Problem 29 - 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.) $$ f(x)=1 /(1+x), x=0.2, \text { error }<10^{-2} $$

Step-by-Step Solution

Answer

A Taylor polynomial of degree 3 or higher is needed.

Step 1: Identify the function and the context

We have the function \( f(x) = \frac{1}{1+x} \) and we need to approximate it using a Taylor polynomial at \( a=0 \). The interval given is \([0, 0.2]\) and we are to ensure the error is less than \(10^{-2}\). We need to determine the degree \( n \) such that the error bound condition is satisfied.

Step 2: Find the derivatives and error term, formulate the problem

For \( f(x) = \frac{1}{1+x} \), the \( n+1 \)th derivative is given by \( f^{(n+1)}(x) = (-1)^{(n+1)}(n+1)!/(1+x)^{n+2} \). The maximum occurs at \( t=0 \), so \( K = |f^{(n+1)}(0)| = (n+1)! \). Thus, the error term is \( \left|R_{n+1}(x)\right| \leq \frac{|x|^{n+1}}{(1+x)^{n+2}(n+1)!} \). For \( x = 0.2 \), \( K = (n+1)! , x^{n+1} = 0.2^{n+1} \).

Step 3: Set up the inequality for error

Plug in known values and simplify the inequality: \[ \frac{K|x|^{n+1}}{(n+1)!} = \frac{(n+1)! \cdot 0.2^{n+1}}{(1.2)^{n+2} \cdot (n+1)!} < 10^{-2} \] simplifying gives \( \left(\frac{0.2}{1.2}\right)^{n+1} < 10^{-2} \).

Step 4: Solve for the minimum degree n

To find \( n \), we solve \( \left(\frac{1}{6}\right)^{n+1} < 10^{-2} \). Take logarithms on both sides: \( (n+1)\log\left(\frac{1}{6}\right) < \log(10^{-2}) \). This simplifies to \( n+1 > \frac{2 \log(10)}{\log(6)} \approx 3.41 \), thus \( n \geq 3 \). Therefore, a Taylor polynomial of degree at least 3 is required.

Key Concepts

Error Term

The error term in a Taylor polynomial is essential because it tells us how much the polynomial approximation deviates from the actual function. It is usually expressed in a form like \[|R_{n+1}(x)| \leq \frac{K|x|^{n+1}}{(n+1)!},\] where the term \(K\) is the largest value of the \((n+1)\)th derivative over the interval being considered. This expression helps us understand and control how accurate the approximation is for a given function.

In practice, when you are working with a specific function like \(f(x) = \frac{1}{1+x}\), finding the optimal degree involves calculating these derivatives and determining the value of \(K\) to assess error bounds. The smaller this error term, the more accurate the polynomial. In our exercise, the error term guides how much we can 'trust' the polynomial over the interval \([0, 0.2]\).

The goal is to ensure the error is less than a defined threshold, in this case, \(10^{-2}\). By analyzing this, we can decide on defining which polynomial degree to use to achieve the desired precision.

Degree of Polynomial

The degree of a Taylor polynomial refers to the highest degree of the term you include in your approximation of a function. In mathematical terms, a higher degree often implies a closer approximation. However, this usually comes at the cost of more complex calculations.

When determining the appropriate degree \(n\) for our Taylor polynomial, we balance accuracy with computational simplicity. The higher \(n\) is, the smaller the error between the polynomial and the actual function. For our problem, to keep the error below \(10^{-2}\), we needed at least a 3rd-degree polynomial.

The process involves setting up an inequality to solve for \(n\), ensuring the error term's condition is satisfied. Here, after plugging values in, the simplification \((\frac{0.2}{1.2})^{n+1} < 10^{-2}\) emerges. Solving this inequality revealed that \(n\) must be at least 3, since the calculated minimum for \(n+1\) was approximately 3.41.

• This method helps avoid excessive computations by using just enough terms to meet the desired precision.
• Deciding on the polynomial degree is crucial for efficiency and precision.

Derivative Calculation

The calculation of derivatives is a foundational skill when working with Taylor polynomials. It helps derive the coefficients of the polynomial terms and is instrumental in determining the accuracy via the error term.

For the function \(f(x) = \frac{1}{1+x},\) the derivatives follow a specific pattern. Each derivative involves increasing factorials and alternate signs due to the function's nature, expressed for \(n+1\)th derivative as:\[f^{(n+1)}(x) = (-1)^{(n+1)}\frac{(n+1)!}{(1+x)^{n+2}}.\]When calculating these derivatives:

• Sign alternates with each derivative due to \((-1)^{(n+1)}\).
• Factorials grow with derivative order, handling complexity, especially at higher degrees.
• Derivative values directly contribute to finding the value of \(K\).

Understanding and computing these derivatives allows us to determine \(K\), which is crucial to calculate error terms effectively. This processing leads to more robust polynomial approximations.