Problem 21 - Calculus: Early Transcendentals

Question

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = x \sin x, \) \( a = 0, \) \( n = 4, \) \( - 1 \le x \le 1 \)

Step-by-Step Solution

Answer

The Taylor polynomial is \( T_4(x) = x^2 - \frac{x^4}{6} \), and the error is \(|R_4(x)| \leq \frac{1}{20} |x|^5 \).

Step 1: Calculate Derivatives

Find the first four derivatives of the function \( f(x) = x \sin x \). Since \( n = 4 \), we need derivatives up to \( f^{(4)}(x) \).- \( f(x) = x \sin x \)- \( f'(x) = \sin x + x \cos x \)- \( f''(x) = 2 \cos x - x \sin x \)- \( f'''(x) = -3 \sin x - x \cos x \)- \( f^{(4)}(x) = -4 \cos x + x \sin x \)

Step 2: Evaluate Derivatives at a

Evaluate each derivative at \( a = 0 \).- \( f(0) = 0 \)- \( f'(0) = 0 \)- \( f''(0) = 2 \)- \( f'''(0) = 0 \)- \( f^{(4)}(0) = -4 \)

Step 3: Construct the Taylor Polynomial

Using the Taylor series formula, construct the polynomial.\[ T_4(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f^{(4)}(0)}{4!}x^4 \]Substituting the values:\[ T_4(x) = 0 + 0 \cdot x + \frac{2}{2}x^2 + 0 \cdot x^3 + \frac{-4}{24}x^4 \]Simplifying gives:\[ T_4(x) = x^2 - \frac{x^4}{6} \]

Step 4: Apply Taylor's Inequality

Taylor's Inequality states the error \( |R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!} \), where \( M \) is an upper bound for \( |f^{(n+1)}(x)| \) on the interval.For \( f^{(5)}(x) = 5 \sin x + x \cos x \), the bound on \( [-1, 1] \) occurs when \( \sin x \to 1 \). Thus, \( M \approx 6 \).Then: \[|R_4(x)| \leq \frac{6 |x|^5}{5!}\] \[|R_4(x)| \leq \frac{6}{120} |x|^5 = \frac{1}{20} |x|^5\]

Step 5: Verify with Graph of \(|R_n(x)|\)

Graph \( |R_4(x)| = \left| x \sin x - \left( x^2 - \frac{x^4}{6} \right) \right| \) over the interval \(-1 \leq x \leq 1\) to confirm the estimated bound.The graph should show that \( |R_4(x)| \leq \frac{1}{20} |x|^5 \), which validates the accuracy of our approximation calculated in Step 4.

Key Concepts

Taylor's Inequality

Taylor's Inequality is a mathematical tool used to estimate the error involved in approximating a function using its Taylor polynomial. This inequality helps us understand how "off" our approximation can be. It states that if a function is approximated by a Taylor polynomial of degree \( n \) at point \( a \), the error \( |R_n(x)| \) is bounded by:

• \( \left| R_n(x) \right| \leq \frac{M |x-a|^{n+1}}{(n+1)!} \) where \( M \) is the maximum value of the absolute value of the \((n+1)\)th derivative over the interval in question.

In practice, we find an upper bound \( M \) for the highest derivative within the given interval to use this inequality effectively. In our exercise, the error for approximating \( f(x) = x \sin x \) with \( T_4(x) \) is calculated using derivatives up to the fifth order within the interval \(-1 \leq x \leq 1\). By estimating \( |f^{(5)}(x)| \) with a suitable \( M \), the inequality gives us a numerical error bound for any \( x \) in the interval.

Derivatives

Derivatives are fundamental in constructing Taylor polynomials. They reveal the rate at which a function changes, allowing us to stack approximations layer by layer. For a function \( f(x) = x \sin x \), finding the derivative involves applying the product rule and trigonometric identities. Calculating derivatives up to a required degree prepares us to form the Taylor polynomial.

• \( f(x) = x \sin x \)
• \( f'(x) = \sin x + x \cos x \)
• \( f''(x) = 2\cos x - x \sin x \)
• \( f'''(x) = -3\sin x - x \cos x \)
• \( f^{(4)}(x) = -4\cos x + x \sin x \)

Evaluating these derivatives at the expansion point \( a = 0 \) simplifies our Taylor polynomial to just needing constant factors. Accurate derivatives ensure precision in the resulting polynomial approximation.

Error Approximation

Understanding and estimating the error in a function approximation is crucial for assessing its usefulness. It tells us how close our polynomial approximation is to the true function. The error, or residual \( |R_n(x)| \), is estimated using Taylor's Inequality.
Given the example of approximating \( f(x) = x \sin x \), the Taylor polynomial \( T_4(x) = x^2 - \frac{x^4}{6} \) approximates \( f(x) \) with an "error approximation" calculated as:

• \(|R_4(x)| \leq \frac{1}{20} |x|^5\)

This inequality provides a guaranteed bound, claiming the error will not surpass \( \frac{1}{20} |x|^5 \) within the interval \(-1 \leq x \leq 1\). Such estimates and error bounds are vital in practical applications, ensuring that approximations are within acceptable limits.

Graphing Residuals

Visualizing the error or the residual error between a function and its Taylor polynomial can provide intuitive insights into the approximation's accuracy.
Graphing the function \( |R_4(x)| = \left| x \sin x - \left( x^2 - \frac{x^4}{6} \right) \right| \) shows us how close our polynomial comes to approximating \( f(x) = x \sin x \) throughout the interval \(-1 \leq x \leq 1\).

• The graph should lie below the estimated bound \( \frac{1}{20} |x|^5 \).
• A close-to-zero residual indicates a more precise approximation.

Visual validation is a powerful way to confirm analytical error approximations. By looking at this graph, students can visually grasp the accuracy and limitations of Taylor polynomial approximations.