Problem 30 - Calculus for Biology and Medicine

Question

Use the following form of the error term $$ \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1) !} $$ where \(K=\) largest value of \(\left|f^{(n+1)}(t)\right|\) for \(0 \leq t \leq x\), to determine in advance the degree of Taylor polynomial at \(a=0\) that would achieve the indicated accuracy in the interval \([\mathbf{0}, \boldsymbol{x}] .\) (Do not compute the Taylor polynomial.) $$ f(x)=\ln (1+x), x=0.1, \text { error }<10^{-2} $$

Step-by-Step Solution

Answer

A second-degree Taylor polynomial is needed.

Step 1: Derivative Analysis

Identify the derivatives of the function. The function is given by \( f(x) = \ln(1+x) \). The first derivative is \( f'(x) = \frac{1}{1+x} \). Generally, the \( n \)th derivative is given by: \[ f^{(n)}(x) = (-1)^{n+1} (n-1)! (1+x)^{-n} \].

Step 2: Evaluate the (n+1)-th Derivative

Evaluate the \( (n+1) \)-th derivative to determine \( K \), the largest value of \( \left|f^{(n+1)}(t)\right| \) for \( 0 \leq t \leq x \). Given \( f^{(n+1)}(x) = (-1)^{n+2} n! (1+x)^{-(n+1)} \), for \( x = 0.1 \), the largest value occurs at \( t=0 \), so \[ K = n! \].

Step 3: Set Up the Error Bound Inequality

Use the given error bound formula \( \left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1)!} \). Substitute \( K = n! \) and \( x = 0.1 \). Thus, the inequality becomes \[ \frac{n! |0.1|^{n+1}}{(n+1)!} < 10^{-2} \].

Step 4: Simplify the Inequality

Recognize that \( \frac{n!}{(n+1)!} = \frac{1}{n+1} \). Hence the inequality becomes \( \frac{|0.1|^{n+1}}{n+1} < 10^{-2} \).

Step 5: Solve for n

To find \( n \), solve \( \frac{0.1^{n+1}}{n+1} < 0.01 \). Try increasing integer values of \( n \) to find the smallest one that satisfies the inequality: - For \( n = 2 \), \( \frac{0.1^{3}}{3} = \frac{0.001}{3} = 0.000333 < 0.01 \) which holds true. Thus, \( n = 2 \) is sufficient.

Key Concepts

Error Analysis

In the realm of mathematics, particularly when dealing with Taylor polynomials, error analysis plays a crucial role. It helps us determine how accurately a polynomial approximates a function within a given interval. This process involves evaluating how much error or deviation exists between the actual function and the polynomial approximation.The error of a Taylor series approximation is denoted by \(R_{n+1}(x)\), and it gives a bound on the accuracy of the polynomial. We use the formula \(\left| R_{n+1}(x) \right| \leq \frac{K |x|^{n+1}}{(n+1)!}\) to assess this error. Here:

• \( K \) represents the largest value of the \(n+1\)th derivative of the function evaluated at some point \(t\) in the interval \([0, x]\).
• \( x \) is the point where we evaluate the error.
• \( n \) is the degree of the Taylor polynomial, and \(n+1\) accounts for the next derivative required for error calculation.

This formula allows us to predict the number of terms needed in the Taylor polynomial to achieve a desired level of accuracy before actually computing the polynomial.

Derivatives

Derivatives are fundamental in understanding Taylor polynomials. They represent the rate of change of a function with respect to its variables. In the case of Taylor series, derivatives help in constructing polynomial approximations of functions.Given the function \(f(x) = \ln(1+x)\), the derivatives are calculated as follows:

• The first derivative, \(f'(x) = \frac{1}{1+x}\), shows the initial rate of change.
• Generally, the \(n\)th derivative for our function is represented as \(f^{(n)}(x) = (-1)^{n+1} (n-1)! (1+x)^{-n}\).

To evaluate the error and choose the right polynomial degree, we also need to examine the \(n+1\)th derivative. This helps us determine \(K\), the maximum value of this derivative over the interval of interest. Understanding these derivatives is essential not only for calculating \(K\) but also for ensuring the Taylor polynomial's accuracy conforms to required standards.

Taylor Series

A Taylor series is an infinite series of mathematical terms that when summed together approximate a mathematical function. Starting from the function's value at a particular point, a Taylor series can effectively describe the function around that point by adding terms that involve the function's derivatives.The general form of a Taylor series for a function \(f\) about point \(a\) is:\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)(x-a)^2}{2!} + \frac{f'''(a)(x-a)^3}{3!} + \ldots\]For functions centered at \(a=0\), the series is called a Maclaurin series. For \(f(x) = \ln(1+x)\), which is centered at \(a=0\), the series expansion helps us approximate the function within the desired error margin by using only a limited number of terms.Taylor series are critically important because they allow complex functions to be expressed as sums of polynomial terms. This makes calculations more manageable and computationally efficient, particularly in fields such as engineering and physics where approximate solutions are often required.