Problem 16 - Essential Calculus Early Transcendentals

Question

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number \(a\) . (b) Use Taylor's Formula to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=x \ln x, \quad a=1, \quad n=3, \quad 0.5 \leqslant x \leqslant 1.5$$

Step-by-Step Solution

Answer

Taylor polynomial: \(T_3(x) = (x-1) + \frac{1}{2}(x-1)^2 - \frac{1}{6}(x-1)^3\). Error \(\leq 0.0104\).

Step 1: Find the Taylor Polynomial

The Taylor polynomial of degree 3, with center at \(a = 1\), for the function \(f(x) = x \ln x\) is given by: \[ T_3(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3. \]First, we need to calculate \(f(1)\), \(f'(1)\), \(f''(1)\), and \(f'''(1)\). 1. \(f(1) = 1 \ln 1 = 0.\)2. \(f'(x) = 1 + \ln x\), thus \(f'(1) = 1.\)3. \(f''(x) = \frac{1}{x}\), thus \(f''(1) = 1.\)4. \(f'''(x) = -\frac{1}{x^2}\), thus \(f'''(1) = -1.\)The Taylor polynomial is:\[ T_3(x) = 0 + 1(x-1) + \frac{1}{2}(x-1)^2 - \frac{1}{6}(x-1)^3. \]Simplifying gives:\[ T_3(x) = (x-1) + \frac{1}{2}(x-1)^2 - \frac{1}{6}(x-1)^3. \]

Step 2: Use Taylor's Formula to Estimate the Error

Taylor's remainder for degree 3 is given by:\[ R_3(x) = \frac{f^{(4)}(c)}{4!}(x-1)^4 \]for some \(c\) between \(x\) and \(1\). For \(f(x)=x \ln x\), calculate \(f^{(4)}(x)\):\[ f^{(4)}(x) = \frac{2}{x^3}.\]The maximum value of \(\left| f^{(4)}(x) \right|\) in the interval \(0.5 \leq x \leq 1.5\) occurs at \(x = 0.5\), giving the maximum value:\[ \left| f^{(4)}(0.5) \right| = \frac{2}{(0.5)^3} = 16. \]Thus, the error estimate is:\[ \left| R_3(x) \right| \leq \frac{16}{24}(0.5)^4 = \frac{1}{6}(0.0625) = 0.0104. \]

Step 3: Verify Error Estimation by Graphing

Graph \(|R_3(x)| = \left| \frac{f^{(4)}(c)}{24}(x-1)^4 \right| \) over \(0.5 \leq x \leq 1.5\).Since the maximum possible error was estimated in step 2 to be 0.0104, use graphing software or a calculator to plot values of \(|R_3(x)|\) and ensure that the maximum does not exceed 0.0104.This helps to visually confirm the error estimation accuracy over the interval.

Key Concepts

Taylor's Formula

Taylor's Formula is a powerful tool in mathematics used to approximate complex functions with simpler polynomial expressions known as Taylor polynomials. The formula provides a way to express a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For a function \(f(x)\) expanded around the point \(a\), the Taylor polynomial of degree \(n\) is:

• \(T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n\)

Each term of the polynomial includes derivatives of \(f\), giving us a polynomial that closely resembles \(f\) near \(a\). In practice, only a few initial terms are used, which defines the degree of the Taylor polynomial, noting that higher degrees provide better approximation.
In the given exercise, the task is to approximate \(f(x)=x \ln x\) at \(a=1\) using a Taylor polynomial of degree 3. Calculating the necessary derivatives at \(x=1\), we constructed a Taylor polynomial:

• \(T_3(x) = (x-1) + \frac{1}{2}(x-1)^2 - \frac{1}{6}(x-1)^3\)

This expression serves as a localized approximation for \(f(x)\) in a nearby interval around \(a=1\).

Error Estimation

Estimating the error in a Taylor polynomial approximation is crucial to determine how close the polynomial is to the actual function. The error is often expressed using the remainder of the Taylor series. For a degree \(n\) polynomial, the Taylor's error estimate, often noted as \(R_n(x)\), is given by:

• \( R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \)

Here, \(c\) is some value in the interval between \(x\) and \(a\).
The error estimation essentially tells us how well the Taylor polynomial fits the function \(f(x)\) over a specific range. In our example, using \(n=3\) and \(a=1\), the expression for the remainder becomes:

• \( R_3(x) = \frac{f^{(4)}(c)}{24}(x-1)^4 \)

By finding the maximum possible value for \( |f^{(4)}(x)| \) in the interval \(0.5 \leq x \leq 1.5\), we calculated the error bound as approximately 0.0104. This means that the true function \(f(x)\) and the polynomial \(T_3(x)\) will not differ by more than this amount over the specified interval.

Remainder Term

The remainder term in the Taylor polynomial approximation plays a crucial role in understanding the accuracy of the approximation. It gives us the difference between the function and the polynomial over an interval and is expressed as \(R_n(x)\). The primary goal is to minimize this discrepancy.
In the Taylor series, the remainder can be viewed as an error bound that helps gauge our polynomial's closeness to \(f(x)\). For example, finding \(R_3(x)\) means calculating :

• \( R_3(x) = \frac{f^{(4)}(c)}{24}(x-1)^4 \)

where \(c\) is some point between \(x\) and the center \(a\).
In practical applications, as seen in the exercise, the true value of the remainder term cannot be exactly pinpointed because \(c\) is not explicitly known. However, estimating its maximum value within a given interval provides a useful guideline on the approximation's precision. Graphical representations of \( |R_3(x)| \) help verify that the calculated bound holds true, visually confirming that the approximation does not exceed the predicted error across the interval.