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The Taylor series is a polynomial of infinite degree used to represent functions like sine, cube roots, and the exponential function. They're how some calculators (and Physicists) make approximations.

Find the error bound for the \(4^\text{th}\) degree polynomial of

\[f(x)=\sin x ~\text{ centered at }~ x=0 ~\text{ on the interval }~ [0,2\pi].\]

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Find the error bound for the \(4^\text{th}\) degree polynomial of

\[f(x)=\cos x ~\text{ centered at }~ x=0 ~\text{ on the interval }~ [0,2\pi].\]

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What is the maximum possible error of the \(10^\text{th}\) degree polynomial of

\[f(x)=e^{x} ~\text{ centered at }~ x=0 ~\text{ on the interval }~ [-2,2]?\]

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What is the maximum possible error of the \(1^\text{st}\) degree polynomial of

\[f(x)=\sqrt{1+x} ~\text{ centered at }~ x=0 ~\text{ on the interval }~ [-0.01,0.01]?\]

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What is the maximum possible error of the \(2^\text{nd}\) degree polynomial of

\[f(x)=\ln(1+x) ~\text{ centered at }~ x=0 ~\text{ on the interval }~ [-0.1,0.1]?\]

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