Hoofdstuk 12 Differentiëren *: Stijgen, dalen, extreme waarden *
Tangent lines
A geometric interpretation of the derivative is the following.
If #C# is a curve in the flat plane and #P# is a point on that curve, then a line #l# through #P# is a tangent to #C# if
- #P# is the only point that #C# and #l# have in common in a small area of #P#, and
- the points of #l# in a small area of #P# all lie at the same side of #C#.
The tangent line #l# therefore does not intersect #C# in points near to but distinct from #P#.
Let #f# be a differentiable function on an open interval #(a,b)# containing the number #p#. The tangent to the graph of #f# at #P=\rv{p,f(p)}# is the best approximation of #f#by a line in a small area of #P#. Instead of the tangent line to the graph of #f# at #P#, we also speak of the tangent line to #f# at #p#.
Below, the graph of a quadratic function #f# and the tangent to #f# at a point #\rv{p,f(p)}# are drawn. You can move the points #\rv{a,f(a)}# and #\rv{b,f(b)}# to see how the tangent moves along.
The statement below enables us to calculate the tangent. Since, once we know a point of a line and the slope of that line, we know the entire line.
If #f# is a differentiable function on an open interval #I# that contains the number #p# and #l# is a tangent line at #p# to the graph of #f#, then #f'(p)# is the slope of #l#.
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