{ "cells": [ { "cell_type": "markdown", "id": "f05a0cde", "metadata": {}, "source": [ "[![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/MeiraDavidson/math_module2/blob/main/notebooks/ch7_taylor.ipynb)\n\n", "# Chapter 7 — Companion Notebook\n", "## Building a Function From Its Derivatives\n", "*a curve is a line, then a parabola, then a polynomial — and you choose how close*\n", "\n", "Four live pictures of the chapter's one machine: replace a smooth curve, near a point, with a **polynomial** that matches it to whatever order you like. Push the degree up and watch the Taylor polynomial hug the curve over a wider window; shade the *remainder* to see exactly where the approximation can be trusted; watch the geometric series track inside $|x|<1$ and explode past the walls; and finally watch $e^{ix}$ trace the unit circle, its shadows drawing $\\cos x$ and $\\sin x$ — Euler's formula, in motion.\n", "\n", "> **How to use this notebook.** Run every cell from the top (Shift+Enter). Each widget below is live — drag the sliders and watch the picture respond. Requires `ipywidgets` and `matplotlib`." ] }, { "cell_type": "code", "execution_count": null, "id": "98ac1b1b", "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from matplotlib import animation\n", "from ipywidgets import (interact, interactive, fixed, Dropdown, Checkbox,\n", " Button, Output, VBox, HBox,\n", " FloatSlider, IntSlider, FloatLogSlider)\n", "from IPython.display import HTML, display, Markdown\n", "\n", "# Book 2 palette — matches the printed figures in figures/png/\n", "BLUE=\"#1f4e79\"; RED=\"#c0392b\"; GREEN=\"#2e8b57\"; ORANGE=\"#e08e0b\"\n", "PURPLE=\"#6c3483\"; TEAL=\"#1b9e9e\"; GRAY=\"#7f8c8d\"; DARK=\"#2c3e50\"\n", "SHADE=\"#9fc5e8\"; SHADE2=\"#f4b6ad\"; SHADEG=\"#a9dfbf\"; LIGHT=\"#d6dbdf\"\n", "\n", "plt.rcParams.update({\n", " \"figure.figsize\": (7.2, 4.5), \"figure.dpi\": 96,\n", " \"font.family\": \"serif\", \"mathtext.fontset\": \"cm\", \"font.size\": 12,\n", " \"axes.spines.top\": False, \"axes.spines.right\": False,\n", " \"axes.grid\": True, \"grid.color\": LIGHT, \"grid.linewidth\": 0.7,\n", " \"lines.linewidth\": 2.2, \"lines.solid_capstyle\": \"round\",\n", "})\n", "\n", "def axes(ax=None, title=None):\n", " '''A clean axes with a light grid; returns the axes.'''\n", " if ax is None:\n", " _, ax = plt.subplots()\n", " if title:\n", " ax.set_title(title, color=DARK)\n", " return ax\n" ] }, { "cell_type": "code", "execution_count": null, "id": "6c13a021", "metadata": {}, "outputs": [], "source": [ "\n", "from math import factorial\n", "\n", "# Maclaurin coefficient c_k (coefficient of x^k) for each flagship function.\n", "def _coef_exp(k): return 1.0 / factorial(k) # e^x\n", "def _coef_sin(k): return 0.0 if k % 2 == 0 else (-1.0)**((k - 1)//2) / factorial(k)\n", "def _coef_cos(k): return 0.0 if k % 2 == 1 else (-1.0)**(k//2) / factorial(k)\n", "def _coef_geo(k): return 1.0 # 1/(1-x)\n", "def _coef_log(k): return 0.0 if k == 0 else (-1.0)**(k - 1) / k # ln(1+x)\n", "\n", "# name -> (f(x), coefficient generator, (xlo, xhi), (ylo, yhi))\n", "FUNCS = {\n", " 'e^x': (lambda x: np.exp(x), _coef_exp, (-4.0, 4.0), (-2.0, 12.0)),\n", " 'sin x': (lambda x: np.sin(x), _coef_sin, (-2*np.pi, 2*np.pi), (-2.5, 2.5)),\n", " 'cos x': (lambda x: np.cos(x), _coef_cos, (-2*np.pi, 2*np.pi), (-2.5, 2.5)),\n", " '1/(1-x)': (lambda x: 1.0/(1.0 - x), _coef_geo, (-0.95, 0.95), (-3.0, 12.0)),\n", " 'ln(1+x)': (lambda x: np.log1p(x), _coef_log, (-0.9, 0.9), (-3.0, 2.0)),\n", "}\n", "\n", "def taylor_coeffs(name, n):\n", " '''The list [c_0, c_1, ..., c_n] for the named function, about 0.'''