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
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:
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.