- WELCOME TO A LESSON ON AVERAGE RATE OF CHANGE OF A FUNCTION.
THE GOALS OF THIS VIDEO ARE TO DEFINE
THE AVERAGE RATE OF CHANGE
AND THEN DETERMINE THE AVERAGE RATE OF CHANGE OF A FUNCTION.
LOOKING AT THIS GRAPH HERE,
THIS BLACK LINE IS A SECANT LINE.
A SECANT LINE IS A LINEAR APPROXIMATION
OF A FUNCTION BETWEEN TWO POINTS.
SO IN THIS GRAPH,
THIS IS OUR FIRST POINT AND THIS IS OUR SECOND POINT.
THE SLOPE OF THE SECANT LINE
REPRESENTS THE AVERAGE RATE OF CHANGE OF THE FUNCTION
BETWEEN THESE TWO POINTS.
SO WHEN YOU'RE ASKED TO DETERMINE
THE AVERAGE RATE OF CHANGE OF A FUNCTION
YOU'RE REALLY JUST FINDING THE SLOPE OF A LINE.
REMEMBER THE EQUATION FOR THE SLOPE OF A LINE
GIVEN TWO POINTS
IS M = Y SUB 2 - Y SUB 1 DIVIDED BY X SUB 2 - X SUB 1.
SO AGAIN, WHEN YOU'RE FINDING
THE AVERAGE RATE OF CHANGE OF A FUNCTION
YOU'RE REALLY JUST DETERMINING THE SLOPE OF A SECANT LINE.
TO GET A BETTER FEEL FOR THIS
LET'S TAKE A LOOK AT AN ANIMATION.
THIS RED SECANT LINE REPRESENTS THE AVERAGE RATE OF CHANGE
OF THIS BLACK FUNCTION BETWEEN THESE TWO POINTS.
SO NOTICE THAT THE SLOPE OF THIS RED LINE IS NEGATIVE
NOTICE HOW THE SLOPE OF THIS SECANT LINE
IS APPROXIMATELY -37.
WHICH MEANS THE AVERAGE RATE OF CHANGE OF THE BLACK FUNCTION
FROM THIS FIRST POINT TO THE SECOND POINT
IS APPROXIMATELY -37,
AND AS WE ANIMATE THIS SECANT LINE
YOU CAN SEE THE SLOPE OF THE SECANT LINE CHANGES
MEANING THE AVERAGE RATE OF CHANGE OF THE FUNCTION
IS ALSO CHANGING BETWEEN THOSE TWO POINTS.
SO YOU CAN SEE AT THIS LOCATION HERE THE AVERAGE RATE OF CHANGE
OF THE FUNCTION BETWEEN THOSE TWO POINTS IS ALMOST -3,
BUT IF WE PROGRESS A LITTLE BIT FURTHER ALONG THIS FUNCTION,
NOTICE HOW NOW THE AVERAGE RATE OF CHANGE IS APPROXIMATELY +2.
AGAIN, THE SLOPE OF THIS SECANT LINE REPRESENTS
THE FUNCTION'S AVERAGE RATE OF CHANGE
BETWEEN THOSE TWO POINTS.
TO GET A BETTER FEEL FOR AVERAGE RATE OF CHANGE
LET'S SAY YOU TAKE A TRIP FROM PHOENIX, ARIZONA
TO LOS ANGELES, CALIFORNIA
AND LET'S SAY YOU TRAVEL THE 380 MILE TRIP IN 5 1/2 HOURS.
WE KNOW FROM ALGEBRA
THAT THE DISTANCE IS EQUAL TO RATE x TIME
SO IF WE SOLVE THIS EQUATION FOR R OR RATE,
WE WOULD HAVE THE RATE IS EQUAL TO DISTANCE DIVIDED BY TIME
AND THIS REPRESENTS AN AVERAGE RATE.
SO IF WE TAKE THE DISTANCE AND DIVIDE BY THE TIME,
OUR AVERAGE SPEED OR AVERAGE RATE OF CHANGE ON THIS TRIP
WOULD BE APPROXIMATELY 69 MILES PER HOUR.
IF WE COMPARE THE TIME AND DISTANCE TRAVELED ON OUR TRIP,
THE GRAPH MIGHT LOOK SOMETHING LIKE THIS.
NOTICE HOW AT CERTAIN TIMES THE DISTANCE TRAVELED
IS LESS THAN AT OTHER TIMES.
MEANING WHEN THE GRAPH IS STEEP WE'RE TRAVELING FASTER
AND WHEN THE GRAPH FLATTENS OUT OR DOESN'T INCREASE AS FAST
OUR SPEED WOULD BE SLOWER.
SO IF WE PICK TWO POINTS ON THIS GRAPH AND SKETCH THE SECANT LINE
THE SLOPE OF THAT LINE WOULD BE
OUR AVERAGE RATE OF CHANGE ON THAT GIVEN INTERVAL OF TIME.
SO FOR EXAMPLE IF WE LOOK AT
THE SECANT LINE FROM WHERE WE STARTED TO WHERE WE END
THE SLOPE OF THIS RED SECANT LINE,
WILL BE APPROXIMATELY 69 MILES PER HOUR.
BUT IF WERE TO PICK TWO DIFFERENT TIMES ON THIS TRIP,
LET'S SAY AT TWO HOURS AND THEN AT THREE HOURS,
THE SLOPE OF THIS SECANT LINE WOULD GIVE US OUR AVERAGE SPEED
DURING THIS TIME INTERVAL.
NOTICE THAT OUR AVERAGE SPEED FROM TWO TO THREE HOURS
WOULD BE LESS THAN THE AVERAGE SPEED OF THE ENTIRE TRIP,
AND YOU CAN SEE BECAUSE IT LOOKS LIKE
WE MAY HAVE STOPPED TO GET GAS
OR MAYBE AT A REST STOP SOMEWHERE AROUND THIS TIME.
