Question

If the nth derivative of a function at a point x0 is known, describe how you would use it to find an estimate of the error when approximating the function using a Taylor series.

Ask by Stuart Welch. in Ghana
Mar 15,2025

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To estimate the error when approximating a function using a Taylor series: 1. Write the Taylor series up to the nth term. 2. Use Taylor’s theorem to find the remainder term \( R_{n+1}(x) \). 3. If you know the magnitude of the \( (n+1) \)th derivative, use it to bound the error: \[ \left|R_{n+1}(x)\right| \le \frac{M}{(n+1)!}\left|x-x_0\right|^{n+1} \] where \( M \) is the bound on the derivative. This gives you an estimate of the error in the approximation.

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To estimate the error when approximating a function using a Taylor series, one can utilize the remainder term of the Taylor series. This involves calculating the (n+1)th derivative of the function at the point x0, which can be denoted as \( f^{(n+1)}(c) \) for some value \( c \) between \( x0 \) and the point where the approximation is evaluated. The error can then be framed using the formula: \[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - x0)^{(n+1)} \] This gives a bound on how much the Taylor polynomial deviates from the actual function, helping you assess the approximation's accuracy. Additionally, understanding that the remainder term behaves differently depending on the function's characteristics can enhance your estimates. For smooth functions, the remainder shrinks rapidly as x approaches x0. However, if the (n+1)th derivative grows rapidly, it could lead to larger errors. So, examining the behavior of derivatives can provide deeper insight into how reliable your Taylor approximation will be!

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