Problem 56 - Calculus Early Transcendentals

Question

Use the remainder term to estimate the absolute error in approximating the following quantities with the nth-order Taylor polynomial centered at \(0 .\) Estimates are not unique. $$\cos 0.45 ; n=3$$

Step-by-Step Solution

Answer

The approximate error in estimating the cosine of 0.45 using a 3rd order Taylor polynomial centered at 0 is approximately 0.000222.

Step 1: Identify the Taylor polynomial for the cosine function

The Taylor polynomial for the cosine function centered at \(0\) is given by: $$ T_n(x) = \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} \frac{(-1)^k x^{2k}}{(2k)!} $$ For this problem, we have \(n=3\), thus, the \(3^{rd}\) order Taylor polynomial for the cosine function is: $$ T_3(x) = \sum_{k=0}^{\lfloor\frac{3}{2}\rfloor} \frac{(-1)^k x^{2k}}{(2k)!} = 1 - \frac{x^2}{2} $$

Step 2: Find the \((n+1)^{th}\) derivative of the cosine function

In this case, the \((n+1)^{th}\) derivative corresponds to the \(4^{th}\) derivative of the cosine function. The derivatives of the cosine function follow a cyclical pattern: 1. Derivative: \(-\sin x\) 2. Derivative: \(-\cos x\) 3. Derivative: \(\sin x\) 4. Derivative: \(\cos x\) Thus, the \(4^{th}\) derivative of the cosine function is again the cosine function: $$ f^{(4)}(x) = \cos x $$

Step 3: Identify the interval and find the maximum value for the \(4^{th}\) derivative

Since we are using the Taylor polynomial centered at \(0\) to approximate the cosine function at \(0.45\), we need to find the maximum value for the \(4^{th}\) derivative in the interval \([0, 0.45]\). We know that the cosine function varies between \(-1\) and \(1\). Therefore, the maximum value for the \(4^{th}\) derivative in the given interval is: $$ M = |\cos x|_{max} = 1 $$

Step 4: Apply Lagrange's form of the remainder to estimate the error

The remainder term, \(R_n(x)\), in Lagrange's form can be written as: $$ R_n(x) = \frac{M |x - 0|^4}{(n+1)!} $$ Calculate the absolute value of the remainder term for estimating the error using \(n=3\) and \(x=0.45\). $$ |R_3(0.45)| = \frac{1 \cdot |0.45|^4}{(3+1)!} = \frac{|0.45|^4}{4!} \approx 0.000222 $$ By analyzing the question and solving it step-by-step, we found that the absolute error in approximating \(\cos(0.45)\) using a \(3^{rd}\) order Taylor polynomial centered at \(0\) is approximately \(0.000222\).

Key Concepts

Absolute Error

When approximating functions using polynomials, it's important to evaluate how accurate that approximation truly is. **Absolute error** is a metric used to estimate this accuracy. It measures the difference between the actual value and the approximated value given by a mathematical method like a Taylor polynomial.
The formula for calculating absolute error given the Taylor polynomial's remainder term, also known as Lagrange's remainder, is:

• \( R_n(x) = \frac{M |x|^{n+1}}{(n+1)!} \)

Here, \(M\) is the maximum absolute value of the derivative \(f^{(n+1)}(x)\) over the interval. It's crucial because it gives us the biggest possible error.
This approach gives us a good estimate for how much our approximation might differ from reality. Estimating the absolute error helps students understand the limits of mathematical models and the precision needed for various calculations.

Taylor Series

A Taylor series is a powerful mathematical tool that expresses a function as an infinite sum of terms. Each of these terms is calculated using the function's derivatives at a particular point, typically around 0 or another convenient point.
The Taylor polynomial is a finite version of this series, offering an approximation when calculating or computing functions, like the cosine function, where exact values are challenging to derive by hand.
To form the Taylor polynomial for the cosine function around 0, for instance, we use the Taylor series formula:

• \[ T_n(x) = \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} \frac{(-1)^k x^{2k}}{(2k)!} \]

This pattern comes from the derivatives of the cosine function that repeat predictably, making it straightforward to calculate terms to a specific order, or \(n\).
Understanding Taylor series and how to construct Taylor polynomials is instrumental in various science and engineering fields, enabling approximations where direct calculations would be complex or impossible.

Cosine Function

The **cosine function** is a fundamental concept in trigonometry, associated with angles and periodic phenomena. It's denoted as \( \cos(x) \) and forms the basis for cyclical patterns often seen in nature and engineering.
When approximating the cosine function, such as \( \cos(0.45) \), Taylor polynomials come into play. These approximations leverage the cyclical nature of the cosine function's derivatives, which follow this sequence:

• 1st Derivative: \(-\sin x\)
• 2nd Derivative: \(-\cos x\)
• 3rd Derivative: \(\sin x\)
• 4th Derivative: \(\cos x\)

The pattern resets every four derivatives, which aids in forming Taylor polynomials efficiently.
Knowing the behavior and properties of the cosine function aids in understanding its representation through Taylor series and evaluating the approximation accuracy using polynomials.