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Chapter 11: Problem 44

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

Short Answer

Expert verified
Answer: The general expression for the remainder term, \(R_n(x)\), is: $$R_{n}(x) = \begin{cases} \frac{\cos \xi (x-\frac{\pi}{2})^{n+1}}{(n+1)!}, & \text{if}\ (n+1) \equiv 0 \pmod{4}, \\ \frac{-\sin \xi (x-\frac{\pi}{2})^{n+1}}{(n+1)!}, & \text{if}\ (n+1) \equiv 1 \pmod{4}, \\ \frac{-\cos \xi (x-\frac{\pi}{2})^{n+1}}{(n+1)!}, & \text{if}\ (n+1) \equiv 2 \pmod{4}, \\ \frac{\sin \xi (x-\frac{\pi}{2})^{n+1}}{(n+1)!}, & \text{if}\ (n+1) \equiv 3 \pmod{4}. \end{cases}$$

Step by step solution

01

Taylor's Theorem

According to Taylor's theorem, the nth-degree Taylor polynomial of a function \(f(x)\) centered at \(a\) can be written as: $$P_n(x)=f(a)+f'(a)(x-a)+\frac{f''(a)(x-a)^2}{2!}+\cdots+\frac{f^{(n)}(a)(x-a)^n}{n!}.$$ With the remainder term denoted as \(R_n(x)\), $$f(x)=P_n(x)+R_n(x).$$ Moreover, the remainder term can be approximately written as: $$R_{n}(x) = \frac{f^{(n+1)}(\xi)(x-a)^{n+1}}{(n+1)!},$$ for some \(\xi\) between \(a\) and \(x\).
02

Calculate Derivatives up to the (n+1)th Order for the \(f(x)=\cos x\) Function

We don't have a specific value for \(n\), so we need to find a general expression for the \((n+1)\)th derivative of \(f(x)\), given \(f(x) = \cos x\). We'll follow the pattern by calculating the first few derivatives: 1. First derivative: \(f'(x) = -\sin x\) 2. Second derivative: \(f''(x) = -\cos x\) 3. Third derivative: \(f^{(3)}(x) = \sin x\) 4. Fourth derivative: \(f^{(4)}(x) = \cos x\) From the pattern above, we can see that the first and third derivatives have \(\sin x\) function, while the second and fourth have \(\cos x\). We can generalize the derivatives as follows: $$f^{(n+1)}(x) = \begin{cases} \,\, \cos x, & \text{if}\ (n+1) \equiv 0 \pmod{4}, \\ -\sin x, & \text{if}\ (n+1) \equiv 1 \pmod{4}, \\ -\cos x, & \text{if}\ (n+1) \equiv 2 \pmod{4}, \\ \,\, \sin x, & \text{if}\ (n+1) \equiv 3 \pmod{4}. \end{cases}$$
03

Determine the Remainder Term \(R_{n}(x)\) by Substituting the (n+1)th Derivative and Given Center \(a=\frac{\pi}{2}\)

We'll now substitute \(f^{(n+1)}(\xi)\) and \(a = \frac{\pi}{2}\) into the remainder term formula: $$R_{n}(x) = \frac{f^{(n+1)}(\xi)(x-\frac{\pi}{2})^{n+1}}{(n+1)!}$$ Depending on the congruence class of \((n+1)\) modulo 4, the remainder term \(R_{n}(x)\) can be expressed as: $$R_{n}(x) = \begin{cases} \frac{\cos \xi (x-\frac{\pi}{2})^{n+1}}{(n+1)!}, & \text{if}\ (n+1) \equiv 0 \pmod{4}, \\ \frac{-\sin \xi (x-\frac{\pi}{2})^{n+1}}{(n+1)!}, & \text{if}\ (n+1) \equiv 1 \pmod{4}, \\ \frac{-\cos \xi (x-\frac{\pi}{2})^{n+1}}{(n+1)!}, & \text{if}\ (n+1) \equiv 2 \pmod{4}, \\ \frac{\sin \xi (x-\frac{\pi}{2})^{n+1}}{(n+1)!}, & \text{if}\ (n+1) \equiv 3 \pmod{4}. \end{cases}$$ This is the general expression for the remainder term \(R_{n}(x)\) for the nth-order Taylor polynomial of \(f(x) = \cos x\), centered at \(a = \frac{\pi}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taylor's Theorem
Taylor's theorem is a powerful tool in calculus that deals with the approximation of functions using polynomials. It states that any function that is infinitely differentiable at a point can be expressed as a Taylor series around that point. The theorem gives us an nth-degree Taylor polynomial plus a remainder term which represents the error in the polynomial's approximation to the function. The remainder, often denoted as \( R_n(x) \), is crucial as it helps us understand how good our approximation is.

