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Chapter 10: Problem 40

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. $$\cos (-0.2)$$

Short Answer

Expert verified
Answer: The approximation of cos(-0.2) using a Taylor polynomial with n=3 is 0.98. The absolute error of this approximation is 0.0001.

Step by step solution

01

Taylor Polynomial of Cosine Function

The Taylor polynomial of a function is given by: $$P_{n}(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)(x-a)^k}{k!}$$ For the cosine function, we have \(f(x) = \cos(x)\), and its derivatives are: \(f^{(1)}(x) = -\sin(x)\) \(f^{(2)}(x) = -\cos(x)\) \(f^{(3)}(x) = \sin(x)\) Since we are looking at the Taylor polynomial around \(x=0\), we can set \(a=0\). Therefore, we have: \(P_{3}(x) = \sum_{k=0}^{3} \frac{f^{(k)}(0)(x-a)^k}{k!}\) \(x=0\) gives us \(f(0) = \cos(0) = 1\), \(f^{(1)}(0) = -\sin(0) = 0\), \(f^{(2)}(0) = -\cos(0) = -1\), and \(f^{(3)}(0) = \sin(0) = 0\). Hence, we get the Taylor polynomial of degree 3 as: \(P_{3}(x) = 1 - \frac{1}{2}x^2\)
02

Approximate \(\cos(-0.2)\) Using \(P_{3}(x)\)

Now, let's plug in \(x = -0.2\) into the Taylor polynomial we found: $$P_{3}(-0.2) = 1 - \frac{1}{2}(-0.2)^2 = 1 - 0.1\cdot0.2^2$$ $$P_{3}(-0.2) = 1 - 0.02 = 0.98$$ So our approximation of \(\cos(-0.2)\) using the Taylor polynomial with \(n=3\) is \(0.98\).
03

Compute the Absolute Error

To compute the absolute error, we need to subtract the approximation from the exact value obtained using a calculator. A calculator gives us the exact value for \(\cos(-0.2) \approx 0.9801\). The absolute error is given by: $$|Exact\;Value - Approximation| = |0.9801 - 0.98| = 0.0001$$ The absolute error in the approximation is \(0.0001\).

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Most popular questions from this chapter

Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$\frac{1}{(3+4 x)^{2}}$$

Explain why the Mean Value Theorem is a special case of Taylor's Theorem.

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than \(10^{-3}\) ? (The answer depends on your choice of a center.) $$\sin 0.2$$

Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty}(-1)^{k} \frac{k x^{k+1}}{3^{k}}$$

Exponential function In Section 3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty < x < \infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=e^{2 x}$$
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