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Chapter 10: Problem 48

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$\sin 1$$

Short Answer

Expert verified
Answer: The first four nonzero terms of the Taylor series for sin(1) are: 1. \(x = 1\) 2. \(-\frac{1}{6}x^3 = -\frac{1}{6}\) 3. \(\frac{1}{120}x^5 = \frac{1}{120}\) 4. \(-\frac{1}{5040}x^7 = -\frac{1}{5040}\) The series can be approximated as: $$\sin(1) \approx 1 - \frac{1}{6} + \frac{1}{120} - \frac{1}{5040}$$

Step by step solution

01

Identify the function and its derivatives

We know that for the sine function, we have the following derivatives: 1. \(f(x) = \sin(x)\) 2. \(f'(x) = \cos(x)\) 3. \(f''(x) = -\sin(x)\) 4. \(f'''(x) = -\cos(x)\) 5. \(f^{(4)}(x) = \sin(x)\) 6. \(f^{(5)}(x) = \cos(x)\) The pattern repeats every four derivatives.
02

Calculate the first four nonzero terms

For the sine function's Taylor series, we need to calculate the first four nonzero terms using the general formula: $$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ To find the first four nonzero terms, we need to plug in the values for \(n = 0, 1, 2, 3\): 1. For \(n=0\): \(\frac{(-1)^0}{(2\cdot0+1)!}x^{2\cdot0+1} = \frac{1}{1!}x^1 = x\) 2. For \(n=1\): \(\frac{(-1)^1}{(2\cdot1+1)!}x^{2\cdot1+1} = -\frac{1}{3!}x^3 = -\frac{1}{6}x^3\) 3. For \(n=2\): \(\frac{(-1)^2}{(2\cdot2+1)!}x^{2\cdot2+1} = \frac{1}{5!}x^5 = \frac{1}{120}x^5\) 4. For \(n=3\): \(\frac{(-1)^3}{(2\cdot3+1)!}x^{2\cdot3+1} = -\frac{1}{7!}x^7 = -\frac{1}{5040}x^7\)
03

Substitute \(x=1\) and find the value of the series

Now, we need to substitute \(x=1\) to find the value of the first four nonzero terms: 1. \(x = 1^1 = 1\) 2. \(-\frac{1}{6}x^3 = -\frac{1}{6}(1)^3 = -\frac{1}{6}\) 3. \(\frac{1}{120}x^5 = \frac{1}{120}(1)^5 = \frac{1}{120}\) 4. \(-\frac{1}{5040}x^7 = -\frac{1}{5040}(1)^7 = -\frac{1}{5040}\) So, the first four nonzero terms of the Taylor series for \(\sin(1)\) are: $$\sin(1) \approx 1 - \frac{1}{6} + \frac{1}{120} - \frac{1}{5040}$$

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Most popular questions from this chapter

Use composition of series to find the first three terms of the Maclaurin series for the following functions. a. \(e^{\sin x}\) b. \(e^{\tan x}\) c. \(\sqrt{1+\sin ^{2} x}\)

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