
||||
(/usr/share/texmf-dist/tex/latex/graphics/trig.sty
|
||||
Package: trig 2021/08/11 v1.11 sin cos tan (DPC)
|
||||
) (/usr/share/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
|
||||
File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration
|
||||
)
|
||||
Package graphics Info: Driver file: pdftex.def on input line 107.
|
||||
(/usr/share/texmf-dist/tex/latex/graphics-def/pdftex.def
|
||||
File: pdftex.def 2022/09/22 v1.2b Graphics/color driver for pdftex
|
||||
))
|
||||
\Gin@req@height=\dimen141
|
||||
\Gin@req@width=\dimen142
|
||||
)
|
||||
*\usepackage{amsmath,amssymb}
|
||||
|
||||
*
|
||||
(/usr/share/texmf-dist/tex/latex/amsmath/amsmath.sty
|
||||
Package: amsmath 2022/04/08 v2.17n AMS math features
|
||||
\@mathmargin=\skip50
|
||||
|
||||
For additional information on amsmath, use the `?' option.
|
||||
(/usr/share/texmf-dist/tex/latex/amsmath/amstext.sty
|
||||
Package: amstext 2021/08/26 v2.01 AMS text
|
||||
(/usr/share/texmf-dist/tex/latex/amsmath/amsgen.sty
|
||||
File: amsgen.sty 1999/11/30 v2.0 generic functions
|
||||
\@emptytoks=\toks17
|
||||
\ex@=\dimen143
|
||||
)) (/usr/share/texmf-dist/tex/latex/amsmath/amsbsy.sty
|
||||
Package: amsbsy 1999/11/29 v1.2d Bold Symbols
|
||||
\pmbraise@=\dimen144
|
||||
) (/usr/share/texmf-dist/tex/latex/amsmath/amsopn.sty
|
||||
Package: amsopn 2022/04/08 v2.04 operator names
|
||||
)
|
||||
\inf@bad=\count193
|
||||
LaTeX Info: Redefining \frac on input line 234.
|
||||
\uproot@=\count194
|
||||
\leftroot@=\count195
|
||||
LaTeX Info: Redefining \overline on input line 399.
|
||||
LaTeX Info: Redefining \colon on input line 410.
|
||||
\classnum@=\count196
|
||||
\DOTSCASE@=\count197
|
||||
LaTeX Info: Redefining \ldots on input line 496.
|
||||
LaTeX Info: Redefining \dots on input line 499.
|
||||
LaTeX Info: Redefining \cdots on input line 620.
|
||||
\Mathstrutbox@=\box51
|
||||
\strutbox@=\box52
|
||||
LaTeX Info: Redefining \big on input line 722.
|
||||
LaTeX Info: Redefining \Big on input line 723.
|
||||
LaTeX Info: Redefining \bigg on input line 724.
|
||||
LaTeX Info: Redefining \Bigg on input line 725.
|
||||
\big@size=\dimen145
|
||||
LaTeX Font Info: Redeclaring font encoding OML on input line 743.
|
||||
LaTeX Font Info: Redeclaring font encoding OMS on input line 744.
|
||||
\macc@depth=\count198
|
||||
LaTeX Info: Redefining \bmod on input line 905.
|
||||
LaTeX Info: Redefining \pmod on input line 910.
|
||||
LaTeX Info: Redefining \smash on input line 940.
|
||||
LaTeX Info: Redefining \relbar on input line 970.
|
||||
LaTeX Info: Redefining \Relbar on input line 971.
|
||||
\c@MaxMatrixCols=\count199
|
||||
\dotsspace@=\muskip16
|
||||
\c@parentequation=\count266
|
||||
\dspbrk@lvl=\count267
|
||||
\tag@help=\toks18
|
||||
\row@=\count268
|
||||
\column@=\count269
|
||||
\maxfields@=\count270
|
||||
\andhelp@=\toks19
|
||||
\eqnshift@=\dimen146
|
||||
\alignsep@=\dimen147
|
||||
\tagshift@=\dimen148
|
||||
\tagwidth@=\dimen149
|
||||
\totwidth@=\dimen150
|
||||
\lineht@=\dimen151
|
||||
\@envbody=\toks20
|
||||
\multlinegap=\skip51
|
||||
\multlinetaggap=\skip52
|
||||
\mathdisplay@stack=\toks21
|
||||
LaTeX Info: Redefining \[ on input line 2953.
|
||||
LaTeX Info: Redefining \] on input line 2954.
