374 lines
10 KiB
TeX
374 lines
10 KiB
TeX
\documentclass{article}
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\usepackage{mathtools}
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\usepackage{amssymb}
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\usepackage{pgfplots}
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\usepgfplotslibrary{external}
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\tikzexternalize
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\tikzset{%
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external/system call={pdflatex \tikzexternalcheckshellescape --halt-on-error --interaction=batchmode --output-directory=./build --jobname "\image" "\texsource"},
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/pgf/images/include external/.code={%
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\includegraphics{build/#1}%
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},
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}
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\title{Analyse 1 - TD 1 \\ NON CORRIGÉ}
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\author{Timéo Pochin}
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\begin{document}
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\maketitle
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\section*{Exercice 1}
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\subsection*{a)}
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\begin{align}
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& 5x+2\geq-3 \nonumber \\
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\iff\quad & 5x\geq-5 \nonumber \\
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\iff\quad & x\geq-1 \nonumber \\
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\iff\quad & x=\left[-1,+\infty\right[ \nonumber
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\end{align}
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\subsection*{b)}
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\begin{align}
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& 2x-1<4x+3\leq-x+6 \nonumber \\
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\iff\quad & (2x-1<4x+3)\land(4x+3\leq-x+6) \nonumber \\
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\iff\quad & (-4<2x)\land(5x\leq 3) \nonumber \\
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\iff\quad & (-2<x)\land(x\leq\frac{3}{5}) \nonumber \\
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\iff\quad & x=\left]-2,\frac{3}{5}\right] \nonumber
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\end{align}
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\subsection*{c)}
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\begin{align}
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& |x-1|<4 \nonumber \\
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\iff\quad & (x-1<4)\land(x-1>-4) \nonumber \\
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\iff\quad & (x<5)\land(x>-3) \nonumber \\
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\iff\quad & x=\left]-3,5\right[ \nonumber
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\end{align}
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\subsection*{d)}
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\begin{align}
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& |x-2|\geq 3 \nonumber \\
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\iff\quad & (x-2\geq 3)\lor(x-2\leq-3) \nonumber \\
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\iff\quad & (x\geq 5)\lor(x\leq-1) \nonumber \\
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\iff\quad & x=\left]-\infty,-1\right]\ \cup\ \left[5,+\infty\right[ \nonumber
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\end{align}
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\subsection*{e)}
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\begin{align}
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& |x-2|\leq|x| \nonumber \\
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\iff\quad & (x-2)^2\leq x^2 \nonumber \\
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\iff\quad & x^2-4x+4\leq x^2 \nonumber \\
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\iff\quad & 4\leq 4x \nonumber \\
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\iff\quad & 1\leq x \nonumber \\
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\iff\quad & x=\left[1,+\infty\right[ \nonumber
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\end{align}
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\subsection*{f)}
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\subsection*{g)}
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\begin{align}
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& \sqrt{x+1}<2 \nonumber \\
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\iff\quad & (x+1<4)\land(x+1\geq0) \nonumber \\
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\iff\quad & (x<3)\land(x\geq-1) \nonumber \\
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\iff\quad & x=\left[-1,3\right[ \nonumber
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\end{align}
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\subsection*{h)}
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\begin{align}
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& x^2+1\leq 3 \nonumber \\
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\iff\quad & x^2\leq 2 \nonumber \\
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\iff\quad & (x\leq\sqrt{2})\land(x\geq-\sqrt{2}) \nonumber \\
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\iff\quad & x=\left[-\sqrt{2},\sqrt{2}\right] \nonumber
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\end{align}
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\subsection*{i)}
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\begin{align}
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& x^2+3x<4 \nonumber \\
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\iff\quad & x^2+3x-4<0 \nonumber \\
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\iff\quad & (x-1)(x+4)<0 \nonumber \\
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\iff\quad & (x<1)\land(x>-4) \nonumber \\
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\iff\quad & x=\left]-4,1\right[ \nonumber
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\end{align}
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\subsection*{j)}
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\begin{align}
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& x^3-3x^2+2x\geq 0 \nonumber \\
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\iff\quad & x(x^2-3x+2)\geq 0 \nonumber \\
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\iff\quad & x(x-1)(x-2)\geq 0 \nonumber \\
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\iff\quad & \big((x\geq 0)\land(x\leq 1)\big)\lor(x\geq 2) \nonumber \\
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\iff\quad & x=\left[0,1\right]\cup\left[2,+\infty\right[ \nonumber
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\end{align}
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\subsection*{k)}
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\section*{Exercice 2}
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\[
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f:x\mapsto ax+b
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\]
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\begin{align}
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& \big(|f(-1)|=3\big)\land\big(|f(2)=2|\big) \nonumber \\
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\iff\quad & (|b-a|=3)\land(|2a+b|=2) \nonumber \\
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\iff\quad & (b-a=3\lor b-a=-3)\land(2a+b=2\lor 2a+b=-2) \nonumber \\
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\iff\quad & (b-a=3\land 2a+b=2) \nonumber \\
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\lor & (b-a=3\land 2a+b=-2) \nonumber \\
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\lor & (b-a=-3\land 2a+b=2) \nonumber \\
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\lor & (b-a=-3\land 2a+b=-2) \nonumber
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\end{align}
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\begin{align}
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& b-a=3\land 2a+b=2 \nonumber \\
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\iff\quad & 3a=-1\land 3b=8 \nonumber \\
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\iff\quad & a=-\frac{1}{3}\land b=\frac{8}{3} \nonumber
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\end{align}
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\begin{align}
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& b-a=3\land 2a+b=-2 \nonumber \\
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\iff\quad & 3a=-5\land 3b=4 \nonumber \\
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\iff\quad & a=-\frac{5}{3}\land b=\frac{4}{3} \nonumber
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\end{align}
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\begin{align}
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& b-a=-3\land 2a+b=2 \nonumber \\
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\iff\quad & 3a=5\land 3b=-4 \nonumber \\
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\iff\quad & a=\frac{5}{3}\land b=-\frac{4}{3} \nonumber
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\end{align}
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\begin{align}
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& b-a=-3\land 2a+b=-2 \nonumber \\
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\iff\quad & 3a=1\land 3b=-8 \nonumber \\
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\iff\quad & a=\frac{1}{3}\land b=-\frac{8}{3} \nonumber
