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Taylor Series

Taylor Series - Error Bounds

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

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

\[\frac{4\pi^{5}}{15} \] \[\frac{1}{5!} \] \[\frac{32\pi^{5}}{5} \] \[\frac{2\pi^{4}}{3} \]

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by Brilliant Staff

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

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

\[\frac{4\pi^{5}}{15} \] \[\frac{32\pi^{5}}{5} \] \[\frac{2\pi^{4}}{3} \] \[\frac{1}{5!} \]

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by Brilliant Staff

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

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

\[\frac{e^2 \cdot 2^{11}}{11!} \] \[\frac{e^2 \cdot 2^{10}}{10!} \] \[\frac{e^{11} \cdot 2^{10}}{10!} \] \[\frac{e^{11} \cdot 2^{11}}{11!} \]

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by Brilliant Staff

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

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

\[\dfrac{0.01^2}{\sqrt{0.99^3}} \] \[\dfrac{0.01^2}{8\sqrt{0.99^2}} \] \[\dfrac{0.01^3}{\sqrt{0.99^3}} \] \[\dfrac{0.01^2}{8\sqrt{0.99^3}} \]

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by Brilliant Staff

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

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

\[\dfrac{0.1^3}{3!(0.9^3)} \] \[\dfrac{0.1^3}{3(0.9^3)} \] \[\dfrac{0.1^3}{3(1.1^3)} \] \[\dfrac{0.1^3}{3!(1.1^3)} \]

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by Brilliant Staff