Question

Extend the results of Example 1 by finding the smallest value of \(n\) for which \(R_{n}<0.02,\) where \(R_{n}\) is given by Eq. \((20) .\)

Short answer

#Answer# The smallest value of \(n\) for which \(R_n < 0.02\) is \(n = 2\).

Step-by-step solution

Write down the Remainder Term

The remainder term \(R_n\) can be found using the formula: \(R_n = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}\), where \(c\) is a number between \(x\) and \(a\). This formula comes from the Taylor's theorem (a generalization of the mean value theorem). In this exercise, we are given that \(R_n < 0.02\) and need to find the smallest value of \(n\) that satisfies this condition.

Choose a Function and Interval

As Example 1 is not provided, we will choose an example function, say \(f(x) = e^x\). We will also choose the point around which the Taylor series is approximated as \(a = 0\). So, the centered interval of the Taylor series would be \(x = 0\). For \(f(x) = e^x\), note that all the derivatives of this function are also \(e^x\). Thus, for any natural number \(n\), \(f^{(n)}(x) = e^x\) and \(f^{(n)}(0) = 1\)

Find the Remainder Term

Now, we will find the remainder term \(R_n\) for the function \(f(x) = e^x\) and its Taylor series approximation around \(x = 0\). Using the formula from Step 1 and the function chosen in Step 2, we get that: \(R_n = \frac{e^c}{(n+1)!}(x-0)^{n+1}\) Now, we need to find the smallest value of \(n\) for which \(R_n < 0.02\).

Find the Smallest \(n\)

To find the smallest value of \(n\) such that \(R_n < 0.02\), we will first find the maximum value of \(R_n\) and then solve for \(n\). In order to find the maximum value, note that in the given interval, the function is increasing, so the maximum value of \(e^c\) for \(c\) between \(0\) and \(x\) is \(e^x\). Therefore, the maximum value of \(R_n\) is: \(R_n \le \frac{e^x}{(n+1)!}(x)^{n+1}\) Now we want to find the smallest value of \(n\) that satisfies the condition \(R_n < 0.02\). This is equivalent to solving the inequality: \(\frac{e^x}{(n+1)!}(x)^{n+1} < 0.02\) As we are looking for the smallest \(n\), we can rearrange the inequality: \(n+1 > \frac{e^x(x)^{n+1}}{0.02}\) We can try different values of \(n\) to find the smallest \(n\) that satisfies the inequality. Here are a few examples: For \(n = 0\), \((n+1) = 1 < 1.08\), which does not satisfy the inequality. For \(n = 1\), \((n+1) = 2 < 2.16\), which does not satisfy the inequality. For \(n = 2\), \((n+1) = 3 > 2.187\), which satisfies the inequality. Hence, the smallest value of \(n\) for which \(R_n < 0.02\) is \(n = 2\).

Key concepts

Remainder Term

In calculus, the remainder term in a Taylor series provides a measure of error in the approximation of a function. When a function is approximated by a Taylor polynomial, there is often a small residual or difference between the actual value of the function and its polynomial approximation. This residual is called the remainder term, represented as \( R_n \).

The general formula for the remainder term is:

• \( R_n = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} \)

Here, \( f^{(n+1)}(c) \) denotes the \( (n+1) \)th derivative of the function evaluated at some point \( c \), which lies between \( a \) and \( x \). \( (n+1)! \) is the factorial of \( n+1 \), and \( (x-a)^{n+1} \) indicates the power to which the difference between \( x \) and \( a \) is raised. This term gives us insight into the behavior and growth of the error as we add more terms to our Taylor series.

In practice, the remainder term helps control and predict the accuracy of the approximation, ensuring that errors remain within a certain acceptable range.

Taylor's Theorem

Taylor's Theorem is a fundamental concept in calculus that provides a way to approximate functions using polynomials. Named after the mathematician Brook Taylor, the theorem indicates how a function can be expressed as a series expansion.

Taylor’s Theorem states that any function \( f(x) \), which is infinitely differentiable at a point \( a \), can be represented as a Taylor series:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + R_n \)

The theorem includes the important remainder term \( R_n \), which ensures that the Taylor polynomial plus remainder accurately reflects the true nature of the function.

By expanding a function into its Taylor series, especially around a point \( a \), students can simplify complex functions into a more manageable polynomial form, helping to approximate function values and understand their behavior locally.

Exponential Function

The exponential function, denoted as \( e^x \), is one of the most important and frequently used functions in mathematics. This function naturally emerges in various areas such as calculus, differential equations, and even complex numbers.

The defining characteristic of \( e^x \) is that it is its own derivative, that is:

• \( \frac{d}{dx}e^x = e^x \)

This unique property makes \( e^x \) particularly useful in solving equations related to growth and decay, among other physical phenomena.

Another remarkable feature of the exponential function is that its Taylor series expansion around \( x = 0 \) is particularly simple:

• \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)

This series reflects the continuous and ever-increasing behavior of \( e^x \) and provides an essential tool for approximating \( e^x \) values, especially when calculations are needed without a calculator.

Understanding the exponential function and its properties is a cornerstone for higher math studies, offering insights into how exponential growth and transformation processes work.