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Chapter 1: Problem 30

Find a theoretical upper bound, as a function of \(x,\) for the absolute error in using \(T_{4}(x)\) to approximate \(f(x)\). (a) \(e^{x} \sin x ; x_{0}=0\). (b) \(e^{-x^{2}} ; x_{0}=0 .\) (c) \(\frac{10}{x}+\sin (10 x) ; x_{0}=\pi\).

Short Answer

Expert verified
(a) \( \leq \frac{e^x}{5!} |x|^5 \), (b) bounded by poly terms, (c) affected by \( |x-\pi|^5 \).

Step by step solution

01

Taylor Series Overview

The n-th order Taylor polynomial, \( T_n(x) \), for a function \( f(x) \) expanded around \( x_0 \) is given by: \[ T_n(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n. \] The error or remainder \( R_n(x) \) for the Taylor series is given by the Lagrange form: \[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}, \] for some \( c \) between \( x_0 \) and \( x \). The goal is to find the theoretical upper bound for \(|R_4(x)|\).
02

Approximating Error for Part (a)

For \( f(x) = e^x \sin x \), the derivatives become complex combinations of exponentials and trigonometric functions, but the key is to bound the \((n+1)\)-th derivative. The 5th derivative, in general terms, is cyclic for \( \sin x \) and grows with \( e^x \). We use the fact that \( |\sin x| \leq 1 \) and bound \( e^x \) over the relevant domain. Thus, \[ \left|R_4(x)\right| \leq \frac{e^x}{5!} |x|^5. \]
03

Approximating Error for Part (b)

For \( f(x) = e^{-x^2} \), the derivatives involve polynomials multiplied by \( e^{-x^2} \). The 5th derivative at \( x_0 = 0 \) can be bounded because \( e^{-x^2} \) is always less than or equal to 1. Hence, \[ \left|R_4(x)\right| \leq \frac{|P(x)|}{5!} |x|^5, \] where \( P(x) \) represents the polynomials in derivatives, leading to a bounded form \( |P(x)| = |x|^{2k} ext{ for some } k. \) Based on computations, it simplifies to a manageable form dependent on \( x \).
04

Approximating Error for Part (c)

For \( f(x) = \frac{10}{x} + \sin(10x) \), compute the derivatives and realize the component \( \sin(10x) \) contributes bounded derivatives, while \( \frac{10}{x} \) leads to more complex terms. However, the behavior at \( x_0 = \pi \) simplifies our goal with respect to \( x-\pi \), leading us to \[ \left|R_4(x)\right| \leq C |x-\pi|^5 \] where \( C \) bounds the derivative terms, facilitating a careful assessment.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lagrange Remainder
When working with Taylor series, it鈥檚 important to understand the concept of the Lagrange remainder. The Lagrange remainder gives us an approximation of the error involved when we use a Taylor polynomial to estimate a function.
  • The Taylor polynomial, being a sum of the first few terms of a function鈥檚 series expansion, can only approximate a function up to a certain degree of accuracy.
  • The Lagrange remainder helps us quantify this approximation error by showing us how much more we need to add to our Taylor polynomial to get the actual function value.
  • It is expressed as: \[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}, \] where \( c \) is a point between \( x_0 \) and \( x \).
Finding the upper bound for this remainder is crucial since it tells us how far off our polynomial approximation could be from the true value. Remember, the key determinant here is the \((n+1)\)-th derivative of the function, which gives insight into the curve's behavior beyond the polynomial's degree.
Error Analysis
Error analysis in Taylor series involves investigating the potential deviation between a function and its polynomial approximation. By understanding this, we can judge how effectively our Taylor series is approximating the function.
  • The error or "remainder" is a reminder that no finite Taylor polynomial can perfect any function across its domain.
  • The strategy of "bounding" helps estimate how large the error could be. Here, the most influential aspect is the magnitude of successive derivatives.
  • For example, in part (a), with \(f(x) = e^x \sin x\), the challenge comes from how fast \(e^x\) grows and the behavior of \(\sin x\). We ensure the upper limit of this growth is within controllable bounds.
  • In cases like part (b), \(f(x) = e^{-x^2}\), error analysis treats the damping effect of the exponential as beneficial, since it naturally diminishes the potential error.
Ultimately, the precise understanding and handling of how functions behave, via their higher derivatives, ensure that our approximations hold true and the errors remain manageable.
Polynomial Approximation
Polynomial approximation using Taylor series is a powerful technique in calculus that allows us to simplify complex functions into polynomials, making them easier to work with.
  • The fundamental idea is to represent a function as a sum of polynomial terms centered around a specific point.
  • This is given by the formula: \[ T_n(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n. \]
  • Approximating a function using a polynomial means we can perform various operations like differentiation and integration more simply, as working with polynomials is straightforward.
  • In Step 3 and Step 4, these approximations help break down complex behaviors of functions like \(\frac{10}{x}\) and trigonometric expressions into something more manageable around a point like \(x_0 = \pi\).
By choosing the order \(n\) and the point \(x_0\) strategically, Taylor polynomials become incredibly useful tools to model forms of real-world phenomena, providing both clarity and computational convenience.

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Most popular questions from this chapter

The Octave function foo \((x)\) is shown below. $$ \begin{array}{l} \text { function res }=\text { foo }(x) \\ \text { if }(x<1) \\ \text { res }=0 \end{array} $$ else half \(=x / 2\) floorhalf \(=\) floor (half); if (half \(==\) floorhalf) res \(=0+\) foo (floorhalf) else res \(=1+\) foo (floorhalf) end\%if end\%if end\%function (a) Find foo(2). (b) Find foo(23).

Find the rates of convergence of the following sequences as \(n \rightarrow \infty\) (a) \(\lim _{n \rightarrow \infty} \sin \frac{1}{n}=0\) (b) \(\lim _{n \rightarrow \infty} \sin \frac{1}{n^{2}}=0\) (c) \(\lim _{n \rightarrow \infty}\left(\sin \frac{1}{n}\right)^{2}=0\) (d) \(\lim _{n \rightarrow \infty}[\ln (n+1)-\ln (n)]=0\) For questions \(8-12,\) use the following definition for rate of convergence for a function. For a function \(f(h),\) we say \(\lim _{h \rightarrow a} f(h)=L\) with rate of convergence \(g(h)\) if \(|f(h)-L| \leq \lambda|g(h)|\) for some \(\lambda>0\) and all sufficiently small \(|h-a|\)

Besides round-off error, how may the accuracy of a numerical calculation be adversely affected?

Write an Octave program (.m file) that uses a loop and the disp() command to output the first 10 powers of 5 starting with \(5^{\circ}\).

All of these equations are mathematically true. Nonetheless, floating point error causes some of them to be false according to Octave. Which ones? HINT: Use the boolean operator wo to check. For example, to check if \(\sin (0)=0,\) type \(\sin (0)==0\) into Octave. ans=1 means true (the two sides are equal according to Octave - no round-off error) and ans=0 means false (the two sides are not equal according to Octave round-off error). (a) \((2)(12)=9^{2}-4(9)-21\) (b) \(e^{3 \ln (2)}=8\) (c) \(\ln (10)=\ln (5)+\ln (2)\) (d) \(g\left(\frac{1+\sqrt{5}}{2}\right)=\frac{1+\sqrt{5}}{2}\) where \(g(x)=\sqrt[3]{x^{2}+x}\) (e) \(\lfloor 153465 / 3\rfloor=153465 / 3\) (f) \(3 \pi^{3}+7 \pi^{2}-2 \pi+8=((3 \pi+7) \pi-2) \pi+8\)
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