|
||||
) (/usr/share/texmf-dist/tex/latex/amsfonts/amssymb.sty
|
||||
Package: amssymb 2013/01/14 v3.01 AMS font symbols
|
||||
(/usr/share/texmf-dist/tex/latex/amsfonts/amsfonts.sty
|
||||
Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
|
||||
\symAMSa=\mathgroup4
|
||||
\symAMSb=\mathgroup5
|
||||
LaTeX Font Info: Redeclaring math symbol \hbar on input line 98.
|
||||
LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
(Font) U/euf/m/n --> U/euf/b/n on input line 106.
|
||||
))
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage[utf8]{inputenc}
|
||||
|
||||
*
|
||||
(/usr/share/texmf-dist/tex/latex/base/inputenc.sty
|
||||
Package: inputenc 2021/02/14 v1.3d Input encoding file
|
||||
\inpenc@prehook=\toks22
|
||||
\inpenc@posthook=\toks23
|
||||
)
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage{icomma}
|
||||
|
||||
*
|
||||
(/usr/share/texmf-dist/tex/latex/was/icomma.sty
|
||||
Package: icomma 2002/03/10 v2.0 (WaS)
|
||||
)
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage{amsmath,amssymb}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage[utf8]{inputenc}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage{icomma}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage{esvect}
|
||||
|
||||
*
|
||||
(/usr/share/texmf-dist/tex/latex/esvect/esvect.sty
|
||||
Package: esvect
|
||||
\symesvector=\mathgroup6
|
||||
)
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage{mathrsfs}
|
||||
|
||||
*
|
||||
(/usr/share/texmf-dist/tex/latex/jknapltx/mathrsfs.sty
|
||||
Package: mathrsfs 1996/01/01 Math RSFS package v1.0 (jk)
|
||||
\symrsfs=\mathgroup7
|
||||
)
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage{mathrsfs}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\usepackage{type1cm}
|
||||
|
||||
*
|
||||
(/usr/share/texmf-dist/tex/latex/type1cm/type1cm.sty
|
||||
Package: type1cm 2002/09/05 v0.04 BlueSky/Y&Y Type1 CM font definitions (DPC, patched RF)
|
||||
)
|
||||
(Please type a command or say `\end')
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\newbox\ASYbox
|
||||
\ASYbox=\box53
|
||||
|
||||
*\newdimen\ASYdimen
|
||||
\ASYdimen=\dimen152
|
||||
|
||||
*\def\ASYprefix{}
|
||||
|
||||
*\long\def\ASYbase#1#2{\leavevmode\setbox\ASYbox=\hbox{#1}%\ASYdimen=\ht\ASYbox%
|
||||
|
||||
*\setbox\ASYbox=\hbox{#2}\lower\ASYdimen\box\ASYbox}
|
||||
|
||||
*\begin{document}
|
||||
(/usr/share/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
File: l3backend-pdftex.def 2023-01-16 L3 backend support: PDF output (pdfTeX)
|
||||
\l__color_backend_stack_int=\count271
|
||||
\l__pdf_internal_box=\box54
|
||||
) (build/prof_-_calcul_integral_-_2023_1.aux
|
||||
! Undefined control sequence.
|
||||
l.27 \reset@newl@bel
|
||||
|
||||
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.28 \xpglanginauxtrue
|
||||
|
||||
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.
|
||||
|
||||
! Not loaded.
|
||||
\selectlanguage ...ing it}\errmessage {Not loaded}
|
||||
|
||||
l.29 \selectlanguage
|
||||
*{french}
|
||||
Selecting a language requires a package supporting it
|
||||
|
||||
|
||||
! LaTeX Error: Missing \begin{document}.
|
||||
|
||||
See the LaTeX manual or LaTeX Companion for explanation.
|
||||
Type H <return> for immediate help.
|
||||
...
|
||||
|
||||
l.29 \selectlanguage *
|
||||
{french}
|
||||
You're in trouble here. Try typing <return> to proceed.
|
||||
If that doesn't work, type X <return> to quit.
|
||||
|
||||
! Undefined control sequence.
|
||||
l.31 \xpglanginauxfalse
|
||||
|
||||
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.34 \savepicturepage
|
||||
{pgfid1}{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.35 \pgfsyspdfmark
|
||||
{pgfid1}{1305270}{27538278}
|
||||
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.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. 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.
|
||||
|
||||
|
||||
! LaTeX Error: No counter 'pagesLTS.arabic.1.local.cnt' defined.
|
||||
|
||||
See the LaTeX manual or LaTeX Companion for explanation.
|
||||
Type H <return> for immediate help.
|
||||
...
