With the notations of Definition 1.4, we have:

An important use of differentials is the approximation of the value of a function at a point . Actually, we can rewrite Definition 2.1 as follows:

When is arbitrarily close to 0, we replace by and by or (these notations have the same meaning). Then we have:

Compare this with the ``actual value'', which is 1.070599167. Not so far! We will learn later how to have an estimation of the error, when computing an approximation with the differential instead of the actual value. Remember that if an scientist makes measurements providing data with an accuracy of , for some positive integer , then a computation leading to results with a greater accuracy (i.e. , with ) is meaningless.

Noah Dana-Picard 2007-12-28