We began this arc of posts with a potted history of the differential calculus; from its origin in the infinitesimals of the 17

We then covered Taylor’s theorem and went on to use it in a comprehensive analysis of finite difference approximations to the derivative in which we discovered that their accuracy is a balance between approximation error and cancellation error, that it always depends upon the unknown behaviour of higher derivatives of the function and that improving accuracy by increasing the number of terms in the approximation is a rather tedious exercise.

Of these issues, the last rather stands out; from tedious to automated is often but a simple matter of programming. Of course we shall first have to figure out an algorithm, but fortunately we shall be able to do so with relative ease using, you guessed it, Taylor’s theorem.

^{th}century, through its formalisation with Analysis in the 19^{th}and the eventual bringing of rigour to the infinitesimals in the 20^{th}.We then covered Taylor’s theorem and went on to use it in a comprehensive analysis of finite difference approximations to the derivative in which we discovered that their accuracy is a balance between approximation error and cancellation error, that it always depends upon the unknown behaviour of higher derivatives of the function and that improving accuracy by increasing the number of terms in the approximation is a rather tedious exercise.

Of these issues, the last rather stands out; from tedious to automated is often but a simple matter of programming. Of course we shall first have to figure out an algorithm, but fortunately we shall be able to do so with relative ease using, you guessed it, Taylor’s theorem.

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