Activity 8.5.2.
We have just seen that the \(n\)th order Taylor polynomial centered at \(a = 0\) for the exponential function \(e^x\) is
\begin{equation*}
\sum_{k=0}^{n} \frac{x^k}{k!}\text{.}
\end{equation*}
In this activity, we determine small order Taylor polynomials for several other familiar functions, and look for general patterns.
(a)
Let \(f(x) = \frac{1}{1-x}\text{.}\)
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Calculate the first four derivatives of \(f(x)\) at \(x=0\text{.}\) Then find the fourth order Taylor polynomial \(P_4(x)\) for \(\frac{1}{1-x}\) centered at \(0\text{.}\)
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Based on your results from part (i), determine a general formula for \(f^{(k)}(0)\text{.}\)
(b)
Let \(f(x) = \cos(x)\text{.}\)
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Calculate the first four derivatives of \(f(x)\) at \(x=0\text{.}\) Then find the fourth order Taylor polynomial \(P_4(x)\) for \(\cos(x)\) centered at \(0\text{.}\)
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Based on your results from part (i), find a general formula for \(f^{(k)}(0)\text{.}\) (Think about how \(k\) being even or odd affects the value of the \(k\)th derivative.)
(c)
Let \(f(x) = \sin(x)\text{.}\)
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Calculate the first four derivatives of \(f(x)\) at \(x=0\text{.}\) Then find the fourth order Taylor polynomial \(P_4(x)\) for \(\sin(x)\) centered at \(0\text{.}\)
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Based on your results from part (i), find a general formula for \(f^{(k)}(0)\text{.}\) (Think about how \(k\) being even or odd affects the value of the \(k\)th derivative.)
