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True or False If \(f\) is a function with both first and second derivatives defined on an interval containing \(x_{0}\), then \(f\) can be approximated by the Taylor Polynomial \(P_{2}( x) =f( x_{0}) +f^\prime ( x_{0}) ( x-x_{0}) +f^{\prime \prime} ( x_{0}) ( x-x_{0}) ^{2}.\)

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True or False A Taylor Polynomial approximation \(P_{n}\) for a function \(f\) at \(x_{0}\) has the following properties: \(P_{n}( x_{0}) =f( x_{0}) \) and \(P_{n}^{k}( x_{0}) =f^{k}( x_{0}) \) for derivatives of all orders from \(k=1\) to \(k=n\).

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\(f(x)=3x^{3}+2x^{2}-6x+5\) at \(x_{0}=1\)

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\(f(x)=4x^{3}-2x^{2}-4\) at \(x_{0}=1\)

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\(f(x)=2x^{4}-6x^{3}+x\) at \(x_{0}=-1\)

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\(f(x)=-3x^{4}+2x^{2}-5\) at \(x_{0}=-1\)

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\(f(x)=x^{5}\) at \(x_{0}=2\)

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\(f(x)=x^{6}\) at \(x_{0}=3\)

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\(f(x)=\ln x\) at \(x_{0}=1\)

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\(f(x)=\ln (1+x)\) at \(x_{0}=0\)

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\(f(x)=\dfrac{1}{x}\) at \(x_{0}=1\)

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\(f(x)=\dfrac{1}{x^{2}}\) at \(x_{0}=1\)

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\(f(x)=\cos x\) at \(x_{0}=0\)

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\(f(x)=\sin x\) at \(x_{0}=\dfrac{\pi }{4}\)

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\(f( x) =e^{2x}\) at \(x_{0}=0\)

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\(f(x)=e^{-x}\) at \(x_{0}=0\)

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\(f(x)=\dfrac{1}{1-x}\) at \(x_{0}=0\)

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\(f(x)=\dfrac{1}{1+x}\) at \(x_{0}=0\)

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\(f(x)=\dfrac{1}{(1+x)^{2}}\) at \(x_{0}=0\)

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\(f(x)=\dfrac{1}{1+x^{2}}\) at \(x_{0}=0\)

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\(f(x)=x\ln x\) at \(x_{0}=1\)

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\(f(x)=xe^{x}\) at \(x_{0}=1\)

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\(f(x)=\sqrt{3+x^{2}}\) at \(x_{0}=1\)

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\(f(x)=\sqrt{1+x}\) at \(x_{0}=0\)

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\(f(x)=\tan x\) at \(x_{0}=\dfrac{\pi }{4}\)

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\(f(x)=\sec x\) at \(x_{0}=0\)

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\(f(x)=\tan ^{-1}x\) at \(x_{0}=0\)

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\(f(x)=\sin ^{-1}x\) at \(x_{0}=0\)

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\(f(x)=3x^{2}-6x+4\)

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\(f(x)=4x^{2}-x+1\)

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\(f(x)=x^{3}+x^{2}-8\)

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\(f(x)=x^{4}+1\)

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\(f( x) =\sin ^{-1}x\) at \(x_{0}=\dfrac{1}{2}\), \(P_{4}(x)\)

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\(f(x)=\tan ^{-1}x\) at \(x_{0}=1\), \(P_{5}(x)\)

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\(f( x) =\dfrac{x}{\sqrt{x^{2}+3}}\) at \(x_{0}=1\), \(P_{5}(x)\)

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\(f( x) =x\sqrt[3]{x^{2}+5}\) at \(x_{0}=2\), \(P_{5}(x)\)

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Uninhibited Decay \(f( x) =0.34e^{[(-\ln 2)/5600] x}\) at \(x=0\), \(P_{4}(x)\)

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Uninhibited Growth \(f( x) =5000e^{0.04x}\) at \(x=0\), \(P_{4}(x)\)

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Logistic Population Growth Model \(f( x) =\dfrac{100}{1+30.2e^{-0.2x}}\) at \(x_{0}=0\), \(P_{3}(x)\)

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Gompertz Population Growth Model \(f(x)=100e^{-3e^{-0.2x}}\) at \(x_{0}=0\), \(P_{3}(x)\)

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The Taylor Polynomial \(P_{7}( x)\) for \(f( x) =\sin x\) at \(0\) has only terms with odd powers. See Example 2.

1. Discuss why this is true. Does this property hold for the Taylor Polynomial \(P_{n}( x)\) for any number \(n\)?
2. Investigate the Taylor Polynomial \(P_{7}( x) \) for \(f\) at \(\frac{\pi}{2}\). Does this Taylor Polynomial have only odd terms? Explain why or why not.

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The Taylor Polynomial \(P_{6}( x) \) for \(f( x) =\cos x\) at \(0\) has only terms with even powers. See Problem 13.

1. Discuss why this is true. Does this property hold for the Taylor Polynomial \(P_{n}( x) \) for any \(n?\)
2. Investigate the Taylor Polynomial \(P_{6}( x) \) for \(f\) at \(\frac{\pi}{2}\). Does this Taylor Polynomial have only even terms? Explain why or why not.

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The graphs of \(y=\sin x\) and \(y=\lambda x\) intersect near \(x=\pi \) if \(\lambda \) is small. Let \(f(x)=\sin x-\lambda x\). Find the Taylor Polynomial \(P_{2}( x)\) for \(f\) at \(\pi\), and use it to show that an approximate solution of the equation \(\sin x=\lambda x\) is \(x=\dfrac{\pi }{1+\lambda}\).

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The graphs of \(y=\cot x\) and \(y=\lambda x\) intersect near \(x=\frac{\pi}{2} \) if \(\lambda \) is small. Let \(f(x)=\cot x-\lambda x\). Find the Taylor Polynomial \(P_{2}( x)\) for \(f\) at \(\dfrac{\pi }{2},\) and use it to find an approximate solution of the equation \(\cot x=\lambda x\).