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\end{align}
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Donc
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\[
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(a,b)\in
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\left\{
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\left(-\frac{1}{3},\frac{8}{3}\right),
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\left(-\frac{5}{3},\frac{4}{3}\right),
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\left(\frac{5}{3},-\frac{4}{3}\right),
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\left(\frac{1}{3},-\frac{8}{3}\right)
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\right\}
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\]
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -4.9, xmax = 4.9,
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ymin = -4.9, ymax = 4.9,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth,
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height = \textwidth,
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]
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\addplot[
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color = black
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]{-(1/3)*x+(8/3)} node[above right,pos=0.83] {$-\frac{1}{3}x+\frac{8}{3}$};
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\addplot[
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color = black
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]{-(5/3)*x+(4/3)} node[above right,pos=0.85] {$-\frac{5}{3}x+\frac{4}{3}$};
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\addplot[
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color = black
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]{(1/3)*x-(8/3)} node[above left,pos=0.95] {$\frac{1}{3}x-\frac{8}{3}$};
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\addplot[
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color = black
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]{(5/3)*x-(4/3)} node[above left,pos=0.8] {$\frac{5}{3}x-\frac{4}{3}$};
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\end{axis}
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\end{tikzpicture}
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\section*{Exercice 3}
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\subsection*{a)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -4.9, xmax = 4.9,
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ymin = -0.9, ymax = 8.9,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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color = black,
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samples = 1000
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]{abs(x)+abs(2*x-4)};
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\addplot[
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ultra thin,
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color = red,
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samples = 1000
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]{abs(x)};
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\addplot[
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ultra thin,
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color = blue,
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samples = 1000
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]{abs(2*x-4)};
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\end{axis}
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\end{tikzpicture}
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\subsection*{b)}
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L’ensemble $f(\mathbb{R})$ est égal à $\left[2,+\infty\right[$.
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%
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\\
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La fonction $f$ est minorée et elle n’est pas majorée.
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\subsection*{c)}
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Les antécédents par $f$ de $3$ sont $1$ et $\frac{7}{3}$.
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\\
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$1$ n’a pas d’antécédents par $f$.
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\\
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L’antécédent par $f$ de $2$ est $2$.
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\section*{Exercice 4}
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\subsection*{a)}
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\subsubsection*{i)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -3.5, xmax = 3.5,
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ymin = -3.5, ymax = 3.5,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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smooth,
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color = red,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x-1};
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\addplot[
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smooth,
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color = black,
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samples = 100
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][
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domain=-1:2
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]{-(0.5*x^2-x-1)};
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\end{axis}
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\end{tikzpicture}
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\subsubsection*{ii)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -3.5, xmax = 3.5,
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ymin = -3.5, ymax = 3.5,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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smooth,
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color = red,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x-1};
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\addplot[
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smooth,
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color = black,
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samples = 100
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][
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domain=-2:1
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]{0.5*(-x)^2-(-x)-1};
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\end{axis}
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\end{tikzpicture}
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\subsubsection*{iii)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -3.5, xmax = 3.5,
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ymin = -3.5, ymax = 3.5,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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smooth,
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color = red,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x-1};
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\addplot[
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smooth,
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color = black,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x+1};
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\end{axis}
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\end{tikzpicture}
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\subsubsection*{iv)}
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\begin{tikzpicture}
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\begin{axis}[
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xmin = -3.5, xmax = 3.5,
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ymin = -3.5, ymax = 3.5,
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axis x line = middle,
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axis y line = middle,
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xtick distance = 1,
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ytick distance = 1,
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grid = both,
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minor tick num = 5,
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major grid style = {lightgray},
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minor grid style = {lightgray!25},
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width = \textwidth/3*2,
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height = \textwidth/3*2,
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]
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\addplot[
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thick,
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smooth,
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color = red,
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samples = 100
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][
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domain=-1:2
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]{0.5*x^2-x-1};
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\addplot[
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smooth,
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color = black,
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samples = 100
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][
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domain=-3:0
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]{0.5*(x+2)^2-(x+2)-1};
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\end{axis}
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\end{tikzpicture}
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\end{document}
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