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Chapter 1: Problem 2

Find the degree 2 Taylor polynomial for \(f(x)=e^{x} \sin (x)\), about the point \(a=0\). Bound the error in this approximation when \(-\pi / 4 \leq x \leq \pi / 4\).

Short Answer

Expert verified
The degree 2 Taylor polynomial is \(x + x^2\). The error bound is \(\frac{e^{\pi/4} (1 + 3/\sqrt{2})}{3!} \left(\frac{\pi}{4}\right)^3\).

Step by step solution

01

Find the function and its derivatives at the given point

Given function: \[f(x) = e^x \sin(x)\] Calculate the derivatives at the point \(a=0\): First derivative: \[f'(x) = e^x \sin(x) + e^x \cos(x)\] Evaluating at \(x=0\): \[f'(0) = e^0 \sin(0) + e^0 \cos(0) = 0 + 1 = 1\] Second derivative: \[f''(x) = e^x (\sin(x) + 2 \cos(x))\] Evaluating at \(x=0\): \[f''(0) = e^0 (\sin(0) + 2 \cos(0)) = 0 + 2 = 2\]
02

Formulate the Taylor polynomial

The Taylor polynomial of degree 2 for the function \(f(x)\) around \(a=0\) is given by: \[P_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\] Using the values obtained: \[P_2(x) = e^0 \sin(0) + 1x + \frac{2}{2!}x^2 = x + x^2\]
03

Bound the error in the approximation

The error in the Taylor polynomial approximation is given by the Lagrange remainder term: \[R_2(x) = \frac{f^{(3)}(c)}{3!} x^3\] where \(c\) is some value between 0 and \(x\). Third derivative of the function: \[f^{(3)}(x) = e^x (\sin(x) + 3 \cos(x))\] The maximum value of \(e^x (\sin(x) + 3 \cos(x))\) in the interval \([-\frac{\pi}{4}, \frac{\pi}{4}]\) occurs at \(e^{\pi/4} (1 + 3/\sqrt{2})\), approximately. Therefore, the error bound is: \[R_2(x) = \frac{e^{\pi/4} (1 + 3/\sqrt{2})}{3!} \left(\frac{\pi}{4}\right)^3\]

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Key Concepts

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

degree 2 Taylor polynomial
A Taylor polynomial is a useful tool for approximating functions. The degree 2 Taylor polynomial provides an approximation using up to the second derivative of the function. To find it for the function \(f(x)=e^{x} \sin (x)\), we start by evaluating the function and its first two derivatives at the point \(a=0\). The formulas are provided by: \[P_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\]
We found the derivatives to be: \[f'(0) = 1 \text{ and }\ f''(0) = 2\] Thus, the degree 2 Taylor polynomial becomes: \[P_2(x) = x + x^2\]
Lagrange remainder term
The Lagrange remainder term gives us the error in our Taylor polynomial approximation. It measures how far our polynomial approximation is from the actual function. For the degree 2 Taylor polynomial, the remainder term \(R_2(x)\) is given by:
\[R_2(x) = \frac{f^{(3)}(c)}{3!} x^3\] where \(c\) is some number between 0 and \(x\). By determining \(R_2(x)\), we can understand how accurate our approximation is.
error bound in approximation
To give a bound for the error in our degree 2 Taylor polynomial approximation, we look at the maximum value of the third derivative of the function in the interval \([-\frac{\pi}{4}, \frac{\pi}{4}]\). The third derivative of our function is: \[f^{(3)}(x) = e^x (\sin(x) + 3 \cos(x))\] For \(x\) in \([-\frac{\pi}{4}, \frac{\pi}{4}]\), this maximum value roughly equals \[e^{\pi/4} (1 + 3/\sqrt{2})\]. Therefore, the error can be bounded as:
\[R_2(x) = \frac{e^{\pi/4} (1 + 3/\sqrt{2})}{3!} \left(\frac{\pi}{4}\right)^3\]
function derivatives
Derivatives are essential in constructing a Taylor polynomial. They provide information about the rate of change of the function. To build our degree 2 Taylor polynomial, we found:
  • First derivative \((f'(x) = e^x \sin(x) + e^x \cos(x))\)
  • Second derivative \((f''(x) = e^x (\sin(x) + 2 \cos(x)))\)
  • Third derivative \((f^{(3)}(x) = e^x (\sin(x) + 3 \cos(x)))\)
Evaluating these derivatives at \(x=0\) gives us the coefficients needed for our polynomial. Properly finding these derivatives ensures our polynomial accurately represents the original function close to \(x=0\).

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Most popular questions from this chapter

Produce a general formula for the degree \(n\) Taylor polynomials for the following functions, all using \(a=0\) as the point of approximation. (a) \(1 /(1-x)\) (b) \(\sin (x)\) (c) \(\sqrt{1+x}\) (d) \(\cos (x)\) (e) \((1+x)^{1 / 3}\)

Define \(h(x)=f(x) g(x)\). Let the Taylor polynomials of degree \(n\) for \(f(x)\) and \(g(x)\) be given by $$ p_{n}(x)=\sum_{t=0}^{n} a_{i} x^{i}, \quad q_{n}(x)=\sum_{j=0}^{n} b_{j} x^{j} $$ Let \(r_{n}(x)\) be obtained by first multiplying \(p_{n}(x) q_{n}(x)\) and then dropping all terms of degree greater than \(n\). (a) For \(n=2\), show that the Taylor polynomial of degree 2 for \(h(x)\) equals \(r_{2}(x) .\) (b) For general \(n \geq 1\), show that the Taylor polynomial of degree \(n\) for \(h(x)\) equals \(r_{n}(x)\). Hint: For repeated differentiation of the product \(f(x) g(x)\), use the Leibniz formula: $$ \frac{d^{k}}{d x^{k}}[f(x) g(x)]=\sum_{j=0}^{k}\left(\begin{array}{l} k \\ j \end{array}\right) f^{(j)}(x) g^{(k-j)}(x) $$

For \(f(x)=e^{x}\), find a Taylor approximation that is in error by at most \(10^{-7}\) on \([-1,1]\). Using this approximation, write a function program to evaluate \(e^{x} .\) Compare it to the standard value of \(e^{x}\) obtained from the MATLAB function \(\exp (x)\); calculate the difference between your approximation and \(\exp (x)\) at 21 evenly spaced points in \([-1,1]\).

(a) Produce the Taylor polynomials of degrees \(1,2,3,4\) for \(f(x)=e^{x^{2}}\) with \(a=0\) the point of approximation. (b) Using the Taylor polynomials for \(e^{t}\), substitute \(t=x^{2}\) to obtain polynomial approximations for \(e^{x^{2}}\). Compare with the results in (a).

Produce the linear and quadratic Taylor polynomials for the following cases. Graph the function and these Taylor polynomials. (a) \(f(x)=\sqrt{x}, a=1\) (b) \(\quad f(x)=\sin (x), a=\pi / 4\) (c) \(f(x)=e^{\cos (x)}, a=0\) (d) \(\quad f(x)=\log \left(1+e^{x}\right), a=0\)
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