- NOW WE'LL LOOK 
AT SEVERAL EXAMPLES
OF DETERMINING WHETHER 
THE FIRST DERIVATIVE
IS ZERO, POSITIVE OR NEGATIVE 
AT A GIVEN POINT ON A FUNCTION.
THE FIRST DERIVATIVE 
FUNCTION VALUE
GIVES US A SLOPE OF THE TANGENT 
LINE AT A GIVEN VALUE OF X.
SO LOOKING AT THIS FIRST GRAPH
IF WE SKETCH A TANGENT LINE 
AT THIS RED POINT
NOTICE HOW THE SLOPE WOULD 
BE POSITIVE
AND THEREFORE THE FIRST 
DERIVATIVE IS POSITIVE
OR GREATER THAN 0 AT THAT POINT.
ALSO NOTICE THAT OVER THIS 
INTERVAL HERE
THE FUNCTION IS INCREASING UNTIL 
IT REACHES THIS HIGH POINT
AND THEREFORE THE FUNCTION IS 
INCREASING OVER THIS INTERVAL
AND THEREFORE THE FIRST 
DERIVATIVE WOULD BE POSITIVE
OVER THIS ENTIRE INTERVAL 
AS WELL.
LOOKING AT OUR SECOND EXAMPLE
IF WE SKETCH AT A TANGENT LINE 
AT THIS RED POINT
NOTICE HOW THE SLOPE OF THE 
TANGENT LINE WOULD BE NEGATIVE,
WHICH MEANS THE FIRST DERIVATIVE 
IS LESS THAN 0 AT THAT POINT.
ALSO NOTICE THIS FUNCTION
IS DECREASING OVER 
THIS INTERVAL HERE
WHICH MEANS THE FIRST DERIVATIVE 
WOULD BE NEGATIVE
OVER THIS ENTIRE INTERVAL.
LET'S TAKE A LOOK AT 
TWO MORE EXAMPLES.
IF WE SKETCH A TANGENT LINE 
AT THIS RED POINT,
NOTICE HOW THE SLOPE OF THE 
TANGENT LINE WOULD BE NEGATIVE
AND THEREFORE
THE FIRST DERIVATIVE IS NEGATIVE 
AT THAT POINT.
AND ONCE AGAIN NOTICE HOW 
THE FUNCTION IS DECREASING
OVER THIS ENTIRE INTERVAL HERE
GOING DOWNHILL FROM LEFT 
TO RIGHT,
THEREFORE THE FIRST DERIVATIVE 
WOULD BE NEGATIVE
OVER THIS ENTIRE INTERVAL.
NOW, FOR OUR LAST EXAMPLE
WE SKETCH A TANGENT LINE 
AT THIS POINT HERE.
NOTICE HOW THIS POINT 
IS A LOW POINT ON THE GRAPH
AND THEREFORE THE SLOPE 
OF THE TANGENT LINE
WOULD ACTUALLY BE HORIZONTAL.
THE SLOPE OF ANY HORIZONTAL LINE 
IS 0, AND THEREFORE,
THE FIRST DERIVATIVE = 0 
AT THAT POINT.
THAT ALSO MEANS THE X COORDINATE 
OF THIS POINT
WOULD BE A CRITICAL NUMBER.
I HOPE YOU FOUND THIS HELPFUL.