\n", " cgen = FUNCS[name][1]\n", " return [cgen(k) for k in range(n + 1)]\n", "\n", "def taylor_eval(name, n, x):\n", " '''Evaluate the degree-n Taylor polynomial T_n at the array x (Horner).'''\n", " cs = taylor_coeffs(name, n)\n", " out = np.zeros_like(np.asarray(x, dtype=float))\n", " for c in reversed(cs): # Horner: ((c_n)x + c_{n-1})x + ...\n", " out = out * x + c\n", " return out\n", "\n", "def taylor_latex(name, n, maxterms=6):\n", " '''A short LaTeX string for T_n (first few nonzero terms, then dots).'''\n", " cs = taylor_coeffs(name, n)\n", " terms = []\n", " for k, c in enumerate(cs):\n", " if abs(c) < 1e-12:\n", " continue\n", " # render the coefficient as a clean fraction where we can\n", " from fractions import Fraction\n", " frac = Fraction(c).limit_denominator(10**6)\n", " mag = abs(frac)\n", " sign = '-' if c < 0 else '+'\n", " if k == 0:\n", " body = f'{mag}'\n", " else:\n", " xpow = 'x' if k == 1 else f'x^{{{k}}}'\n", " body = xpow if mag == 1 else (\n", " f'\\\\tfrac{{{mag.numerator}}}{{{mag.denominator}}}{xpow}'\n", " if mag.denominator != 1 else f'{mag.numerator}{xpow}')\n", " terms.append((sign, body))\n", " if len(terms) >= maxterms:\n", " break\n", " if not terms:\n", " return '0'\n", " head = terms[0][1] if terms[0][0] == '+' else '-' + terms[0][1]\n", " rest = ''.join(f' {s} {b}' for s, b in terms[1:])\n", " tail = ' + \\\\cdots' if any(abs(c) > 1e-12 for c in taylor_coeffs(name, n + 4)[n+1:]) else ''\n", " return head + rest + tail\n" ] }, { "cell_type": "markdown", "id": "12bddec0", "metadata": {}, "source": [ "## 1. The Taylor slider\n", "\n", "The whole machine in one control. The **Taylor polynomial** of degree $n$ for $f$ about $0$ is\n", "\n", "$$T_n(x) = \\sum_{k=0}^{n}\\frac{f^{(k)}(0)}{k!}\\,x^k,$$\n", "\n", "and for our five flagships the coefficients $\\frac{f^{(k)}(0)}{k!}$ have a clean closed form — $\\frac{1}{k!}$ for $e^x$, the alternating odd/even terms for $\\sin$ and $\\cos$, all $1$'s for $\\frac{1}{1-x}$, and $\\frac{(-1)^{k-1}}{k}$ for $\\ln(1+x)$ — so we never need to differentiate anything numerically.\n", "\n", "Pick a function (blue) and push the degree $n$ up. The polynomial $T_n$ (red) starts as the tangent line, becomes a parabola, then bend after bend it **clings to the curve over a wider and wider window** — *(Figure 7.3)*." ] }, { "cell_type": "code", "execution_count": null, "id": "44a145ac", "metadata": {}, "outputs": [], "source": [ "def _taylor_slider(func='e^x', n=1):\n", " f, _, (xlo, xhi), (ylo, yhi) = FUNCS[func]\n", " xs = np.linspace(xlo, xhi, 800)\n", " fig, ax = plt.subplots()\n", " ax.plot(xs, f(xs), color=BLUE, lw=2.6, label=f'$f(x) = {func}$',\n", " zorder=3)\n", " ax.plot(xs, taylor_eval(func, n, xs), color=RED, lw=2.4, ls='--',\n", " label=f'$T_{{{n}}}(x)$', zorder=4)\n", " ax.scatter([0], [f(0.0)], s=55, color=DARK, zorder=6)\n", " ax.axhline(0, color=GRAY, lw=0.8, ls=':')\n", " ax.set_xlim(xlo, xhi); ax.set_ylim(ylo, yhi)\n", " ax.set_xlabel('$x$'); ax.set_ylabel('$y$')\n", " ax.set_title(f'$T_{{{n}}}(x) = {taylor_latex(func, n)}$',\n", " color=DARK, fontsize=12)\n", " ax.legend(loc='upper left', framealpha=0.9, fontsize=11)\n", " plt.show()\n", "\n", "interact(_taylor_slider,\n", " func=Dropdown(options=list(FUNCS), value='e^x',\n", " description='f(x)'),\n", " n=IntSlider(value=1, min=0, max=18, step=1,\n", " description='degree n'));\n" ] }, { "cell_type": "markdown", "id": "539a55c6", "metadata": {}, "source": [ "## 2. How wrong is it? The remainder, shaded\n", "\n", "An approximation you can't bound is a guess. Write $f(x) = T_n(x) + R_n(x)$, where the **remainder** $R_n(x) = f(x) - T_n(x)$ is everything the polynomial missed. Below, the salmon region is the *actual* error $|f(x) - T_n(x)|$ as $x$ moves away from the center.\n", "\n", "Watch its shape: the error is essentially $0$ in a window around $0$ — flat against the axis, the zone you can trust — and then **peels upward** as you leave that window. Raise $n$ and the trusty window widens; for $e^x$, $\\sin$, $\\cos$ it widens without bound, while for $\\frac{1}{1-x}$ and $\\ln(1+x)$ the error stays stubborn past the wall at $x=1$ no matter how high you climb. The lower panel shows the same error on a log scale, where the trustworthy valley is unmistakable." ] }, { "cell_type": "code", "execution_count": null, "id": "9353acbd", "metadata": {}, "outputs": [], "source": [ "def _remainder(func='e^x', n=3):\n", " f, _, (xlo, xhi), _ = FUNCS[func]\n", " xs = np.linspace(xlo, xhi, 800)\n", " err = np.abs(f(xs) - taylor_eval(func, n, xs))\n", "\n", " fig, (ax, axlog) = plt.subplots(2, 1, figsize=(7.2, 6.0),\n", " sharex=True,\n", " gridspec_kw={'height_ratios': [2, 1]})\n", " # Top: f vs T_n with the error shaded between them.\n", " ax.plot(xs, f(xs), color=BLUE, lw=2.4, label=f'$f(x) = {func}$',\n", " zorder=3)\n", " ax.plot(xs, taylor_eval(func, n, xs), color=RED, lw=2.2, ls='--',\n", " label=f'$T_{{{n}}}(x)$', zorder=4)\n", " ax.fill_between(xs, f(xs), taylor_eval(func, n, xs), color=SHADE2,\n", " alpha=0.85, label='$|f - T_n|$', zorder=2)\n", " ax.axvline(0, color=GRAY, lw=0.8, ls=':')\n", " f0 = f(0.0)\n", " ax.set_ylim(f0 - 6, f0 + 6)\n", " ax.set_ylabel('$y$')\n", " ax.set_title(f'Degree {n}: where $T_n$ is trustworthy, and where '\n", " f'it peels away', color=DARK, fontsize=12)\n", " ax.legend(loc='upper left', framealpha=0.9, fontsize=10)\n", " # Bottom: the error magnitude itself, shaded to the axis.