LET'S TAKE A LOOK AT A FEW OF OUR OWN EXAMPLES.
LET'S DETERMINE THE AVERAGE RATE OF CHANGE OF THE GIVEN FUNCTION
FROM X = 0 TO X = 3.
WELL THE FIRST THING WE'RE GOING TO DO
IS DETERMINE THE COORDINATES OF THE POINTS
WHEN X = 0 AND WHEN X = 3.
ONCE WE HAVE OUR TWO POINTS
WE CAN THEN JUST USE THE SLOPE FORMULA
TO DETERMINE OUR AVERAGE RATE OF CHANGE.
SO WE'LL HAVE X AND F OF X. SO X= 0 AND X = 3.
WELL WHEN X = 0,
F OF 0 WOULD BE 0 SQUARED + 1, SO F OF X WOULD BE 1.
REMEMBER F OF X IS THE SAME AS Y,
AND WHEN X = 3, WE'D HAVE 3 SQUARED + 1, THAT WOULD BE 10,
AND WE CAN CHECK THAT OVER HERE ON OUR GRAPH.
HERE'S THE POINT (0, 1) AND HERE'S THE POINT (3, 10).
SO THE AVERAGE RATE OF CHANGE OF THIS FUNCTION
WILL BE THE SLOPE OF THE LINE CONTAINING THIS SEGMENT HERE.
SO THE AVERAGE RATE OF CHANGE AGAIN IS Y2 - Y1/X2 - X1
AND JUST TO KEEP THINGS ORGANIZED
I'LL CALL THIS (X SUB 1, Y SUB 1)
AND THIS WILL BE (X SUB 2, Y SUB 2).
SO WE'RE GOING TO HAVE 10 - 1 IN THE NUMERATOR
AND THEN X2 - X1 IS GOING TO BE 3 - 0.
SO WE'LL HAVE 9 DIVIDED BY 3 WHICH IS EQUAL TO 3.
SO THE AVERAGE RATE OF CHANGE FROM X = 0 TO X = 3 IS 3.
TO DETERMINE THE MEANING OF THIS IT MAY BE HELPFUL
TO WRITE IT AS A FRACTION.
REMEMBER SLOPE TELLS US THE CHANGE IN Y
WITH RESPECT TO THE CHANGE OF X.
SO ON THE INTERVAL FROM 0 TO 3
THE AVERAGE RATE OF CHANGE OF THIS FUNCTION
IS AS X INCREASES BY 1, Y WILL INCREASE BY 3.
REMEMBER X IS THE INDEPENDENT OR CONTROL VARIABLE
AND Y IS THE DEPENDENT VARIABLE.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE
WHERE WE HAVE THE SAME FUNCTION
BUT NOW WE WANT THE AVERAGE RATE OF CHANGE
FROM X = -2 TO X = 1.
SO LET'S START BY FILLING OUT OUR T TABLE FOR X = -2 AND X = 1
AND THEN WE'LL FIND F OF X, WHICH AGAIN IS THE SAME AS Y.
SO WHEN X = -2, WE'LL HAVE -2 SQUARED,
THAT'S 4 + 1, SO Y WILL BE 5
AND THEN WHEN X IS 1, WE'LL HAVE 1 SQUARED + 1.
THAT WILL GIVE US 2.
AGAIN, WE CAN VERIFY THAT OVER HERE ON THE GRAPH.
HERE'S THE POINT (-2, 5) AND HERE'S THE POINT (1, 2).
NOTICE HOW THE SLOPE OF THIS SEGMENT,
THAT WOULD BE CONTAINED WITHIN THE SECANT LINE,
IS NOW GOING DOWNHILL AND THEREFORE NEGATIVE.
THEREFORE, THE AVERAGE RATE OF CHANGE OF THIS FUNCTION
FROM -2 TO 1 IS NEGATIVE.
SO NOW WE JUST NEED TO DETERMINE THE SLOPE OF THE SECANT LINE
PASSING THROUGH THESE TWO POINTS
AND AGAIN TO KEEP THINGS ORGANIZED,
I'LL CALL THIS (X SUB 1, Y SUB 1)
AND THIS POINT (X SUB 2, Y SUB 2).
SO Y SUB 2 - Y SUB 1 IS GOING TO BE 2 - 5
AND THE DENOMINATOR IS GOING TO BE X SUB 2 - X SUB 1.
WELL 1 - -2 HERE WE'RE GOING TO HAVE -3 OVER--
THIS IS GOING TO BE +3.
SO THE AVERAGE RATE OF CHANGE IS EQUAL TO -1
BUT TO ADD SOME MEANING TO THIS
WE'LL GO AHEAD AND WRITE THIS AS A FRACTION -1/1.
THAT TELLS US THAT FROM X = -2 TO X = 1,
AS X INCREASES BY 1, Y DECREASES BY 1.
SO EVEN THOUGH IN FRACTION FORM
IT'S MORE MEANINGFUL WHEN GIVING YOUR ANSWER
YOU ALWAYS WANT TO HAVE IT IN THE MOST SIMPLEST FORM
MEANING YOUR AVERAGE RATE OF CHANGE
WOULD BE -1 FOR THIS PROBLEM
AND THE SAME THING ON THE PREVIOUS EXAMPLE.
EVEN THOUGH IT'S MORE MEANINGFUL TO WRITE IT AS 3/1
YOU SHOULD EXPRESS THE AVERAGE RATE OF CHANGE AS +3.
I HOPE YOU FOUND THIS HELPFUL. THANK YOU FOR WATCHING.