Problem 52 - Thomas Calculus

Question

A cubic approximation Use Taylor's formula with \(a=0\) and \(n=3\) to find the standard cubic approximation of \(f(x)=\) 1\(/(1-x)\) at \(x=0 .\) Give an upper bound for the magnitude of the error in the approximation when \(|x| \leq 0.1\)

Step-by-Step Solution

Answer

The cubic approximation of \( f(x) = \frac{1}{1-x} \) at \( x=0 \) is \( 1 + x + x^2 + x^3 \) with an error bound of approximately 0.0001024 for \(|x| \leq 0.1\).

Step 1: Calculate Derivatives

The function given is \( f(x) = \frac{1}{1-x} \). First, we need to calculate the derivatives of \( f(x) \) up to the third order since \( n = 3 \).- The first derivative, \( f'(x) = \frac{1}{(1-x)^2} \).- The second derivative, \( f''(x) = \frac{2}{(1-x)^3} \).- The third derivative, \( f'''(x) = \frac{6}{(1-x)^4} \).

Step 2: Evaluate Derivatives at \(a=0\)

Substitute \( x = 0 \) into each of our derivatives and the original function to obtain the coefficients for the Taylor series expansion.- \( f(0) = \frac{1}{1-0} = 1 \)- \( f'(0) = \frac{1}{(1-0)^2} = 1 \)- \( f''(0) = \frac{2}{(1-0)^3} = 2 \)- \( f'''(0) = \frac{6}{(1-0)^4} = 6 \)

Step 3: Write Taylor Series

Using Taylor's formula, express the cubic approximation of \( f(x) \) around \( x = 0 \):\[ P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 \]Substituting the coefficients, we have:\[ P_3(x) = 1 + x + \frac{2}{2}x^2 + \frac{6}{6}x^3 \ = 1 + x + x^2 + x^3 \]

Step 4: Determine Error Bound

The remainder term, \( R_3(x) \), for a Taylor series at \( n = 3 \) is:\[ R_3(x) = \frac{f^{(4)}(c)}{4!}x^4 \]where \( c \) is some value between 0 and \( x \). The fourth derivative of \( f(x) \) is \( f^{(4)}(x) = \frac{24}{(1-x)^5} \).For \( |x| \leq 0.1 \), the maximum value of \( |f^{(4)}(c)| \) where \( c \) is in \([-0.1, 0.1]\) is approximately \( \frac{24}{(1-0.1)^5} \approx 24.5763 \). Therefore, the error bound is:\[ |R_3(x)| \leq \frac{24.5763}{4!}(0.1)^4 = \frac{24.5763}{24}(0.0001) \approx 0.0001024 \]

Key Concepts

Cubic Approximation

A cubic approximation is a mathematical method used to estimate the value of a function, particularly for functions that can be difficult to calculate directly. Essentially, it involves creating a polynomial expression that closely follows the behavior of the function near a chosen point, known as the point of approximation.

• A cubic approximation uses up to the third derivative of the function.
• This type of approximation is often used because it balances simplicity and accuracy for small intervals.
• It involves evaluating the function and its first three derivatives at a specific point.

In this exercise, the function \( f(x) = \frac{1}{1-x} \) is approximated around \( x = 0 \) to produce a polynomial, \( P_3(x) = 1 + x + x^2 + x^3 \). The expression accurately follows the function's behavior around the origin, making it straightforward to approximate values within a small range around this point by using this polynomial.

Error Bound

The error bound is an important concept in approximating functions using series, like the Taylor series. When approximating a function, it is crucial to understand how close the approximation is to the actual function values.

• The error bound gives you the maximum possible error between the true function value and the approximation.
• In the context of Taylor series, this value is expressed by the remainder term.
• We compute the error bound to understand the approximation accuracy within a specific interval.

For our exercise, given that \( |x| \leq 0.1 \), the error bound for the cubic approximation \( P_3(x) \) was estimated. By computing the bound, we find that the maximum error we can expect is approximately 0.0001024, ensuring that the approximation is highly accurate for inputs within this range.

Derivatives

Derivatives play a fundamental role in constructing Taylor series, particularly in deciding the coefficients of the polynomial terms in the series expansion.
- The derivative of a function describes the rate at which the function value changes.Here are the derivatives calculated for the function \( f(x) = \frac{1}{1-x} \):

• The first derivative \( f'(x) = \frac{1}{(1-x)^2} \)
• The second derivative \( f''(x) = \frac{2}{(1-x)^3} \)
• The third derivative \( f'''(x) = \frac{6}{(1-x)^4} \)

These derivatives help in getting the terms required to build the cubic approximation. They are evaluated at a specific point (in this case, \( x = 0 \)), setting up the polynomial's structure and forming its coefficients.

Remainder Term

The remainder term is a crucial part of understanding how accurate our Taylor series approximation truly is. It tells us how much of the function's behavior is not captured by the polynomial approximation.

• The remainder term represents the error of the approximation.
• It is given by the formula: \( R_3(x) = \frac{f^{(4)}(c)}{4!}x^4 \)
• Where \( c \) is a value between the point of expansion and the value of \( x \).

In this problem, the fourth derivative \( f^{(4)}(x) \) of the function \( f(x) = \frac{1}{1-x} \) is used to express the remainder term. Computing this term helps us estimate how the approximation differs from the true function values, ensuring we acknowledge and factor in this potential discrepancy. For the interval \( |x| \leq 0.1 \), the maximum value of the remainder term provides a robust estimate of the approximation's potential error, as shown by the previous calculation of about 0.0001024.