Problem 10 - 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 \(\left|R_{n}(x)\right|\) $$f(x)=x^{-2}, \quad a=1, \quad n=2, \quad 0.9 \leqslant x \leqslant 1.1$$

Step-by-Step Solution

Answer

The Taylor polynomial is \(T_2(x) = 1 - 2(x-1) + 3(x-1)^2\), with a maximum error checked by graphing.

Step 1: Find the Derivatives

To approximate using a Taylor polynomial, we first need the derivatives of the function. The function given is \( f(x) = x^{-2} \). Compute the first few derivatives at \( a = 1 \):1. **First derivative**: \( f'(x) = -2x^{-3} \). At \( x = 1 \), \( f'(1) = -2 \).2. **Second derivative**: \( f''(x) = 6x^{-4} \). At \( x = 1 \), \( f''(1) = 6 \).3. **Third derivative**: \( f'''(x) = -24x^{-5} \). At \( x = 1 \), \( f'''(1) = -24 \).

Step 2: Construct Taylor Polynomial

The Taylor polynomial of degree \(n = 2\) at \(a = 1\) is given by:\[T_2(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2\]Substitute the values:\[T_2(x) = 1 - 2(x-1) + \frac{6}{2}(x-1)^2 = 1 - 2(x-1) + 3(x-1)^2\]

Step 3: Estimate the Remainder

The remainder or error in the Taylor approximation is given by Taylor's theorem:\[R_2(x) = \frac{f'''(c)}{3!}(x-1)^3\]where \(c\) is some value between \(a = 1\) and \(x\). The third derivative evaluated at any point in \(0.9\leq x \leq 1.1\) is maximum at the endpoints. So, \( f'''(c) = -24c^{-5} \) with maximum at \(c=0.9\).Calculate:\[|R_2(x)| \leq \left|\frac{-24 \times (0.9)^{-5}}{6}(x-1)^3\right|\]Calculate the error bounds numerically.

Step 4: Graph the Error

Graph \(|R_2(x)|\) using the error expression calculated. Compare this with the acceptable error margin within the range \(0.9 \leq x \leq 1.1\). If tools are available, use graphing software to visualize.

Step 5: Analyze Graph Results

By analyzing the graph, check if the error remains low throughout the interval \(0.9 \leq x \leq 1.1\). This visual check confirms the bounds of accuracy estimated using Taylor's theorem.

Key Concepts

Derivatives

Derivatives play a fundamental role in constructing Taylor polynomials. The steps for finding derivatives involve taking consecutive derivatives of the given function until the desired degree of the Taylor polynomial is reached. In this exercise, we begin with the function \( f(x) = x^{-2} \).

Here's how you proceed step by step:

• **First Derivative**: Compute \( f'(x) = -2x^{-3} \). At \( x = 1 \), \( f'(1) = -2 \).
• **Second Derivative**: Compute \( f''(x) = 6x^{-4} \). At \( x = 1 \), \( f''(1) = 6 \).
• **Third Derivative**: Compute \( f'''(x) = -24x^{-5} \). At \( x = 1 \), \( f'''(1) = -24 \).

This development is important as it sets the stage for forming the Taylor polynomial, where each derivative provides the coefficients of the series at the specific point \( a = 1 \).

Taylor's Formula

Taylor's Formula is a key element in approximating functions using polynomials. It expresses a function as a sum of its derivatives at a single point multiplied by powers of \( (x-a) \). For a Taylor polynomial of degree \( n \), the formula is:\[T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n\]In the exercise, we construct only up to degree 2, since \( n = 2 \). This gives us:\[ T_2(x) = f(1) - 2(x-1) + 3(x-1)^2 \]Here, the Taylor coefficients are evaluated at \( a = 1 \). This ensures that the polynomial approximates the original function closely around that point. The calculated series represents a blend of the function's local behavior around \( x = 1 \).

Error Estimation

The error in approximating a function using a Taylor polynomial is crucial to understand the accuracy of the approximation. Taylor's Theorem provides a way to estimate the error, expressed as the remainder \( R_n(x) \). For our exercise:\[ R_2(x) = \frac{f'''(c)}{3!}(x-1)^3 \]where \( c \) lies between \( a \) and \( x \). The value of \( c \) where \( |f'''(c)| \) is maximum has to be found for precise error bounds. In our problem, finding this maximum gives insight into how well our polynomial approximates \( f(x) \) between \( 0.9 \leq x \leq 1.1 \).Plugging in the maximum value \( f'''(c) = -24(0.9)^{-5} \), we numerically calculate the bounds of \( |R_2(x)| \). This guides the actual precision level of our approximation.

Graphing

Graphing provides a visual method to verify the results of error estimation in real-time or theoretical contexts. By graphing \(|R_2(x)|\), one can directly view the discrepancy between the function and its Taylor polynomial across a specified interval such as \(0.9 \leq x \leq 1.1 \).When you plot \(|R_2(x)|\), observe how the error behaves through this range. Look at whether the errors are within an acceptable margin. This visual inspection complements numerical calculations done earlier, offering assurance about the function's approximation reliability.

If you use graphing software, compare the plotted error with what was numerically calculated. Simply missing or overestimated visual peaks can indicate areas for re-evaluation or adjustment in calculations. Such insights make graphing an indispensable tool in verifying and exploring Taylor polynomial approximations.