Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the third-order Maclaurin polynomial for $$ (1+x)^{-1 / 2} $$ and bound the error \(R_{3}(x)\) if \(-0.05 \leq x \leq 0.05\).

Step-by-Step Solution

Answer

The third-order polynomial is \( P_3(x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 \), with error bound \(|R_3(x)| \leq 3.2268 \times 10^{-7}\).

Step 1: Review the Maclaurin Series

The Maclaurin series is a specific type of Taylor series expansion of a function \( f(x) \) about 0. It is given by: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots\] Our task is to find the third-order polynomial, which includes terms up to \( x^3 \).

Step 2: Calculate Derivatives of the Function

The function under consideration is \( g(x) = (1+x)^{-1/2} \). We first find the derivatives:1. \( g(x) = (1+x)^{-1/2} \)2. \( g'(x) = -\frac{1}{2}(1+x)^{-3/2} \)3. \( g''(x) = \frac{3}{4}(1+x)^{-5/2} \)4. \( g'''(x) = -\frac{15}{8}(1+x)^{-7/2} \)

Step 3: Evaluate Derivatives at x = 0

Evaluate each derivative at \( x = 0 \): - \( g(0) = 1 \) - \( g'(0) = -\frac{1}{2} \) - \( g''(0) = \frac{3}{4} \) - \( g'''(0) = -\frac{15}{8} \)

Step 4: Construct the Third-Order Polynomial

Substitute the evaluated derivatives into the Maclaurin series formula up to the third order:\[ P_3(x) = 1 - \frac{1}{2}x + \frac{3}{4}\frac{x^2}{2!} - \frac{15}{8}\frac{x^3}{3!}\] Simplify this expression:\[ P_3(x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3\]

Step 5: Error Bound for the Maclaurin Polynomial

The error for the Maclaurin polynomial of order \( n \) is given by the remainder term \( R_n(x) \) which can be expressed as \[ R_3(x) = \frac{g^{(4)}(c)}{4!}x^4\]where \( c \) is some value between 0 and \( x \). Find \( g^{(4)}(x) = \frac{105}{16}(1+x)^{-9/2} \). Evaluate it for bounds \( -0.05 \leq x \leq 0.05 \). The maximum occurs at \( x = -0.05 \).

Step 6: Calculate Maximum Error

The absolute maximum of \( g^{(4)}(x) \) over \( -0.05 \leq x \leq 0.05 \) is \( \frac{105}{16}(1+0.05)^{-9/2} \approx 6.2129 \). Thus, \[ |R_3(x)| \leq \frac{6.2129}{24} (0.05)^4 \approx 3.2268 \times 10^{-7} \]. This represents the maximum error when \(-0.05 \leq x \leq 0.05\).

Key Concepts

Taylor Series Expansion

The Taylor series is a powerful tool used in mathematics to approximate functions by polynomials. This expansion expresses a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The Maclaurin series is a special case where this point is zero. For a given function, the Maclaurin series can be written as:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \)

This formula takes advantage of a function’s behavior near the origin, which is extremely useful for approximation purposes in physics, engineering, and other scientific fields. The idea is to truncate the series after a desired number of terms. This provides a polynomial approximation of the function. Each added term makes the approximation closer to the actual function. It is important to note the difference from other polynomial expansions—a Taylor series accounts specifically for the behavior of derivatives, allowing for more accurate approximation. Understanding the Taylor series' foundations can enrich your comprehension of mathematical functions and their applications.

Third-order Polynomial

The third-order polynomial refers to truncating the Maclaurin series expansion at the third derivative term. This means the polynomial includes all terms from the series up to and including the one containing \( x^3 \). For the function \( g(x) = (1+x)^{-1/2} \), the third-order polynomial approximation is achieved by evaluating the first, second, and third derivatives at the point \( x = 0 \).

• First, evaluate \( g(0) = 1 \).
• Next, find \( g'(0) = -\frac{1}{2} \).
• Then, \( g''(0) = \frac{3}{4} \).
• Finally, evaluate \( g'''(0) = -\frac{15}{8} \).

Using these derivative values, you substitute them into the Maclaurin series formula up to \( x^3 \):

• \( P_3(x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2 - \frac{5}{16}x^3 \)

This polynomial provides a good approximation for small values of \( x \) and forms the basis for understanding how to construct these approximations for other functions.

Error Bound

When approximating functions using polynomial series like the Maclaurin series, understanding the error bound is crucial. The error bound is an estimate of how far the polynomial approximation deviates from the actual function. For the third-order Maclaurin polynomial, this error is denoted by the remainder term \( R_3(x) \). The formula for computing the error bound is:

• \( R_3(x) = \frac{g^{(4)}(c)}{4!}x^4 \)

where \( c \) is some value between 0 and \( x \). To find the maximum error, you evaluate the fourth derivative of the function over a specified interval—in this case, \(-0.05 \leq x \leq 0.05\). The maximum occurs at \( x = -0.05 \).This gives a calculated maximum error of approximately \( 3.2268 \times 10^{-7} \).Knowing the error bound helps in assessing the reliability of the polynomial approximation. It informs us how closely our third-order polynomial represents the original function within a specific range of \( x \).

Derivatives Evaluation

Evaluating derivatives is a key step in constructing a Taylor or Maclaurin series. Let's take the function \( g(x) = (1+x)^{-1/2} \) as an example. To find its third-order Maclaurin polynomial, we need to determine the first few derivatives at \( x = 0 \).

• The first derivative is \( g'(x) = -\frac{1}{2}(1+x)^{-3/2} \).
• The second derivative is \( g''(x) = \frac{3}{4}(1+x)^{-5/2} \).
• The third derivative is \( g'''(x) = -\frac{15}{8}(1+x)^{-7/2} \).

These expressions are then evaluated at \( x = 0 \):

• \( g(0) = 1 \)
• \( g'(0) = -\frac{1}{2} \)
• \( g''(0) = \frac{3}{4} \)
• \( g'''(0) = -\frac{15}{8} \)

Calculating derivatives up to the required order ensures that each term in the polynomial corresponds to the original function's behavior at the origin. This process is indispensable for creating accurate polynomial approximations and understanding the nuances of more complex functions.