# The linear approximation of a function at one point.

Definition 6.2.1   Let be a function defined in a neighborhood of the real . The function has a linear approximation at if there exist a real number and a function defined on a neighborhood of 0, such that

Example 6.2.2   Let and . Then we have:

Example 6.2.3   Let and . Then we have:

With the notations of Definition 1.4, we have:

Definition 6.2.4   The expression is called the differential of at .

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:

Example 6.2.5   With the settings of Example 2.3, we wish to compute . Let and ; 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