\n", " axlog.fill_between(xs, 1e-16, err + 1e-16, color=SHADE2, alpha=0.9,\n", " zorder=2)\n", " axlog.plot(xs, err + 1e-16, color=RED, lw=1.6, zorder=3)\n", " axlog.set_yscale('log')\n", " axlog.set_ylim(1e-14, 1e3)\n", " axlog.set_xlabel('$x$'); axlog.set_ylabel('$|R_n(x)|$')\n", " axlog.set_title('the same error, log scale (low valley = trustworthy)',\n", " color=DARK, fontsize=10)\n", " plt.tight_layout()\n", " plt.show()\n", "\n", "interact(_remainder,\n", " func=Dropdown(options=list(FUNCS), value='e^x',\n", " description='f(x)'),\n", " n=IntSlider(value=3, min=0, max=16, step=1,\n", " description='degree n'));\n" ] }, { "cell_type": "markdown", "id": "a0e9ea84", "metadata": {}, "source": [ "## 3. The walls at $x = \\pm 1$: a series that diverges\n", "\n", "A Taylor series is a *local* promise — true only where it converges. The cleanest warning is the geometric series\n", "\n", "$$\\frac{1}{1-x} = 1 + x + x^2 + x^3 + \\cdots,$$\n", "\n", "which holds **only for $|x| < 1$**. Inside that band the partial sums $1 + x + \\cdots + x^N$ (red, increasing $N$) crowd in on the true curve $\\frac{1}{1-x}$ (blue). But step past the dashed walls at $x = \\pm 1$ and they don't just lag — they **fly off**, alternating ever more wildly. Plug in $x = 2$ and the left side is $-1$ while the right side is $1 + 2 + 4 + \\cdots = +\\infty$. Push $N$ up and see the convergence tighten inside the band and the explosion worsen outside it." ] }, { "cell_type": "code", "execution_count": null, "id": "43dc6307", "metadata": {}, "outputs": [], "source": [ "def _diverge(N=4):\n", " fig, ax = plt.subplots()\n", " # The true function, drawn on each side of its pole at x=1.\n", " xl = np.linspace(-2.4, 0.985, 400)\n", " xr = np.linspace(1.015, 2.4, 400)\n", " ax.plot(xl, 1.0/(1.0 - xl), color=BLUE, lw=2.8,\n", " label=r'$\\frac{1}{1-x}$', zorder=5)\n", " ax.plot(xr, 1.0/(1.0 - xr), color=BLUE, lw=2.8, zorder=5)\n", " # Partial sums 1 + x + ... + x^N over the whole window.\n", " xs = np.linspace(-2.4, 2.4, 700)\n", " psum = np.zeros_like(xs)\n", " for k in range(N + 1):\n", " psum = psum + xs**k\n", " ax.plot(xs, psum, color=RED, lw=2.2, ls='--',\n", " label=f'$1 + x + \\\\cdots + x^{{{N}}}$', zorder=4)\n", " # The walls of the interval of convergence.\n", " ax.axvspan(-1, 1, color=SHADEG, alpha=0.35, zorder=0,\n", " label='$|x|<1$ (converges)')\n", " ax.axvline(-1, color=DARK, lw=1.2, ls=':')\n", " ax.axvline(1, color=DARK, lw=1.2, ls=':')\n", " ax.axhline(0, color=GRAY, lw=0.8, ls=':')\n", " ax.set_xlim(-2.4, 2.4); ax.set_ylim(-8, 12)\n", " ax.set_xlabel('$x$'); ax.set_ylabel('$y$')\n", " ax.set_title(f'Geometric partial sum, N = {N}: hugs inside, '\n", " f'explodes outside', color=DARK, fontsize=12)\n", " ax.legend(loc='upper center', framealpha=0.9, fontsize=10)\n", " plt.show()\n", "\n", "interact(_diverge,\n", " N=IntSlider(value=4, min=0, max=30, step=1,\n", " description='terms N'));\n" ] }, { "cell_type": "markdown", "id": "22c9d090", "metadata": {}, "source": [ "## 4. Euler's formula, in motion\n", "\n", "Feed an *imaginary* input into the exponential series and the terms sort themselves into the cosine and sine series:\n", "\n", "$$e^{ix} = \\cos x + i\\sin x.