Problem 27 - 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.) $$ f(x)=e^{x}, x=2, \text { error }<10^{-3} $$

Step-by-Step Solution

Answer

The degree needed is 8.

Step 1: Understand the Problem

We are tasked with finding the degree of the Taylor polynomial for the function \(f(x) = e^x\) such that the error term \(R_{n+1}(x)\) is less than \(10^{-3}\) for the interval \([0, 2]\). This requires utilizing the given error bound formula.

Step 2: Identify the Parameter Values

Given the function \(f(x) = e^x\), the derivative of any order is the same function, i.e., \(f^{(n+1)}(x) = e^x\). Therefore, the maximum value of \( |f^{(n+1)}(t)| \) on the interval \([0, 2]\) is \(e^2\). Thus, \(K = e^2\) and \(x = 2\).

Step 3: Express the Error Bound

The error bound formula is \(\left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1)!}\). Plugging in the values, we get \(\left|R_{n+1}(2)\right| \leq \frac{e^2 \times 2^{n+1}}{(n+1)!}\).

Step 4: Set Inequality for Desired Accuracy

To achieve the desired accuracy, we need \(\frac{e^2 \times 2^{n+1}}{(n+1)!} < 10^{-3}\). This inequality will determine the minimum degree \(n\) needed for the Taylor polynomial.

Step 5: Solve the Inequality

Begin testing values of \(n\) to find the smallest integer satisfying the inequality:- For \(n = 6\): \(\frac{e^2 \times 2^7}{7!}\) approximates \(0.0086 > 10^{-3}\).- For \(n = 7\): \(\frac{e^2 \times 2^8}{8!}\) approximates \(0.0018 > 10^{-3}\).- For \(n = 8\): \(\frac{e^2 \times 2^9}{9!}\) approximates \(0.00036 < 10^{-3}\).Thus, the required degree is \(n = 8\).

Step 6: Conclusion

The minimum degree \(n\) for the Taylor polynomial centered at \(a=0\) that ensures the error is less than \(10^{-3}\) on the interval \([0, 2]\) is 8.

Key Concepts

Error Term

In the context of Taylor polynomials, the error term represents the difference between the actual value of a function and its approximation using a Taylor polynomial. The formula for the error term, given as \[\left|R_{n+1}(x)\right| \leq \frac{K|x|^{n+1}}{(n+1)!}\]is crucial in determining how accurate the Taylor polynomial is.

In this formula:

• \(K\) is the largest value of the absolute derivative \(\left|f^{(n+1)}(t)\right|\) within the interval from 0 to \(x\).
• \(n+1\) is the order of the derivative, signifying how many terms are in the polynomial.
• \(x\) is the point at which we are approximating the function.

The goal is to make sure this error term is kept below a certain threshold, which in many problems, like the one you're tackling, is set to a tiny value, such as \(10^{-3}\). By addressing this bound, we ensure our polynomial is a sufficiently close approximation to the actual function.

Degree of Polynomial

The degree of a Taylor polynomial, represented by \(n\), indicates the highest power of \(x\) in the polynomial. It fundamentally affects the accuracy of approximation. A polynomial of higher degree usually provides a better approximation of the actual function since it includes more terms to mimic the curve's behavior.

For our task, this is particularly important, as we need to determine the smallest \(n\) that achieves the desired precision. By integrating the sufficient number of terms determined by trial and error testing within the error inequality, we guarantee the polynomial meets the required accuracy.

• Starting with a low degree, we test and gradually increase it until the error is within bounds, \(< 10^{-3}\).
• In this problem, solving the inequality with test cases showed that a polynomial of degree 8 is necessary.

This iterative testing emphasizes how the degree of the polynomial directly influences error and accuracy.

Interval of Accuracy

The interval of accuracy relates to the range over which the Taylor polynomial closely approximates the original function. It's crucial because it determines where the approximation holds true, hence ensuring reliability over the specified section of the function's domain.

In the given problem, the interval \([0, 2]\) was specified, representing where the Taylor polynomial must maintain an accurate approximation. The characteristics of this interval help guide:

• The determination of \(K\), the maximum derivative value within the interval, ensuring the error estimate is accurate.
• The proper evaluation of the polynomial and the function's behavior over the distance from the center \(a=0\) to \(x=2\).

Understanding and applying the interval aids in solving the problem, ensuring the function approximation remains precise and reliable across the specified range.