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- AFTER DRIVING TWO HOURS, JUAN IS 110 MILES FROM HIS HOME.
AFTER DRIVING 5 HOURS, HE IS 275 MILES FROM HIS HOME.
WHAT IS HIS AVERAGE RATE OF CHANGE?
IF WE LET R EQUAL THE AVERAGE RATE OF CHANGE
R WOULD BE EQUAL TO THE CHANGE IN THE DISTANCE
DIVIDED BY THE CHANGE IN THE TIME.
SO IN THIS CASE
THE CHANGE IN THE DISTANCE WOULD BE 275 MILES - 110 MILES.
LET'S LEAVE OFF THE UNITS FOR RIGHT NOW
AND INCLUDE THEM AT THE END
AND THE CHANGE IN TIME WOULD BE 5 HOURS - 2 HOURS.
WELL, 275 - 110 = 165 THIS WOULD BE MILES
AND THEN 5 - 2 = 3 WHICH WOULD BE 3 HOURS.
165 DIVIDED BY 3 = 55
BUT LET'S LEAVE THIS IN FRACTION FORM
SO WE'LL WRITE 55/1 THIS REPRESENTS 55 MILES
AND THE 1 REPRESENTS 1 HOUR
GIVING US OUR AVERAGE RATE OF CHANGE.
THE AVERAGE RATE OF CHANGE DURING THIS TIME PERIOD
IS 55 MILES PER HOUR. BUT LET'S ALSO LOOK AT THIS GRAPHICALLY.
IF WE LABELED THE X AXIS HOURS AND THE Y AXIS MILES,
WE COULD PLOT THE GIVEN INFORMATION
ON THE COORDINATE PLANE.
TO REPRESENT THAT AFTER 2 HOURS JUAN IS 110 MILES FROM HIS HOME
WE WOULD HAVE THIS POINT HERE WITH COORDINATES (2, 110)
AND TO REPRESENT THAT AFTER 5 HOURS
HE IS 275 MILES FROM HIS HOME
WE'D HAVE THIS POINT HERE WITH COORDINATES (5, 275).
SO NOW IF WE SKETCH THE LINE
PASSING THROUGH THESE TWO POINTS,
WE WOULD HAVE THIS BLUE LINE HERE
AND THE SLOPE OF THIS LINE
WOULD BE THE AVERAGE RATE OF CHANGE.
SO NOW WE COULD USE THE FORMULA M = Y SUB 2 - Y SUB 1
WHICH WOULD GIVE US THE CHANGE IN THE DISTANCE
AND THEN DIVIDE BY X SUB 2 - X SUB 1
WHICH WOULD BE THE CHANGE IN TIME
WHICH WOULD GIVE US OUR AVERAGE RATE OF CHANGE
WHICH WE FOUND TO BE 55 OR 55 MILES PER HOUR.
THE REASON IT'S HELPFUL TO LOOK AT THIS GRAPHICALLY
IS THAT 55 MILES PER HOUR IS THE AVERAGE RATE OF CHANGE
BUT WE KNOW FROM OUR EXPERIENCE
JUAN DID NOT DRIVE 55 MILES PER HOUR
FOR THIS ENTIRE TIME PERIOD.
HE PROBABLY DROVE FASTER AND SLOWER THROUGHOUT THIS TRIP.
SO EVEN THOUGH WE FOUND THE AVERAGE RATE OF CHANGE
IF WE COMPARED THE TIME AND DISTANCE THROUGHOUT THE TRIP
HIS ACTUAL TRIP MIGHT HAVE LOOKED SOMETHING LIKE THIS,
WHERE HE DROVE FASTER AND SLOWER THROUGHOUT THE TRIP.
SO THE ACTUAL DATA MIGHT LOOK SOMETHING LIKE THIS
BUT THE SLOPE OF THIS BLUE LINE
DOES REPRESENT HIS AVERAGE RATE OF CHANGE
NOT HIS INSTANTANEOUS RATE OF CHANGE
THROUGHOUT THE TRIP.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.