Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The second degree Taylor polynomial generated by \(\ln (1+x)\) about \(x=0\) is \(x-\frac{x^{2}}{2}\). Use Taylor s Theorem to nd a good upper bound on the error involved in using this polynomial to approximate the following. (a) \(\ln (1.2)\) (b) \(\ln (0.8)\)

Step-by-Step Solution

Answer

The upper bounds on the errors involved in using the Taylor polynomial to approximate \(\ln (1.2)\) and \(\ln (0.8)\) are approximately \(0.58\) and \(0.17\) respectively.

Step 1: Calculate the Degree of the Polynomial

The degree of the Taylor polynomial found in the problem is \(2\). This can be seen from the highest power in the polynomial. Notably, \(x^2\).

Step 2: Compute the Third Derivative

Taylor's theorem states that the error made by a degree \(n\) Taylor polynomial is determined by the \(n+1\)-th derivative of the function. Since the degree is \(2\) in this case, we need to compute the third derivative of the function \(f(x) = \ln (1+x)\). The first three derivatives are: \(f'(x) = \frac{1}{1+x}\), \(f''(x) = -\frac{1}{(1+x)^2}\), and \(f'''(x) = \frac{2}{(1+x)^3}\).

Step 3: Apply Taylor's Theorem to Find the Error's Upper Bound

According to Taylor's theorem, the error \(E(x)\) is bounded by \(|E(x)|\leq M |x-a|^{n+1}/(n+1)!\) where \(M\) is an upper bound of \(|f^{(n+1)}(x)|\) on the interval from \(a\) to \(x\) and \(a\) is where we are comparing around (in this case, \(a = 0\)). The function in our problem is \(\ln (1+x)\), the degree is \(2\), and our \(a = 0\), hence we can rewrite as \(|E(x)|\leq M |x-0|^{2+1}/(2+1)!\) = \(M|x|^3/6\). As we computed in Step 2, \(f'''(x)\) = \(2/(1+x)^3\). If we find the maximum value of this on the interval between \(0\) and \(x = 1.2\), which happens to be \(1\), we can find \(M\). Hence, \(M = 2\). Thus, the bound on the error for \(\ln (1.2)\) is \(2 * 1.2^3/6\) and for \(\ln (0.8)\) is \(2 * 0.8^3/6\).

Step 4: Calculate Error's Upper Bound

Substituting the range of \(x\) in \(M|x|^3/6\), we obtain the error's upper bound for both cases: \n- For \(\ln (1.2)\), the error's upper bound is \(2 * 1.2^3/6\), which is approximately \(0.58\). \n- For \(\ln (0.8)\), the error's upper bound is \(2 * 0.8^3/6\), which is approximately \(0.17\).

Key Concepts

Taylor's Theorem

Taylor's Theorem is a powerful tool in calculus that provides an approximation of functions using polynomials. It essentially states that functions can be approximated as a sum of their derivatives around a certain point, known as the center of the expansion.
This is particularly useful because it allows us to simplify complex functions down to manageable polynomial expressions.
When applying Taylor's Theorem, it is important to consider the remainder or error term, which helps us quantify how accurate our polynomial approximation is compared to the actual function.
This error is significantly influenced by the degree of the Taylor polynomial and the higher-order derivatives of the function.
The general error bound according to Taylor's Theorem is given by

• \(|E(x)| \leq \frac{M |x-a|^{n+1}}{(n+1)!}\)

where \(n\) is the degree of the polynomial, \(M\) is an upper bound of the \((n+1)\)-th derivative over the interval of interest, and \(a\) is the center of the polynomial expansion.

Polynomial Approximation

Polynomial approximation is the process of estimating a more complex function using a polynomial, which is much simpler to work with.
The Taylor polynomial is one of the most common techniques for polynomial approximation.
By using the derivatives of the function, Taylor polynomials provide us with an approximation that closely resembles the behavior of the original function near a particular point.
In our example, we used a second-degree Taylor polynomial to approximate the natural logarithm function,

• \(\ln(1+x) \approx x - \frac{x^2}{2}\).

This polynomial is generated by calculating the derivatives of \(\ln(1+x)\) and evaluating them at \(x = 0\).
This generates the coefficients for each term of the polynomial. Since higher degree polynomials often yield a more accurate approximation, it's crucial to balance complexity with approximation needs.

Derivative Calculation

Derivative calculation is essential in forming Taylor polynomials as the derivatives of a function provide the coefficients for each term in the polynomial expansion.
Knowing how to derive functions accurately is key to obtaining a precise polynomial approximation.
For the natural logarithm function \(f(x) = \ln(1+x)\), we calculate the derivatives step by step:

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

These derivatives provide the coefficients used in building Taylor polynomials, allowing us to approximate \(\ln(1+x)\) with increasing accuracy as we include more terms from the derivatives.
This step forms the foundation for both the polynomial itself and the evaluation of the accuracy of the resultant approximation.