Problem 53 - Calculus Early Transcendentals

Question

Find the remainder term \(R_{n}\) for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of \(n\) $$f(x)=\sin x ; a=\pi / 2$$

Step-by-Step Solution

Answer

Answer: The remainder term \(R_n(x)\) is given by: $$R_n(x) = \begin{cases} \frac{\sin c}{(n+1)!}(x-\frac{\pi}{2})^{(n+1)} &\text{if} \ n \equiv 0 \pmod{4} \\ \frac{\cos c}{(n+1)!}(x-\frac{\pi}{2})^{(n+1)} &\text{if} \ n \equiv 1 \pmod{4} \\ \frac{-\sin c}{(n+1)!}(x-\frac{\pi}{2})^{(n+1)} &\text{if} \ n \equiv 2 \pmod{4} \\ \frac{-\cos c}{(n+1)!}(x-\frac{\pi}{2})^{(n+1)} &\text{if} \ n \equiv 3 \pmod{4} \\ \end{cases}$$ where \(c\) is a value between \(x\) and \(a=\pi/2\) that maximizes the absolute value of the sine or cosine function, depending on the value of \(n\).

Step 1: Find the (n+1)th derivative of f(x)

To find the (n+1)th derivative of \(f(x) = \sin x\), we will start by calculating the first few derivatives and try to identify the pattern. $$f'(x) = \cos x$$ $$f''(x) = -\sin x$$ $$f'''(x) = -\cos x$$ $$f^{(4)}(x) = \sin x$$ Notice that the derivatives repeat in a cycle with a period of 4. Therefore, we can write a general formula for the (n+1)th derivative: $$f^{(n+1)}(x) = \begin{cases} \sin x &\text{if} \ n \equiv 0 \pmod{4} \\ \cos x &\text{if} \ n \equiv 1 \pmod{4} \\ -\sin x &\text{if} \ n \equiv 2 \pmod{4} \\ -\cos x &\text{if} \ n \equiv 3 \pmod{4} \\ \end{cases}$$

Step 2: Evaluate the (n+1)th derivative at c

Now, we need to evaluate \(f^{(n+1)}(c)\) for a value of \(c\) between \(x\) and \(a=\pi/2\). We know that \(f^{(n+1)}(c)\) has a maximum absolute value when $$c = \begin{cases} 0 &\text{if} \ n \equiv 0 \pmod{4} \\ \pi/2 &\text{if} \ n \equiv 1 \pmod{4} \\ \pi &\text{if} \ n \equiv 2 \pmod{4} \\ 3\pi/2 &\text{if} \ n \equiv 3 \pmod{4} \\ \end{cases}$$ This is because the sine and cosine functions have maximum absolute values at these points in the cycle.

Step 3: Find the remainder term Rn(x)

Using Taylor's theorem formula, we have $$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{(n+1)}$$ Now, we plug in the values for \(f^{(n+1)}(c)\), \(a=\pi/2\), and the (n+1)th derivative at c: $$R_n(x) = \begin{cases} \frac{\sin c}{(n+1)!}(x-\frac{\pi}{2})^{(n+1)} &\text{if} \ n \equiv 0 \pmod{4} \\ \frac{\cos c}{(n+1)!}(x-\frac{\pi}{2})^{(n+1)} &\text{if} \ n \equiv 1 \pmod{4} \\ \frac{-\sin c}{(n+1)!}(x-\frac{\pi}{2})^{(n+1)} &\text{if} \ n \equiv 2 \pmod{4} \\ \frac{-\cos c}{(n+1)!}(x-\frac{\pi}{2})^{(n+1)} &\text{if} \ n \equiv 3 \pmod{4} \\ \end{cases}$$ The remainder term \(R_n(x)\) has been expressed for a general value of \(n\) using the given function \(f(x) = \sin x\) and its nth-order Taylor polynomial centered at \(a=\pi/2\).

Key Concepts

Taylor polynomial

Taylor polynomials are powerful tools in mathematics that approximate functions using sums of their derivatives at a specific point. This concept, introduced by Baron Taylor, is utilized extensively in calculus. At its core, a Taylor polynomial of degree \( n \) for a function \( f(x) \) centered at \( a \) is an approximation represented by a polynomial with terms based on the function's derivatives calculated at \( x = a \). The general form of a Taylor polynomial is:\[T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n\]Key aspects of Taylor polynomials include:

• The degree \( n \) signifies how many derivatives of the function are used in the polynomial.
• Higher degree polynomials tend to provide better approximations of the function.
• The choice of the center \( a \) affects the polynomial's approximation quality within a neighborhood around \( a \).

Understanding Taylor polynomials helps in perceiving complex functions as simpler polynomials in various applications, especially when evaluating local behavior around a point.

remainder term

The remainder term \( R_n(x) \) in a Taylor polynomial plays a crucial role in determining how accurate our approximation is. When we truncate the Taylor series to an \( n \)-th degree polynomial, the remainder captures the error or difference between the actual function and its polynomial approximation. Using Taylor’s theorem, the remainder for a function \( f(x) \) centered at \( a \) can be expressed as:\[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{(n+1)}\]In this expression:

• \( c \) is some value in the interval between \( a \) and \( x \).
• \( f^{(n+1)}(c) \) is the \( (n+1) \)-th derivative evaluated at \( c \).
• \((n+1)!\) is the factorial of \( (n+1) \), and it grows rapidly as \( n \) increases, often reducing the impact of the error term for larger \( n \).

The remainder term is essential for understanding how closely a Taylor polynomial approximates a function, especially when dealing with more precise calculations or when higher accuracy is needed.

trigonometric functions

Trigonometric functions, like sine and cosine, often appear in problems involving Taylor series. These periodic functions are smoother and thus fluctuate predictably, making them suitable for approximation techniques such as Taylor polynomials. For instance, the function \( f(x) = \sin x \) is cyclic with repeating derivatives:

• \( f'(x) = \cos x \)
• \( f''(x) = -\sin x \)
• \( f'''(x) = -\cos x \)
• \( f^{(4)}(x) = \sin x \) - and the pattern repeats every four derivatives.

The cyclical nature of these derivatives simplifies calculating a Taylor series, as the derivatives follow a predictable pattern. This periodicity is why trigonometric functions are excellent candidates for approximation via Taylor polynomials. Understanding these repetitive cycles allows mathematicians and students to easily derive accurate polynomial approximations and analyze functions within a specific range.