2.1 Curve Sketching
===================

Definitions
-----------
:critical point: A point where the first derivative is zero.
:local minimum: A point where the first derivative is zero, and the second
                derivative is *positive* on both sides. :math:`f(a) \le f(x)`
                for all points within a certain region if the minimum is at
                :math:`a`.
:local maximum: A point where the first derivative is zero, and the second
                derivative is *negative* on both sides. :math:`f(a) \ge f(x)`
                for all points within a certain region if the maximum is at
                :math:`a`.
:absolute maximum/minimum: The minimum or maximum for all the points in the
                           range.
:extreme point: A local minimum or a maximum.
:point of inflection: A point where the second derivative changes sign. This
                      can occur where the second derivative is undefined or
                      zero.
:even function: :math:`f(-x) = f(x)`
:odd function: :math:`f(-x) = -f(x)`


Curve Sketching Process
-----------------------

:Step 1: Identify where :math:`f'(x)=0` (Critical points.) The tangent line is
         zero here.
:Step 2: Where :math:`f''(x)>0`, it is concave upward. Where :math:`f''(x)<0`, it
         concave downward. 
:Step 3: Where :math:`f'(x)=0`, If :math:`f''(x)>0`, then it is a local
         minimum point. If :math:`f''(x)<0`, then it is a local maximum point.
:Step 4: If :math:`f''(x)` changes sign (at zero or undefined points) it is a
         point of inflection.
:Step 5: Even and odd functions need only be investigated for positive x. Flip
         or rotate it when done.