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Chapter 8: Problem 1

Find the Taylor polynomials of degree \(n\) approximating \(\cos (3 x)\) for \(x\) near 0 : For \(n=2, P_{2}(x)=\) For \(n=4, P_{4}(x)=\) For \(n=6, P_{6}(x)=\)

Short Answer

Expert verified
For \(n=2\): \(P_2(x) = 1 - \frac{9}{2}x^2\). For \(n=4\): \(P_4(x) = 1 - \frac{9}{2} x^2 + \frac{81}{24} x^4\). For \(n=6\): \(P_6(x) = 1 - \frac{9}{2} x^2 + \frac{81}{24} x^4 - \frac{729}{720} x^6.\)

Step by step solution

01

Identify the function and its derivatives

The function is \(\text{cos}(3x)\). To find the Taylor polynomials, compute the derivatives at \(x = 0\).
02

Compute the first few derivatives

Calculate the derivatives of \(\text{cos}(3x)\): \(\text{First derivative: } f'(x) = -3\text{sin}(3x)\), \(\text{Second derivative: } f''(x) = -9\text{cos}(3x)\). Continue this process up to the 6th derivative.
03

Evaluate derivatives at \(x = 0\)

Evaluate each derivative at \(x = 0\). For example, \(f(0) = \text{cos}(0) = 1\), \(f'(0) = 0\), \(f''(0) = -9\). Continue for higher-order derivatives as needed.
04

Formulate the Taylor polynomial for \(n = 2\)

Using the formula \(P_n(x) = \frac{f(0)}{0!} + \frac{f'(0)}{1!}x + \frac{f''(0)}{2!}x^2 \), the Taylor polynomial of degree 2 is \(P_2(x) = 1 - \frac{9}{2}x^2\).
05

Formulate the Taylor polynomial for \(n = 4\)

The Taylor polynomial of degree 4 includes up to the fourth derivative: \(P_4(x) = 1 - \frac{9}{2} x^2 + \frac{81}{24} x^4 \).
06

Formulate the Taylor polynomial for \(n = 6\)

The Taylor polynomial of degree 6 involves up to the sixth derivative: \(P_6(x) = 1 - \frac{9}{2} x^2 + \frac{81}{24} x^4 - \frac{729}{720} x^6. \)

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Key Concepts

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

Derivatives
Derivatives measure how a function changes as its input changes. For a function like \(\text{cos}(3x)\), we need to find several derivatives to build the Taylor polynomial. The first derivative is the rate of change of the original function. Here, \(f'(x) = -3\text{sin}(3x)\). For the second derivative, we take the derivative of the first derivative. Thus, \(f''(x) = -9\text{cos}(3x)\). This continues for higher derivatives.
Remember, evaluating these derivatives at specific points like \(x = 0\) helps in forming the Taylor series.
Taylor Series
A Taylor series approximates a function as an infinite sum of terms calculated from the values of its derivatives at a single point. For a function \(f(x)\) centered at \(x=0\), the Taylor series is:
\[ P(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + ... \]
In this exercise with \(\text{cos}(3x)\), computing terms up to the 6th derivative helps build the Taylor polynomials of degrees 2, 4, and 6, which are easier to calculate.
Function Approximation
Function approximation involves estimating a function using simpler functions. Taylor polynomials are one way to approximate functions, especially for values near a designated point. Here, \(x=0\). For example, the 2nd-degree polynomial for \( \text{cos}(3x) \):
\[ P_2(x) = 1 - \frac{9}{2} x^2 \]
This approximation gets better as more terms are included, like in the 4th and 6th-degree polynomials:
  • \( P_4(x) = 1 - \frac{9}{2}x^2 + \frac{81}{24}x^4 \)
  • \( P_6(x) = 1 - \frac{9}{2}x^2 + \frac{81}{24}x^4 - \frac{729}{720}x^6 \)
Trigonometric Functions
Trigonometric functions like \(\text{cos}(x)\) and \( \text{sin}(x) \) are fundamental in mathematics. They appear in various fields, including physics and engineering. Here, the function \( \text{cos}(3x) \) is used. When we compute its Taylor polynomial, we use its derivatives, all of which involve \( \text{cos}(3x) \) and \( \text{sin}(3x) \).
The cosine function is especially useful because it is periodic and symmetric, simplifying calculations.

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Most popular questions from this chapter

We have shown that if \(\sum(-1)^{k+1} a_{k}\) is a convergent alternating series, then the sum \(S\) of the series lies between any two consecutive partial sums \(S_{n}\). This suggests that the average \(\frac{S_{n}+S_{n+1}}{2}\) is a better approximation to \(S\) than is \(S_{n}\). a. Show that \(\frac{S_{n}+S_{n+1}}{2}=S_{n}+\frac{1}{2}(-1)^{n+2} a_{n+1}\). b. Use this revised approximation in (a) with \(n=20\) to approximate \(\ln (2)\) given that $$ \ln (2)=\sum_{k=1}^{\infty}(-1)^{k+1} \frac{1}{k} . $$ Compare this to the approximation using just \(S_{20} .\) For your convenience, \(S_{20}=\frac{155685007}{232792560}\).

For the following alternating series, \(\sum_{n=1}^{\infty} a_{n}=0.45-\frac{(0.45)^{3}}{3 !}+\frac{(0.45)^{5}}{5 !}-\frac{(0.45)^{7}}{7 !}+\ldots\) how many terms do you have to compute in order for your approximation (your partial sum) to be within 0.0000001 from the convergent value of that series?

Given: \(A_{n}=\frac{80}{8^{n}}\) Determine: (a) whether \(\sum_{n=1}^{\infty}\left(A_{n}\right)\) is convergent. (b) whether \(\left\\{A_{n}\right\\}\) is convergent. If convergent, enter the limit of convergence. If not, enter DIV.

Compute the value of the following improper integral. If it converges, enter its value. Enter infinity if it diverges to \(\infty\), and -infinity if it diverges to \(-\infty\). Otherwise, enter diverges. $$ \int_{1}^{\infty} \frac{3 d x}{x^{2}+1}= $$ Does the series \(\sum_{n=1}^{\infty} \frac{3}{n^{2}+1}\) converge or diverge? [Choose: converges | diverges to +infinity | diverges to -infinity | diverges]

Based on the examples we have seen, we might expect that the Taylor series for a function \(f\) always converges to the values \(f(x)\) on its interval of convergence. We explore that idea in more detail in this exercise. Let \(f(x)=\left\\{\begin{array}{ll}e^{-1 / x^{2}} & \text { if } x \neq 0, \\ 0 & \text { if } x=0 .\end{array}\right.\) a. Show, using the definition of the derivative, that \(f^{\prime}(0)=0\). b. It can be shown that \(f^{(n)}(0)=0\) for all \(n \geq 2\). Assuming that this is true, find the Taylor series for \(f\) centered at 0 . c. What is the interval of convergence of the Taylor series centered at 0 for \(f ?\) Explain. For which values of \(x\) the interval of convergence of the Taylor series does the Taylor series converge to \(f(x) ?\)
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