When we talk about the nth-degree Taylor polynomial, we're referring to an approximation that includes terms up to the \(n\)th power. The polynomial itself is made up of the function's value and its derivatives at the center point \(a\). Taylor's theorem is fundamental when it comes to analyzing the behavior of functions near a point and is widely used in both theoretical and practical applications in science and engineering.
nth-Degree Taylor Polynomial
The nth-degree Taylor polynomial is a sum of terms that provides an approximation of a function around a specific point, termed the center. Each term in this polynomial incorporates the function's derivatives at this center and is of the form \( \frac{f^{(k)}(a)(x-a)^k}{k!} \) for the \(k\)th term.

The idea behind the nth-degree Taylor polynomial is to create a polynomial, which is easier to work with than the original function, that closely mimics the function's behavior near the center point. For many functions, including the cosine function as seen in the exercise, a higher degree polynomial means a better approximation. This approach allows for complex functions to be simplistically represented, facilitating operations like integration and differentiation.
Derivative Calculation
Derivative calculation is a fundamental process in calculus that involves finding the rate at which a function changes at any given point. It involves applying the rules of differentiation to the function in question. For periodic functions like sine and cosine, the derivatives follow a cyclical pattern, as seen in the pattern of derivatives for the \(f(x) = \text{cos} x\).

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

These patterns are used to predict higher-order derivatives, which are essential for forming the Taylor polynomial and calculating the remainder term.
Cosine Function
The cosine function, denoted as \(\text{cos} x\), is one of the fundamental trigonometric functions. This periodic function is closely related to the circle and it's important in the study of oscillatory motions, waves, and many other areas of mathematics and physics. Since cosine is periodic, its Taylor polynomial and remainder term exhibit characteristic patterns based on the angle's periodicity.

The cosine function is even, meaning that \(\text{cos}(-x) = \text{cos}(x)\), and it has a period of \(2\pi\), which means it repeats every \(2\pi\) radians. In the context of Taylor series approximations, the cosine function's properties provide us with a clear pattern in derivatives and facilitate the estimation of the remainder in polynomial approximations.

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Most popular questions from this chapter

a.Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients. b.Determine the radius of convergence of the series. $$f(x)=\left\\{\begin{array}{ll}\frac{\sin x}{x} & \text { if } x \neq 0 \\\1 & \text { if } x=0\end{array}\right.$$

Probability: tossing for a head The expected (average) number of tosses of a fair coin required to obtain the first head is \(\sum_{i=1}^{\infty} k\left(\frac{1}{2}\right)^{t} .\) Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)

Use the Maclaurin series $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1

Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is $$ J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{2^{2 k}(k !)^{2}} x^{2 k} $$ a. Write out the first four terms of \(J_{0}\) b. Find the radius and interval of convergence of the power series for \(J_{0}\) c. Differentiate \(J_{0}\) twice and show (by keeping terms through \(x^{6}\) ) that \(J_{0}\) satisfies the equation \(x^{2} y^{\prime \prime}(x)+x y^{\prime}(x)+x^{2} y(x)=0\)

Compute the coefficients for the Taylor series for the following functions about the given point \(a\), and then use the first four terms of the series to approximate the given number. $$f(x)=\sqrt{x} \text { with } a=36 ; \text { approximate } \sqrt{39}$$.
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