{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Aproximații cu serii Taylor" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ideea de bază a seriilor Taylor este de a reprezenta orice tip de funcție ca o sumă infinită de funcții polinomiale numită seria lui Taylor. Să presupunem că avem o funcție de o singură variabilă reală $f(x)$ de $n$ ori derivabilă într-un punct oarecare $a$, atunci putem exprima funcția prin următoarea serie infinită:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f(x)=f(a)+f'(a)(x-a)+\\frac{f''(a)}{2!}(x-a)^2+\\frac{f'''(a)}{3!}(x-a)^3+... $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "sau mai compact:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f(x)= \\sum\\limits_{k=0}^{\\infty}\\frac{f^k(a)}{k!}(x-a)^k $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pentru unele funcții seria converge pentru orice valoare a lui $x$. Pentru altele doar pentru anumite valori." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Teorema lui Taylor**, care poate fi găsită sub mai multe enunțuri, arată în esență că o funcție **poate fi aproximată** într-un anumit punct cu o sumă finită de termeni din seria lui Taylor." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "De exemplu aproximația Taylor de ordin $n$ a funcției $f(x)$ în jurul punctului $a$ este:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f(x)\\approx T_{n,a}(x)=\\sum\\limits_{k=0}^{n}\\frac{f^k(a)}{k!}(x-a)^k $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Aproximația este cu atât mai acurată cu cât numărul $x$ este mai apropiat de $a$ si cu cât numărul de termeni $k$ din aproximare este mai mare." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Restul termenilor de la $n$ la $ \\infty$ alcătuiesc **restul aproximării** $R_{n,a}$. Evaluare acetui termen exprimă eroarea aproximării." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$R_{n,a}(x)=\\sum\\limits_{k=n}^{\\infty}\\frac{f^k(a)}{k!}(x-a)^k $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exemplul 1**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Dorim să calculăm dezvoltarea în serie Taylor a funcției $f(x)=sinx$ în jurul punctului $a=0$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f(x)=sinx,\\quad\n", "f(0)=0,$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f'(x)=cosx, \\quad f'(0)=1, $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f''(x)=-sinx,\\quad f''(0)=0 $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f'''(x)=-cosx, \\quad f'''(0)=-1 $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f^4(x)=sinx, \\quad f^4(0)=0 $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f^5(x)=cosx, \\quad f^5(0)=1 $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f(x)=\\frac{0}{0!}x^0+\\frac{1}{1!}(x)^1+\\frac{0}{2!}(x)^2+\\frac{-1}{3!}(x)^3+\\frac{0}{4!}(x)^3+\\frac{1}{5!}(x)^5+.. = x-\\frac{1}{3!}x^3+\\frac{1}{5!}x^5-\\frac{1}{7!}x^7...$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Să considerăm două situații $x=0.3$ mai apropiat de $a=0$ si $x=0.7$ mai departe de $a=0$. Folosind primii 5 termeni ai dezvoltării în serie Taylor se observă mai jos că aproximarea este mult mai bună pentru prima situație când $x$ este mai aproape de $a=0$." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "estimatul lui sin(.3) este 0.29552025\n", "valoarea exacta a lui sin(.3) este 0.295520206661\n", "estimatul lui sin(.7) este 0.564648\n", "valoarea exacta a lui sin(.7) este 0.644217687238\n" ] } ], "source": [ "import numpy as np\n", "x1 = .3\n", "x2 =.6\n", "\n", "estimat1 = x1 - x1**3/6 + x1**5/120\n", "exact1 = np.sin(.3)\n", "estimat2 = x2 - x2**3/6 + x2**5/120\n", "exact2 = np.sin(.7)\n", "\n", "print'estimatul lui sin(.3) este', estimat1\n", "print'valoarea exacta a lui sin(.3) este', exact1\n", "\n", "print'estimatul lui sin(.7) este', estimat2\n", "print'valoarea exacta a lui sin(.7) este', exact2\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Grafic putem vedea cum se comportă funcțiile care reprezintă primii doi , patru, sase termeni din dezvoltarea în serie Taylor care aproximează $f(x)=sinx$. Cu cât numărul de termeni creste cu atât aproximarea e mai bună." ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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OHGglNRRrwTCMoQpwsCiYbTlNAnvIHdGjVt+FVNqNbRlWo/JSu5Xp1q0bK8mI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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "%matplotlib inline\n", "plt.style.use('ggplot')\n", "\n", "x = np.linspace(-8, 8, 160) \n", "y1 = np.sin(x)\n", "y2 = x\n", "y3 = x - x**3/6\n", "y4 = x - x**3/6 + x**5/120\n", "\n", "plt.xlabel('x')\n", "plt.ylabel('f(x)')\n", "plt.ylim(-1.5, 2.5)\n", "plt.title('Aproximarea functiei sin(x)')\n", "\n", "plt.plot(x, y1,'-b', label='sin')\n", "plt.plot(x, y2,'-r', label='f2')\n", "plt.plot(x, y3,'-g', label='f4')\n", "plt.plot(x, y4,'-y', label='f5')\n", "\n", "plt.legend(loc='upper right')\n", "#plt.grid(True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exemplu 2**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Dezvoltarea în serie Taylor a funcției $f(x)=cosx$ în jurul punctului $a=0$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$f(x)=1-\\frac{1}{2!}x^2+\\frac{1}{4!}x^4-\\frac{1}{6!}x^6+...$" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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MI5iV7Tk5Ofj666+Rn58PlmUxaNCgGk8Jt27dwsqVK+Ht7Q1Am+AdN26cPeQ2GYKCgjBj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\n4gnCw8PVjN1i3HOPnf79i9i5U7snj/x8hR07DNWeOJIykug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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "%matplotlib inline\n", "plt.style.use('ggplot')\n", "\n", "x = np.linspace(-8, 8, 160) \n", "y1 = np.cos(x)\n", "#y2 = 1\n", "y3 = 1 - x**2/2\n", "y4 = 1 - x**2/2 + x**4/24\n", "y2 = 1-x**2/2 + x**4/24+x**6/780\n", "\n", "plt.xlabel('x')\n", "plt.ylabel('f(x)')\n", "plt.ylim(-1.5, 2.5)\n", "plt.title('Aproximarea functiei cos(x)')\n", "\n", "plt.plot(x, y1,'-b', label='cos')\n", "plt.plot(x, y2,'-r', label='f7')\n", "plt.plot(x, y3,'-g', label='f3')\n", "plt.plot(x, y4,'-y', label='f5')\n", "\n", "plt.legend(loc='upper right')\n", "#plt.grid(True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Alte exemple**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Pentru $-1" ] }, "execution_count": 196, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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Llui5wiIMAyIiPbl9+zZatWoFQHsf8oyMDHTs2LFc7w0ICMDx48dx//59fZaowzAgItKD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\nAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "\n", "%matplotlib inline\n", "plt.style.use('ggplot')\n", "\n", "x = np.linspace(0, 10)\n", "x1 = np.linspace(4, 5)\n", "y=np.sqrt(x)\n", "plt.axhline(y=2.,color='k',linestyle='dashed')\n", "y1=1+x/4\n", "y2=1+x/4-(x-4)**2/64\n", "y3=abs(y-y2)\n", "\n", "\n", "\n", "plt.xlabel('x')\n", "plt.ylabel('f(x)')\n", "plt.ylim(0, 5)\n", "plt.title('Aproximatii cu serii Taylor ')\n", "\n", "plt.plot(x, y,'-r', label='f(x)')\n", "plt.plot(x, y1,'-b', label='T1',linestyle='dashed')\n", "plt.plot(x, y2,'-g', label='T2',linestyle='dashed')\n", "plt.legend(loc='upper right')\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Eroarea acestei aproximări folosind reprezentarea grafică a restului $|R_n|=(f(x)-T_2)|$ pentru $x$ cuprins între $a=4$ și $5$ este reprezentată\n", "mai jos." ] }, { "cell_type": "code", "execution_count": 199, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 199, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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MTL755hsCAwNZvny5w3gfffQRn3zyCRqNhjFjxnDnnXc6J1EhhPBgfm++Scs559BywQWn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\nNm3aJEXIEygKqoYG1GYz6poa1NXVaGp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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "\n", "%matplotlib inline\n", "plt.style.use('ggplot')\n", "x = np.linspace(4,5,100)\n", "y=np.sqrt(x)\n", "\n", "y2=1+x/4-(x-4)**2/64\n", "y3=abs(y-y2)\n", "\n", "\n", "plt.xlabel('x')\n", "plt.ylabel('R(x)')\n", "plt.title('Eroarea aproximarii ')\n", "plt.plot(x, y3,'-r', label='r(x)')\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "O altă posibilitate să estimăm eroarea aproximării este să calculăm $R_{n,a}=\\frac{M}{(n+1)!}(x-a)^{n+1}$ unde $M$ reprezintă limita superioară a valorii absolute a lui $f^{n+1}(c)$ cu $c$ intre $a$ si $x.$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$ |R_{n,a}|=\\frac{|f^{n+1}(c)|}{(n+1)!}|x-a|^{n+1} $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$|R_{2,4}|=\\frac{|f^{3}(c)|}{(3)!}|5-4|^{3} \\quad a=4, x=5$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$ |R_{2,4}|=\\frac{\\frac{1}{12}\\frac{1}{c^{\\frac{4}{3}}}}{3!}1^3$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Dacă $c$ este între $a=4$ si $5$, valoarea maximă a lui $M = \\frac{1}{12}\\frac{1}{c^{\\frac{4}{3}}}$ corespunde lui $c=4$, de unde eroarea maxima a aproximării este $|R_{2,4}|<0,0021$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Unde utilizăm serii Taylor?**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "În practică aproximațiile sunt utile atunci când avem de rezolvat o problemă care presupune calcule matematice complicate si care în anumite condiții pot fi simplificate pentru a oferi o primă soluție estimativă asupra rezultatului. În fizică în multe situații este convenabil să aproximăm într-un anumit punct o funcție cu o variantă a sa simplificată fără ca rezolvarea problemei să fie afectată semnificativ. Avem mai jos câteva exemple." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "1. Aproximarea unghiurilor mici\n", "2. Forța gravitațională\n", "3. Neglijarea efectelor relativiste" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Aproximarea unghiurilor mici**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "În anumite situații se aproximează funcția $f(x)=sinx\\approx x$, primul termen al dezvoltării în serie Taylor, sau $cos(x)\\approx 1$, sau $tg(x)\\approx x$. Dacă luăm ca exemplu funcția $f(x)=sinx$ si calculăm diferența dintre $x-sin(x)=R(x)$ reprezentând eroarea aproximației obținem graficul de mai sus. Acest grafic arată clar că până în $5,6$ grade eroarea este nesemnificativă si aproximația se poate face." ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "**Forța gravitațională**" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "Forța de atracție gravitațională exercitată de Pământ asupra unui corp de masă $m$ variază cu distanța $h$ față de suprafața Pământului conform relației:\n", " $$F(h)=K\\frac{M_pm}{(R+h)^2}$$ \n", " unde $K$ este constanta atracției gravitaționale, $M_p$ este masa Pământului, $R=6371km$ este raza Pământului." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Folosind aproximația binomială $(1\\pm x)^n\\approx (1\\pm x )$ avem $T(h)= K\\frac{M_p m}{R^2}=F_0\\frac{1}{(1+\\frac{h}{R})^2}=F_0(1+\\frac{h}{R})^{-2}\\approx F_{0}(1-\\frac{2h}{R})$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Variația forței de atracție gravitaționale exercitate de Pământ asupra unui corp cu masa de 1kg pe o distanță \n", "de la suprafața Pământului până la o rază terestră este arătată în graficul de mai jos." ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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w9y6f4cUYq1m4mFSj7t1zMHVqGj74wBlJSWKnYYyxqsPFpJr5+2eiX79s+Psr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\nOXjwIDw8PMSOwVi5deigxc6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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "\n", "%matplotlib inline\n", "plt.style.use('ggplot')\n", "\n", "x = np.linspace(0, 6400) \n", "y = 9.816/(1+x/6371)**2\n", "\n", "\n", "plt.xlabel('h(km)')\n", "plt.ylabel('F(h)')\n", "plt.title('Variatia fortei gravitationala pana la o inaltime Rp ')\n", "plt.plot(x, y,'-b', label='F')\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Dacă se utilizează formula aproximativă dată de formula binomială discutată mai sus si se reprezintă grafic eroarea absolută\n", "$|{f(x)-T(x)}|$ pe o înălțime de 10 km față de suprafața Pământului atunci se observă că eroarea absolută este foarte mică (vezi graficul de mai jos)." ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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++CDOzs5UV1fz+OOPd2pwQgghWvz0k4bJk00880wF0dG13R1Op3HoiKmiooJh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\n1VdOREV5smOHC1u2lDJzZnNREqI3USl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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "\n", "%matplotlib inline\n", "plt.style.use('ggplot')\n", "\n", "x = np.linspace(0, 10) \n", "y1 = 9.816/(1+x/6371)**2\n", "y2 = 9.816*(1-(2*x)/6371)\n", "y=abs(y1-y2)\n", "\n", "\n", "plt.xlabel('h(km)')\n", "plt.ylabel('Eroarea absoluta ')\n", "plt.title('Eroarea absoluta a aproximatiei ')\n", "plt.plot(x, y,'-b', label='E')\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Neglijarea efectelor relativiste**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In Teoria Relativității restrânse apare în multe relații asa numitul factor relativist sau factor Lorentz \n", "$\\gamma=\\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}}=\\frac{1}{\\sqrt{1-\\beta^2}}$. Factorul Lorentz depinde de viteza $v$ așa cum se vede în graficul de mai jos. La viteze mici, numite și nerelativiste, $\\gamma\\approx 1$. La viteze mai mari ca $250000km/s$ factorul relatist devine mai mare ca $2$ și crește semnificativ." ] }, { "cell_type": "code", "execution_count": 124, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 124, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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EzddWLYtmJ4u4uDikpaXh+PHjuHTpEsLCwtCrVy8oFNYi+vbti759+7o8QCIi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\nREQOMVkQEZFDTBZEROQQkwURETn0/wGuBuqxLtoMrQAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "\n", "%matplotlib inline\n", "plt.style.use('ggplot')\n", "\n", "x = np.linspace(0,299000000)\n", "c=300000000\n", "c2=c**2\n", "beta=(x/c)\n", "beta2=beta**2\n", "b=1-beta2\n", "y=np.power(b,-0.5)\n", "\n", "\n", "plt.xlabel('v(m/s)')\n", "plt.ylabel('gamma')\n", "plt.title('Variatia factorului relativist gamma ')\n", "plt.plot(x, y,'-b', label='g')\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Dacăse dezvoltă în serie Taylor folosind formula binomială, $\\gamma=\\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}}=\\frac{1}{\\sqrt{1-\\beta^2}}=(1-\\beta^2)^{-2}=1+\\frac{1}{2}\\frac{v^2}{c^2}-\\frac{3}{8}\\frac{v^4}{c^4}+\\frac{5}{16}\\frac{v^6}{c^6}+...$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Dacă se calculează energia cinetică relativistă $E_c=mc^2-m_0c^2=\\frac{m_0c^2}{\\sqrt{1-\\beta^2}}-m_0c^2\\approx m_0c^2(1+\\frac{1}{2}\\beta-\\frac{3}{8}\\beta^2+\\frac{5}{16}\\beta^3+... -1)\\approx m_0c^2(\\frac{1}{2}\\beta-\\frac{3}{8}\\beta^2+ \\frac{5}{16}\\beta^3+..)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "La viteze mici termenii de doi în $\\beta$ se pot neglija și $E_c\\approx m_0c^2\\frac{\\beta}{2}=\\frac {m_0v^2}{2}$, care este expresia clasică a energiei cinetice." ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.12" } }, "nbformat": 4, "nbformat_minor": 0 }