Problem 16 - Calculus: Early Transcendentals

Question

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = \sin x, \) \( a = \pi/6, \) \( n = 4, \) \( 0 \le x \le \pi/3 \)

Step-by-Step Solution

Answer

The approximation of \( f(x) = \sin x \) by a 4th-degree Taylor polynomial at \( a = \pi/6 \) is accurate within 0.0002 for \( x \) in \([0, \pi/3]\).

Step 1: Write the Taylor Series Formula

The Taylor series of a function \( f(x) \) about \( a \) is given by:\[T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n\]

Step 2: Identify necessary derivatives of sine function

For \( f(x) = \sin(x) \), we have:- \( f(a) = \sin(\pi/6) = 1/2 \)- \( f'(x) = \cos(x) \), \( f'(\pi/6) = \sqrt{3}/2 \)- \( f''(x) = -\sin(x) \), \( f''(\pi/6) = -1/2 \)- \( f'''(x) = -\cos(x) \), \( f'''(\pi/6) = -\sqrt{3}/2 \)- \( f^{(4)}(x) = \sin(x) \), \( f^{(4)}(\pi/6) = 1/2 \)

Step 3: Construct the Taylor Polynomial

Using the Taylor series formula, we construct the 4th degree Taylor polynomial:\[T_4(x) = \frac{1}{2} + \frac{\sqrt{3}}{2}(x - \frac{\pi}{6}) - \frac{1}{2(2!)}(x - \frac{\pi}{6})^2 - \frac{\sqrt{3}}{2(3!)}(x - \frac{\pi}{6})^3 + \frac{1}{2(4!)}(x - \frac{\pi}{6})^4\]

Step 4: Apply Taylor's Inequality

Taylor's Inequality estimates \( |R_n(x)| \), the remainder, as:\[|R_n(x)| \le \frac{M}{(n+1)!} |x-a|^{n+1}\]where \( M \) is an upper bound for \(|f^{n+1}(c)|\) on \([a, x]\). Since \( f^{(5)}(x) = \cos(x) \) and \( |\cos(x)| \le 1 \) on \([0, \pi/3]\), \( M = 1 \). Thus,\[|R_4(x)| \le \frac{1}{5!} |x-\frac{\pi}{6}|^5\]

Step 5: Compute the Error Bound

For \( 0 \leq x \leq \frac{\pi}{3} \), the maximum value of \( |x-\frac{\pi}{6}| \) is \( \frac{\pi}{6} \). Therefore,\[|R_4(x)| \le \frac{1}{120} \left( \frac{\pi}{6} \right)^5 \approx 0.0002\]

Step 6: Graph the Remainder

To verify, plot \( |R_4(x)| = |f(x) - T_4(x)| \) over \( 0 \le x \le \frac{\pi}{3} \) using a graphing tool. The plot should visually confirm that \( |R_4(x)| \leq 0.0002 \) for this interval.

Key Concepts

Taylor Series

A Taylor series is a way of representing a function as an infinite sum of terms, calculated from the values of its derivatives at a single point. This method allows us to approximate different types of functions using simple polynomial expressions, which are easier to work with. When we talk about the Taylor polynomial of degree \( n \), we refer to the partial sum of the first \( n+1 \) terms of the Taylor series:

• It starts with the function value at a point \( a \), denoted as \( f(a) \).
• The first derivative evaluated at \( a \) is multiplied by \( (x-a) \).
• Subsequent terms use higher derivatives, with each term further divided by factorials to adjust the coefficients.

These polynomials provide approximations around the center point \( a \). As \( n \) increases, the Taylor polynomial becomes a more accurate representation of the original function within its radius of convergence.

Taylor's Inequality

To understand the error or difference between our Taylor polynomial \( T_n(x) \) and the actual function, Taylor's Inequality becomes essential. This inequality gives an estimate of the remainder or error term \(|R_n(x)|\). This error is the difference between the real function value and the polynomial approximation:

• \(|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}\)
• Here, \( M \) is determined as an upper bound for the absolute value of the next derivative of the function on a given interval \([a, x]\).

The inequality helps us ascertain how precise our polynomial approximation is over a specific range by placing a cap on the potential error size. Knowing \( M \) ensures developers can predict the accuracy of the approximation for practical uses.

Sine Function

The sine function, written as \( \sin(x) \), is one of the fundamental trigonometric functions. It is periodic and oscillates between -1 and 1. In the context of Taylor series, we can represent \( \sin(x) \) using its derivatives:

• The function value at any point \( a \), \( \sin(a) \).
• The first derivative \( f'(x) = \cos(x) \, \) evaluates to \( \cos(a) \).
• The second derivative follows as \( f''(x) = -\sin(x) \).
• Subsequent derivatives alternate between sine and cosine with changing signs.

For smaller degrees, these derived values allow constructing polynomial approximations of sine, greatly valuable for calculations where high precision of \( \sin(x) \) is not critical.

Error Estimation

When approximating functions with Taylor polynomials, understanding and estimating the error is crucial. Error estimation answers the question of how close our approximation is to the actual function.

• The error of a Taylor polynomial approximation is given by the remainder term \( |R_n(x)| \).
• We apply Taylor's Inequality to estimate its potential magnitude across an interval.
• Calculating this involves finding a reasonable upper bound \( M \) related to the derivatives of the target function.

The smaller the error bound, the closer the approximation is expected to be. For precise applications, ensuring the resultant error stays within acceptable limits is critical, often achieved by choosing a sufficiently high degree for the Taylor polynomial. This estimation is often visually verified using graphing methods to compare the Taylor polynomial's curve to that of the actual function across the interval.