|
||||
|
||||
l.64 \setcounter{pagesLTS.arabic.1.local.cnt}{4}
|
||||
|
||||
Your command was ignored.
|
||||
Type I <command> <return> to replace it with another command,
|
||||
or <return> to continue without it.
|
||||
|
||||
|
||||
! LaTeX Error: No counter 'pagesLTS.pagenr' defined.
|
||||
|
||||
See the LaTeX manual or LaTeX Companion for explanation.
|
||||
Type H <return> for immediate help.
|
||||
...
|
||||
|
||||
l.66 \setcounter{pagesLTS.pagenr}{4}
|
||||
|
||||
Your command was ignored.
|
||||
Type I <command> <return> to replace it with another command,
|
||||
or <return> to continue without it.
|
||||
|
||||
)
|
||||
\openout1 = `prof_-_calcul_integral_-_2023_1.aux'.
|
||||
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
|
||||
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
|
||||
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
|
||||
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
|
||||
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
|
||||
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
|
||||
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 0.
|
||||
LaTeX Font Info: ... okay on input line 0.
|
||||
(/usr/share/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
|
||||
Overfull \hbox (42.78717pt too wide) in paragraph at lines 29--13
|
||||
\OT1/cmr/m/n/12 fid513052704539756 pg-fid62 pg-fid6130527044980978 pg-fid72 pg-fid7130527014660377
|
||||
[]
|
||||
|
||||
[Loading MPS to PDF converter (version 2006.09.02).]
|
||||
\scratchcounter=\count272
|
||||
\scratchdimen=\dimen153
|
||||
\scratchbox=\box55
|
||||
\nofMPsegments=\count273
|
||||
\nofMParguments=\count274
|
||||
\everyMPshowfont=\toks24
|
||||
\MPscratchCnt=\count275
|
||||
\MPscratchDim=\dimen154
|
||||
\MPnumerator=\count276
|
||||
\makeMPintoPDFobject=\count277
|
||||
\everyMPtoPDFconversion=\toks25
|
||||
) (/usr/share/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
|
||||
Package: epstopdf-base 2020-01-24 v2.11 Base part for package epstopdf
|
||||
Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 485.
|
||||
(/usr/share/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
|
||||
File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live
|
||||
))
|
||||
*\makeatletter%
|
||||
|
||||
*\let\ASYencoding\f@encoding%
|
||||
|
||||
*\let\ASYfamily\f@family%
|
||||
|
||||
*\let\ASYseries\f@series%
|
||||
|
||||
*\let\ASYshape\f@shape%
|
||||
|
||||
*\makeatother%
|
||||
|
||||
*\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{}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\immediate\write16{>dim(\the\wd\ASYbox)dim}
|
||||
>dim(0.0pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\ht\ASYbox)dim}
|
||||
>dim(0.0pt)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{$O$}
|
||||
LaTeX Font Info: Trying to load font information for U+esvect on input line 0.
|
||||
(/usr/share/texmf-dist/tex/latex/esvect/uesvect.fd
|
||||
File: uesvect.fd
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+rsfs on input line 0.
|
||||
(/usr/share/texmf-dist/tex/latex/jknapltx/ursfs.fd
|
||||
File: ursfs.fd 1998/03/24 rsfs font definition file (jk)
|
||||
)
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\immediate\write16{>dim(\the\wd\ASYbox)dim}
|
||||
>dim(9.30602pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\ht\ASYbox)dim}
|
||||
>dim(8.2pt)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{$I$}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\immediate\write16{>dim(\the\wd\ASYbox)dim}
|
||||
>dim(6.12566pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\ht\ASYbox)dim}
|
||||
>dim(8.2pt)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{$J$}
|
||||
|
||||
*
|
||||
(Please type a command or say `\end')
|
||||
*\immediate\write16{>dim(\the\wd\ASYbox)dim}
|
||||
>dim(7.60553pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\ht\ASYbox)dim}
|
||||
>dim(8.2pt)dim
|
||||
|
||||
*\immediate\write16{>dim(\the\dp\ASYbox)dim
|
||||
Binary file not shown.
@ -0,0 +1,29 @@
|
||||
% File : prof_-_calcul_integral_-_2023_1.tex
|
||||
% Author : Jeff Lance <email@jefflance.me>
|
||||
% Date : 18.08.2023 12:20:01
|
||||
% Last Modified Date: 18.08.2023 12:20:01
|
||||
% Last Modified By : Jeff Lance <email@jefflance.me>
|
||||
|
||||
\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}
|
||||
Loading…
Reference in New Issue
Block a user