(i) Compute (e^x-e^-x)/(2 x)-1 on a 10 -digit calculator (or...

(i) Compute $\frac{e^{x}-e^{-x}}{2 x}-1$ on a 10 -digit calculator (or similar) for $x=1,10^{-2}, 10^{-4}, 10^{-6}$. (ii) Use the Taylor series to find an expression for the function that is accurate for small values of $x$. (iii) Use this to recompute the function for $x=10^{-2}, 10^{-4}, 10^{-6}$.

(i) Compute $\frac{e^{x}-e^{-x}}{2 x}-1$ on a 10 -digit calculator (or similar) for $x=1,10^{-2}, 10^{-4}, 10^{-6}$.
(ii) Use the Taylor series to find an expression for the function that is accurate for small values of $x$. (iii) Use this to recompute the function for $x=10^{-2}, 10^{-4}, 10^{-6}$.

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Key Concepts

Taylor Series Expansion

The Taylor series is a method for expressing a function as an infinite sum of terms calculated from the function’s derivatives at a specific point. In this problem, it is used to derive an approximation that is particularly accurate for small values of x by expanding the exponential functions and then simplifying the series.

Numerical Precision and Catastrophic Cancellation

When performing computations on finite-precision calculators, subtracting nearly equal numbers (such as those encountered when evaluating functions for very small x) can lead to significant rounding errors known as catastrophic cancellation. Recognizing this issue guides the use of series expansions to obtain a more stable and accurate approximation for the function.

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