Calculus Graphing With the First Derivative
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Graphing With the First Derivative

 
 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) = X2

 
 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

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