



The
derivative is the slope of the equation line at any given point. 





When the derivative is positive, the graph is going up. 





When the
slope is a negative number, the graph is going down. 





When the
slope is zero, the graph is horizontal. 








What
would we do if we had something like: 





F(X) = X^{2}






Since
we're calculus big shots now, 


We would
find the derivative of this, 


in our
heads!... 


F'(X)
= 2X






So the
derivative is negative when X is less than zero: 








the
derivative is zero when X = 0: 





the
derivative is positive when X is greater than zero. 





So from
this information, 


we know that the graph of
F(X) = X
^{2}
looks something like this: 





If we
solve the equation for the point where the slope is zero 


(where X =
0 in this particular case), we can tack this curve to a coordinate
axis: 





The slope
can only change directions 


(like going from positive to negative or
negative to positive), 


if there
is a point where the slope is zero or undefined in between. 


That's
what happened when X = 0 in this problem. 


If you
have an equation where the slope is never zero or undefined, 


it will
either always be going up or always be going down. 





One more
point: 


The slope
CAN change directions where the slope is zero or undefined, 


but it
doesn't have to. 





Wouldn't
it be nice to have a test to know if it does change directions 


without drawing the graph first? 





OK,
here's another question: 


Guess
what's on the next page? 





copyright 2008 Bruce Kirkpatrick 
