Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The second degree Taylor polynomial for \(f(x)=(1+x)^{p}\) is \(1+p x+\frac{p(p-1)}{2 !} x^{2} .\) If the second degree Taylor polynomial is used to approximate \(\sqrt{1+x}\) for \(|x| \leq 0.2\), nd an upper bound for the magnitude of the error. Use the Taylor Inequality; then check your answer by graphing \(R_{2}(x)\).

Step-by-Step Solution

Answer

The upper bound for the magnitude of the error, when using the second degree Taylor polynomial to approximate \(\sqrt{1+x}\), is approximately 0.004.

Step 1: Find the third derivative of \(f(x)\)

Start with the function \(f(x) = (1+x)^p\). The first derivative is \(f'(x) = p(1+x)^{p-1}\) through the power rule. The second derivative is then \(f''(x) = p(p-1)(1+x)^{p-2}\), again using the power rule. The third derivative, which is needed for the Taylor Approximation formula, is \(f'''(x) = p(p-1)(p-2)(1+x)^{p-3}\). In the case of approximating \(\sqrt{1+x}\), set \(p = 1/2\) to get \(f'''(x) = (-3/8)(1+x)^{-5/2}\).

Step 2: Find the maximum of the third derivative in \([-0.2, 0.2]\)

For \(|x|\leq 0.2\), the maximum of the absolute value of the third derivative will occur at the endpoint \(x = -0.2\). Plug this into \(f'''(x)\) to get \(|f'''(x)| = |-3/8(-1.2)^{-5/2}| = 5.8838.\

Step 3: Use Taylor Inequality to approximate the error

The Taylor inequality states that the error of using the nth degree Taylor polynomial to approximate a function is less than or equal to (M / (n+1)!) * |x - a|^(n+1), where M is the maximum of the (n+1)th derivative on the interval and a is the point around which the Taylor polynomial is centered (in this case, a = 0). Here, the Taylor polynomial is second degree, so n = 2 and \(M = 5.8838\). Plugging these in gives an error less than or equal to \(5.8838 / (3!) * 0.2^{3}\), which equals 0.003922533333 or roughly 0.004.

Step 4: Check the answer with a graph of \(R_{2}(x)\)

To confirm whether the theoretical answer is reasonable, check with a graph of \(R_{2}(x)\), which is the remainder term of the second degree Taylor polynomial. If the graph and the calculated error from Step 3 agree (i.e. the maximum value of \(R_{2}(x)\) in \([-0.2, 0.2]\) is around 0.004), then it validates the answer.

Key Concepts

Taylor series

The Taylor series is a mathematical concept used to approximate complex functions with simpler polynomial forms. It expands a function into an infinite sum of terms calculated from the values of its derivatives at a single point. For example, for a function \( f(x) \) centered at \( a \), the Taylor series is given by:

• \( f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \)

In practice, often a finite number of terms is used to get what is known as a Taylor polynomial, which serves as an approximation of the function. Adding more terms increases the accuracy of the approximation. In the given exercise, a second degree Taylor polynomial is used to approximate \( \sqrt{1+x} \), illustrating how even a few terms can provide a useful estimate of a function within a specific interval around \( a = 0 \).

Error estimation

Error estimation in the context of Taylor series is crucial as it allows us to understand how far off our polynomial approximation is from the true value of a function. The error, also known as the remainder, is the difference between the actual function and its Taylor polynomial approximation. An effective way to bound this error is using Taylor's Inequality, which states that:

• The error of an \( n \)-th degree Taylor polynomial is \( \leq \frac{M}{(n+1)!} |x-a|^{n+1} \).

Here \( M \) is the maximum value of the \((n+1)\)-th derivative of \( f \) on the interval of interest. This allows us to estimate an upper bound on the error when approximating the function. In the exercise, this concept is applied to ensure that the error in approximating \( \sqrt{1+x} \) using a second degree polynomial is sufficiently small, assuring us of the approximation's quality.

Power rule

The power rule is a simple and essential rule in calculus for differentiating functions of the form \( x^n \). The rule states:

• The derivative of \( x^n \) is \( n \cdot x^{n-1} \).

This rule makes computing derivatives straightforward and is applied repeatedly when determining the derivatives needed for the Taylor series. In our exercise, the power rule has been used to find the derivatives of \( (1+x)^p \) where \( p = 1/2 \). Using the power rule:

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

These derivatives are essential for constructing the Taylor polynomial and estimating errors effectively.

Derivative calculation

Derivative calculation is a core skill in calculus and is integral for finding both the Taylor series and error estimation. In the process of building a Taylor polynomial, derivatives at a specific point provide the coefficients for each term of the polynomial. In our exercise:

• We begin with the function \( f(x) = (1+x)^p \) and find its higher-order derivatives.
• For \( p = 1/2 \) representing \( \sqrt{1+x} \), we compute the first three derivatives using the power rule.
• Each derivative is evaluated to help form the second degree Taylor polynomial and for calculating the error bound using Taylor's Inequality.

Finding derivatives accurately ensures the Taylor series correctly approximates the function. This practice is pivotal not only in this exercise but in a wide array of mathematical and real-world applications.