$$\n", "\n", "So as $x$ runs from $0$ to $2\\pi$, the point $e^{ix}$ travels once around the **unit circle**. Its two shadows are the trigonometric functions: the horizontal coordinate (the real part) draws $\\cos x$, and the vertical coordinate (the imaginary part) draws $\\sin x$. Run the animation below — the green dot rides the circle on the left while its projections trace the two waves on the right. At $x = \\pi$ the dot sits at $-1$, which is Euler's identity $e^{i\\pi} = -1$ — *(Figure 7.7)*." ] }, { "cell_type": "code", "execution_count": null, "id": "b2f16a7e", "metadata": {}, "outputs": [], "source": [ "_FRAMES = 60\n", "_xmax = 2*np.pi\n", "_theta = np.linspace(0, 2*np.pi, 240)\n", "\n", "fig, (axc, axw) = plt.subplots(1, 2, figsize=(9.6, 4.4),\n", " gridspec_kw={'width_ratios': [1, 1.5]})\n", "\n", "def _frame(i):\n", " x = _xmax * i / (_FRAMES - 1)\n", " c, s = np.cos(x), np.sin(x)\n", "\n", " # ---- Left: the unit circle and the moving point e^{ix}. ----\n", " axc.clear()\n", " axc.plot(np.cos(_theta), np.sin(_theta), color=GRAY, lw=1.6,\n", " zorder=1)\n", " axc.plot([0, c], [0, s], color=PURPLE, lw=2.0, zorder=2)\n", " # dropped lines to each axis -> the two shadows\n", " axc.plot([c, c], [0, s], color=BLUE, lw=1.4, ls=':', zorder=2)\n", " axc.plot([0, c], [s, s], color=RED, lw=1.4, ls=':', zorder=2)\n", " axc.scatter([c], [s], s=90, color=GREEN, zorder=5)\n", " axc.scatter([c], [0], s=45, color=BLUE, zorder=4) # cos shadow\n", " axc.scatter([0], [s], s=45, color=RED, zorder=4) # sin shadow\n", " axc.axhline(0, color=LIGHT, lw=0.8); axc.axvline(0, color=LIGHT, lw=0.8)\n", " axc.set_xlim(-1.4, 1.4); axc.set_ylim(-1.4, 1.4)\n", " axc.set_aspect('equal')\n", " axc.set_title(f'$e^{{ix}}$, $x = {x:.2f}$', color=DARK, fontsize=12)\n", " axc.grid(False)\n", "\n", " # ---- Right: the two shadows tracing cos x and sin x. ----\n", " axw.clear()\n", " xx = np.linspace(0, _xmax, 240)\n", " axw.plot(xx, np.cos(xx), color=LIGHT, lw=1.2, zorder=1)\n", " axw.plot(xx, np.sin(xx), color=LIGHT, lw=1.2, zorder=1)\n", " upto = np.linspace(0, x, max(2, int(240 * (i + 1) / _FRAMES)))\n", " axw.plot(upto, np.cos(upto), color=BLUE, lw=2.4,\n", " label=r'$\\cos x = \\mathrm{Re}\\,e^{ix}$', zorder=3)\n", " axw.plot(upto, np.sin(upto), color=RED, lw=2.4,\n", " label=r'$\\sin x = \\mathrm{Im}\\,e^{ix}$', zorder=3)\n", " axw.scatter([x], [c], s=70, color=BLUE, zorder=5)\n", " axw.scatter([x], [s], s=70, color=RED, zorder=5)\n", " axw.axhline(0, color=GRAY, lw=0.8, ls=':')\n", " axw.set_xlim(0, _xmax); axw.set_ylim(-1.4, 1.4)\n", " axw.set_xlabel('$x$')\n", " axw.set_title('the shadows draw the two waves', color=DARK,\n", " fontsize=12)\n", " axw.legend(loc='upper right', framealpha=0.9, fontsize=10)\n", " return ()\n", "\n", "ani = animation.FuncAnimation(fig, _frame, frames=_FRAMES, interval=80)\n", "plt.close(fig)\n", "HTML(ani.to_jshtml())\n" ] }, { "cell_type": "markdown", "id": "df8d42a9", "metadata": {}, "source": [ "---\n", "\n", "*Four readings of one idea. The slider builds the polynomial; the shaded panel keeps it honest about its error; the walls at $x = \\pm 1$ show that the promise is local; and Euler's circle shows the same series, fed an imaginary input, weaving the exponential and the two trig functions into a single object. Carry the leading-term idea into Chapter 8, where reading the first surviving Taylor term at a critical point *is* the second-derivative test.*" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "pygments_lexer": "ipython3" } }, "nbformat": 4, "nbformat_minor": 5 }