Problem 15 - 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 \( \mid R_n(x) \mid . \) \( f (x) = x^{2/3}, \) \( a = 1, \) \( n = 3, \) \( 0.8 \le x \le 1.2 \)

Step-by-Step Solution

Answer

The Taylor polynomial is \( T_3(x) = 1 + \frac{2}{3}(x-1) - \frac{1}{9}(x-1)^2 + \frac{4}{81}(x-1)^3 \). The error bound is approximately 0.00511.

Step 1: Define the Taylor Series

The Taylor series for a function \( f(x) \) at \( x = a \) is \[T_n(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots + \frac{f^{(n)}(a)}{n!}(x-a)^n.\]

Step 2: Calculate Derivatives

Calculate the derivatives of \( f(x) = x^{2/3} \) up to the third derivative:- \( f(x) = x^{2/3} \)- \( f'(x) = \frac{2}{3}x^{-1/3} \)- \( f''(x) = -\frac{2}{9}x^{-4/3} \)- \( f'''(x) = \frac{8}{27}x^{-7/3} \) Evaluate these derivatives at \( a = 1 \):- \( f(a) = 1 \)- \( f'(a) = \frac{2}{3} \)- \( f''(a) = -\frac{2}{9} \)- \( f'''(a) = \frac{8}{27} \)

Step 3: Form Taylor Polynomial

Substitute the values of the function and its derivatives into the Taylor series formula:\[T_3(x) = 1 + \frac{2}{3}(x-1) - \frac{1}{9}(x-1)^2 + \frac{4}{81}(x-1)^3.\]

Step 4: Taylor's Inequality

Taylor's Inequality tells us the remainder is bounded by:\[|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}\]where \( M = \max_{x \in [0.8, 1.2]} |f^{(n+1)}(x)| \). Evaluate the fourth derivative of \( f(x) = x^{2/3} \):- \( f^{(4)}(x) = -\frac{56}{81}x^{-10/3} \)On \([0.8, 1.2]\), estimate \( M \) from \( f^{(4)}(x) \) (max occurs at \( x=0.8 \)):- \( M \approx 19.69 \)Then,\[|R_3(x)| \leq \frac{19.69}{24}(0.4)^4 \approx 0.00511.\]

Step 5: Graph Remainder

Graph \( |R_3(x)| = |f(x) - T_3(x)| \) over \( 0.8 \le x \le 1.2 \). Verify that the maximum value is consistent with the bound found using Taylor's Inequality, approximately 0.00511.

Key Concepts

Taylor's Inequality

Taylor's Inequality is a powerful tool used to estimate the error or remainder when approximating a function with its Taylor polynomial. This concept essentially provides an upper bound on the absolute value of the remainder, which helps you determine how accurate your approximation is. It is defined by:
\[ |R_n(x)| \leq \frac{M}{(n+1)!} |x-a|^{n+1} \]where:

• \( M \) is the maximum value of the \((n+1)\)th derivative of the function \( f(x) \) over the interval of interest.
• \( |x-a| \) is the distance from the center of the expansion \(a\) to \(x\).
• \( n \) is the degree of the Taylor polynomial.

In our example, the fourth derivative was maximized at \( x = 0.8 \), and the maximum was used in the inequality to find that \(|R_3(x)|\) is approximately bounded by 0.00511 over the given interval. Taylor's Inequality not only shows the error but also gives us an insight into how rapidly or slowly the approximation converges.

Taylor series

Taylor series is an incredibly useful approximation tool used in mathematics to represent complicated functions with polynomials that are much easier to work with. A Taylor series expansion of a function \( f(x) \) around the point \(a\) takes the form:
\[ f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \]where the degree \( n \) determines how many terms are included in the approximation. These terms involve derivatives of \( f(x) \) evaluated at \(x = a\).
The higher the number of terms (higher degree), the better the polynomial approximates the function within a specific interval around \(a\). The Taylor series is widely used in fields like physics and engineering, where precise calculations are necessary. For our exercise, the Taylor polynomial of degree 3 was used to approximate \( f(x) = x^{2/3} \) around \( x = 1 \). This approximation is particularly useful when direct computation of \( f(x) \) is infeasible.

derivatives

Derivatives are foundational in calculus and are essential in forming Taylor polynomials. When you want to create a Taylor polynomial, you need to calculate the derivatives of a function up to the degree of the polynomial you wish to construct. Each derivative gives us information about the curvature and the rate at which the function is changing at a certain point.
For example, if you are constructing a third-degree Taylor polynomial, you will need the original function, its first, second, and third derivatives:

• The function itself gives the initial value.
• The first derivative tells us about the instantaneous rate of change or the slope.
• The second derivative provides information on the function's concavity.
• The third derivative assesses the rate at which the concavity is changing.

In our example, the derivatives up to the third order were used, each calculated at \( a = 1 \). Calculating derivatives correctly is crucial since any mistake can lead to a poor approximation of the original function.

remainder estimation

Remainder estimation reflects the difference between the actual function value and its Taylor polynomial approximation. Understanding the remainder allows us to appreciate how close our polynomial is to the actual function. This difference is called the remainder and is represented by \( R_n(x) \).

Remainder estimation is particularly significant when it needs to be proven that the Taylor polynomial is sufficient for applications demanding high accuracy. By using Taylor's Inequality, you can determine an upper bound for \( |R_n(x)| \), ensuring that your computed error does not exceed a certain threshold.
For our given exercise, the Taylor polynomial aimed for a high accuracy where it showed that \( |R_3(x)| \) is approximately bounded by 0.00511. This means that for the interval given, the Taylor polynomial provides an excellent approximation to \( f(x) \). The graph of the remainder function can further verify this bound, confirming the precision of the Taylor polynomial within the given range.