{ "cells": [ {"cell_type":"markdown","source":"
We begin by loading packagse for plotting, automatic differentiation, and symbolic manipulations:
","metadata":{}}, {"outputs":[],"cell_type":"code","source":["using Plots\nusing Roots\nusing ForwardDiff\nD(f, k=1) = k > 1 ? D(D(f),k-1) : x -> ForwardDiff.derivative(f, float(x))\nusing QuadGK\nusing SymPy"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"A Taylor polynomial of a function $f(x)$ about $x=c$ of degree $n$ is formally defined by
","metadata":{}}, {"cell_type":"markdown","source":"\n$$\nT_n(x) = f(c) + \\frac{f'(c)}{1!}(x-c) \\frac{f''(c)}{2!}(x-c)^2 + \\cdots + \\frac{f^{(n)}(c)}{n!}(x-c)^n.\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"When $n=1$ this is the familiar tangent line approximation. Higher orders, generally yield better approximations.
","metadata":{}}, {"cell_type":"markdown","source":"In julia
we can create this series numerically using the D
function. It is useful to write our function so that it returns a function, as is done with the following. We use a quadratic approximation for the default order and expand around $0$ as the default (the Maclaurin polynomial).
We can see graphically that the Taylor polynomial approximates the function $f(x)$ around the $x=c$. For example, we approximate $\\cos(x)$ at $x=0$ and $x=1$:
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":"Plot(...)","image/png":"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"},"metadata":{"image/png":{"height":480,"width":600}},"execution_count":1}],"cell_type":"code","source":["f(x) = cos(x)\nplot([f, taylor(f, 2, c=0), taylor(f, 2, c=pi/4)], -pi/2, pi/2)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"We can see that taking higher orders can lead to better approximations. In the following we use a comprehension and splatting to avoid having to type plot([f, taylor(f,1), taylor(f,2), taylor(f,3), ..., taylor(f,6) ], 0, 4)
:
You should be able to confirm that the higher-order approximations do a better job.
","metadata":{}}, {"cell_type":"markdown","source":"Mathematically, the above picture is governed by the following error bound for Taylor polynomials:
","metadata":{}}, {"cell_type":"markdown","source":"\n$$\n| f(x) - T_n(x) | \\leq | \\frac{f^{(n+1)}(z^*)}{(n+1)!} \\cdot (x-c)^{n+1} |\n$$\n","metadata":{}}, {"cell_type":"markdown","source":"where $z^*$ is chosen to maximize the above expression between $x$ and $c$. That is, the largest possible value (in absolute value) for the next term in the Taylor polynomial provides a bound for the error.
","metadata":{}}, {"cell_type":"markdown","source":"Verify graphically that the approximation $\\log(1 + x) \\approx x$ comes from using the Maclaurin polynomial of degree 1. The approximation is not good for all $x$. What seems like a reasonable range of $x$ for where it is \"good?\"
\nLet $f(x) = \\tan(x)$. Plot the first $6$ Maclaurin polynomials and $f$ over $[-\\pi/3, \\pi/3]$. Is the approximation good at $x=0$? Are any of the approximations good at $\\pi/3$?
\nLet $f(x) = \\sin(x)$. Plot the first $6$ Maclaurin polynomials and $f$ over $[-\\pi/2, \\pi/2]$. Is the approximation good at $x=0$? Are any of the approximations good at $\\pi/2$?
\nAgain for $f(x) = \\sin(x)$. Plot the function and the Maclaurin polynomials of degree $3$ and $4$. Is the $4$th degree polynomial a better approximation? Why or why not?
\nThe function $f(x) = e^{-x^2}$ does not have an antiderivative, hence can not be integrated by the fundamental theorem of calculus. One can estimate the integral using Riemann sums or one can estimate the function by an easily integrable function and integrate that. Of course, Maclaurin polynomials are easily integrable.
\nPlot several Maclaurin polynomials that approximate $f(x)$ over the interval $[0, 0.5]$. Choose the lowest order one that graphically matches over that interval. Integrate that approximation and compare to the value given by:
\nLet $f(x) = e^x$ a monotone function. Let $T_3(x)$ be the 3rd-order Maclaurin polynomial. Verify graphically that over the interval $[0,4]$ the difference $|f(x) - T_3(x)| < f''''(4)/4! \\cdot (x-0)^4$. (We know that for this function $f''''(4)$ is monotone increasing, so is largest at the right end point, hence the use of $f''''(4)$ in the bound.)
\nIs the bound a \"tight\" bound over the entire interval, in that the actual error is close to the given bound? (Check graphically.)
\nThe following two formulas come from different eras of physics:
\nWhich we can express via:
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/plain":["k2 (generic function with 1 method)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["k1(v) = v^2 / 2\nk2(v; c=10) = c^2 * (1/sqrt(1 - (v/c)^2) - 1)"],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"Both describe the kinetic energy, but one uses Einstein's theory of relativity. For any value of $c > 0$, show that the two agree up to second order approximations by Maclaurin polynomials.
","metadata":{}}, {"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}, {"cell_type":"markdown","source":"Here is a different motivation of the Maclaurin polynomial that can be explored symbolically. (This is a generalization of the tangent line, where the \"polynomial\" is the secant line, and the limiting values there combine to give the tangent line.)
\nFor concreteness let $f(x) = e^x$. For any $b > 0$, there is just one fifth degree polynomial, $p(x)$, satisfying $p(i\\cdot b) = f(i\\cdot b)$ for $i=0,1,2,3,4,5$. This is polynomial interpolation at 6 points.
","metadata":{}}, {"cell_type":"markdown","source":"This particular solution can be found as follows using SymPy
:
At first glance, this seems to have nothing to do with the Maclaurin polynomial for $f(x)$, $T_5(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120$. But wait, let's take the limit as $b$ goes to zero of each value in di
. For example, the fifth term:
The answer is the a5
coefficient of $T_5(x)$. Verify that this is the case for each of a0
, ..., a4
.