Problem 15 - Essential Calculus Early Transcendentals

Question

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number \(a\) . (b) Use Taylor's Formula to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=x \sin x, \quad a=0, \quad n=4, \quad-1 \leqslant x \leqslant 1$$

Step-by-Step Solution

Answer

The Taylor polynomial approximation is \( T_4(x) = x^2 - \frac{1}{6}x^4 \). The error is small within \(-1 \leq x \leq 1\).

Step 1: Understand the Function and Parameters

Identify the function given: \( f(x) = x \sin x \). You are asked to approximate this function using a Taylor polynomial of degree \( n = 4 \) around the point \( a = 0 \) and to assess the accuracy within the interval \( -1 \leq x \leq 1 \).

Step 2: Find Derivatives of the Function

Compute the derivatives of \( f(x) = x \sin x \) up to the 4th derivative at \( x = 0 \):\[f(x) = x \sin x,\quad f(0) = 0 \]First derivative: \( f'(x) = \sin x + x \cos x \),\quad \( f'(0) = 0 \)Second derivative: \( f''(x) = 2\cos x - x \sin x \),\quad \( f''(0) = 2 \)Third derivative: \( f'''(x) = -3\sin x - x\cos x \),\quad \( f'''(0) = 0 \)Fourth derivative: \( f^{(4)}(x) = -4\cos x + x \sin x \),\quad \( f^{(4)}(0) = -4 \)

Step 3: Write the Taylor Polynomial

Write the Taylor polynomial of degree 4 at \( x = 0 \) using the formula:\[T_4(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f^{(4)}(0)}{4!}x^4\]Substitute the derivatives:\[T_4(x) = 0 + 0 \cdot x + \frac{2}{2}x^2 + 0 \cdot x^3 + \frac{-4}{24}x^4 = x^2 - \frac{1}{6}x^4\]

Step 4: Estimate the Accuracy of the Approximation

Use Taylor's formula to determine the remainder \( R_4(x) \) for the approximation, which gives the error:\[R_4(x) = \frac{f^{(5)}(c)}{5!}(x^5)\]Since \( f^{(5)}(x) \) is the fifth derivative, use an upper bound for \( |x| \leq 1 \). You can assume next derivatives will not increase drastically:\[|R_4(x)| \leq \max |f^{(5)}(c)| \cdot \frac{|x|^5}{120} \]

Step 5: Graph the Error Function

Compute or plot \( |R_4(x)| \) over the interval \( -1 \leqslant x \leqslant 1 \) using computational tools to visualize and verify the range of error. This can be accomplished using graphing software, plotting libraries like Matplotlib in Python, or graphing calculators.

Key Concepts

Taylor's Formula

Taylor's Formula is a powerful mathematical tool used to approximate functions with polynomials. It works by expressing a function as the sum of its derivatives at a particular point (known as the center). The formula represents the function as a series of terms calculated from these derivatives. This method can simplify complex functions by approximating them with simpler polynomial forms over a specific interval.

The formula for a Taylor polynomial of degree \(n\) centered at \(a\) is given by:

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

Here, \(f(a)\) is the function value at \(a\), \(f'(a)\) is the first derivative, and so on. Each term adds more precision to the approximation, especially near the center \(a\) where the function and polynomial match exactly up to the nth derivative.

Polynomial Approximation

Polynomial approximation is the process of estimating a function using a polynomial. This is particularly useful when you need an easy-to-calculate form of a function. The Taylor Polynomial is a type of polynomial used in these approximations. It captures the shape and behavior of the original function locally around a given point, called the center. In our exercise, we approximate the function \(f(x) = x \sin x\) with a Taylor polynomial of degree 4.

The resulting polynomial is:
- \(T_4(x) = x^2 - \frac{1}{6}x^4\)

This approximation works well close to the center \(a = 0\), and within the desired interval of \(-1 \leq x \leq 1\). The highest degree polynomial term, \(-\frac{1}{6}x^4\), mainly controls the precision of the approximation within this interval. By choosing the degree and center wisely, polynomials can represent even complex functions effectively within certain limits.

Derivatives

Derivatives help in determining the behavior and characteristics of a function. They are a fundamental concept in calculus, giving us information about the rate of change of a function. In Taylor's series, derivatives are used to calculate the terms of the polynomial.

For \(f(x) = x \sin x\), we need derivatives up to the fourth degree for our approximation:

• First derivative: \(f'(x) = \sin x + x \cos x\), \(f'(0) = 0\)
• Second derivative: \(f''(x) = 2\cos x - x \sin x\), \(f''(0) = 2\)
• Third derivative: \(f'''(x) = -3\sin x - x \cos x\), \(f'''(0) = 0\)
• Fourth derivative: \(f^{(4)}(x) = -4\cos x + x \sin x\), \(f^{(4)}(0) = -4\)

These derivatives are evaluated at \(x = 0\), which is the center of our Taylor series. Only terms up to the fourth derivative are included in the polynomial approximation, which fully incorporates the curvature and immediate changes of the function around the center.

Error Estimation

Error estimation is an essential step in approximations because it informs us about the difference between the true function and its polynomial approximation. In Taylor's approximation, this error is represented by the remainder term, often denoted as \(R_n(x)\). It estimates the accuracy of the approximation and helps in understanding the limits of the approximation.

For a Taylor Polynomial of degree 4, the remainder \(R_4(x)\) can be estimated as:
- \(R_4(x) = \frac{f^{(5)}(c)}{5!}(x-a)^5\)

We use an inequality to bound the error over the interval \(-1 \leq x \leq 1\). Since the next derivatives are unlikely to increase drastically, the terms \(|R_4(x)|\) can be bounded using assumptions about \(|f^{(5)}(c)|\). This provides a sense of how closely the Taylor polynomial matches the actual function within the specified range, and why graphing \(|R_4(x)|\) helps visualize the accuracy of the approximation.