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- WELCOME TO A SECOND LESSON LINEAR GROWTH.
IN THIS LESSON WE WILL EXPRESS LINEAR GROWTH IN RECURSIVE FORM,
AS WELL AS EXPLICIT FORM,
BUT IT IN THIS LESSON WE WONT BE GIVEN D,
THE COMMON DIFFERENCE, OR THE SLOPE. WE'LL HAVE TO FIND IT.
SO FOR REVIEW, IF A QUANTITY STARTS AT SIZE P SUB ZERO
AND GROWS BY D EVERY TIME PERIOD THEN THE QUANTITY
AFTER N TIME PERIODS CAN BE DETERMINED
USING EITHER OF THE TWO RELATIONS EXPRESSED HERE.
IN RECURSIVE FORM, P SUB N = P SUB N - 1 + D, WHERE P SUB N
IS THE POPULATION ONE UNIT OF TIME BEFORE P SUB N.
THE EXPLICIT FORM WOULD BE P SUB N = P SUB 0 + D x N.
NOW THE EXPLICIT FORM HERE IS THE SAME
AS THE SLOPE INTERCEPT FORM OF THE LINE,
OR THE FORM Y = MX + B, WHERE P SUB ZERO WOULD B,
THE Y INTERCEPT, D,
THE COMMON DIFFERENCE WOULD BE THE SAME
AS THE SLOPE N, AND P SUB N IS THE SAME
AS THE OUTPUT OR THE Y VALUE.
SO AGAIN, IN THIS EQUATION D REPRESENTS
THE COMMON DIFFERENCE,
THE AMOUNT THE POPULATION CHANGES EACH TIME
N INCREASES BY ONE.
SO THE COMMON DIFFERENCE D,
WHICH AGAIN IS THE SAME AS THE SLOPE,
WOULD BE EQUAL TO THE CHANGE IN THE FUNCTION OUTPUT
DIVIDED BY THE CHANGE IN THE FUNCTION INPUT.
WELL, IF WE WERE GIVEN TWO POINTS YOU MAY REMEMBER
WE CAN FIND THE SLOPE BY DETERMINING THE CHANGE IN Y
DIVIDED BY THE CHANGE OF X.
LET'S TAKE A LOOK AT AN EXAMPLE.
THE JAVELINA POPULATION OF A LOCAL DESSERT PARK
WAS ESTIMATED AT 200 IN THE YEAR 2004.
IN 2012, THE POPULATION WAS ESTIMATED TO BE 376.
ONE, ASSUMING LINEAR GROWTH,
WE WANT TO FIND A RECURSIVE AND EXPLICIT EQUATION
FOR THE GROWTH.
TWO, WE WANT TO PREDICT THE JAVELINA POPULATION IN 2016.
AND THEN THREE, PREDICT WHEN THE POPULATION WILL REACH 500.
NOTICE FOR BOTH THE RECURSIVE AND EXPLICIT FORMULAS,
WE'LL HAVE TO FIND THE COMMON DIFFERENCE D,
WHICH WOULD ALSO BE THE SLOPE OF THE LINE.
SO LETS BEGIN BY DOING THAT.
REMEMBER D IS GOING TO BE EQUAL TO THE CHANGE IN THE OUTPUT,
OR IN THIS CASE THE CHANGE IN THE POPULATION,
DIVIDED BY THE CHANGE IN THE INPUT,
WHICH IN THIS CASE WOULD BE THE CHANGE IN THE YEARS.
SO AGAIN, D, THE COMMON DIFFERENCE IN THIS CASE,
IS GOING TO BE EQUAL TO THE CHANGE IN THE POPULATION
DIVIDED BY THE CHANGE IN THE YEARS.
WELL, THE CHANGE IN THE POPULATION
WOULD BE 376 - 200 AND THE CHANGE IN THE YEARS
WOULD BE 2012 - 2004.
SO WE'D HAVE 176 DIVIDED 8, WHICH IS EQUAL TO 22.
AND BECAUSE THE COMMON DIFFERENCE IS POSITIVE,
THIS INDICATES THE POPULATION IS INCREASING 22 JAVELINA PER YEAR.
AND BECAUSE WE'RE ASSUMING LINEAR GROWTH,
THIS IS THE CONSTANT RATE OF CHANGE EACH YEAR.
THERE IS ONE MORE THING TO MENTION HERE.
P SUB ZERO IS THE STARTING POPULATION
WHERE THE POPULATION N EQUALS ZERO, WHICH MEANS N EQUALS ZERO,
REPRESENTS THE YEAR 2004.
SO WE CAN SAY N EQUALS THE NUMBER OF YEARS
AFTER 2004.
BEFORE WE FIND THE RECURSIVE AND EXPLICIT EQUATIONS
TO MODEL THIS GROWTH, LETS TAKE A LOOK AT THIS GRAPHICALLY.
KNOWING THE POPULATION WAS 200 IN 2004,
THAT WOULD BE THE POINT (0, 200) WHERE THE FIRST COORDINATE IS N
AND THE SECOND COORDINATE IS P SUB N.
REMEMBER N IS THE NUMBER OF YEARS AFTER 2004.
AND THEREFORE, N IS 0 FOR THE YEAR 2004.
THE SECOND POINT IN 2012 WHEN THE POPULATION IS 376
WOULD BE (8, 376).
NOTICE N IS 8 HERE BECAUSE 2012 IS 8 YEARS AFTER 2004.
ANOTHER WAY TO FIND N IS TO TAKE 2012 AND SUBTRACT 2004.
SO IF WE PLOT THESE TWO POINTS ON THE COORDINATE PLANE,
HERE'S (0, 200) AND HERE'S (8, 376).
BECAUSE WE'RE ASSUMING LINEAR GROWTH,
WHEN WE FIND THE EQUATIONS
IN BOTH RECURSIVE AND EXPLICIT FORM,
BOTH FORMULAS WOULD GIVE US POINTS ON THIS LINE.
SO NOW, LET'S GO BACK AND FIND
THE RECURSIVE AND EXPLICIT EQUATIONS.
IN RECURSIVE FORM P SUB N,
WHERE THE POPULATION N YEARS AFTER 2004,
WOULD BE EQUAL TO P SUB N - 1,
THE POPULATION 1 YEAR BEFORE P SUB N + THE COMMON DIFFERENCE,
WHICH IS 22.
THIS WOULD BE THE RECURSIVE FORMULA FOR OUR LINEAR GROWTH.
AND NOW FOR THE EXPLICIT EQUATION,
WE HAVE P SUB N EQUALS P SUB ZERO + D x N,
WHICH WOULD GIVE US P SUB N = P SUB 0
REPRESENTS THE STARTING POPULATION
OR THE POPULATION IN 2004, WHICH IS 200.
SO WE HAVE 200 + THE COMMON DIFFERENCE x N, OR SO + 22N.
THIS WOULD BE THE EXPLICIT FORM FOR OUR LINEAR GROWTH.
NOW WE WANT TO PREDICT THE JAVELINA POPULATION IN 2016.
TO DO THIS WE COULD USE EITHER FORMULA,
BUT TO USE THE RECURSIVE FORMULA,
WE'D HAVE TO CALCULATE THE POPULATION FOR EACH YEAR
FROM 2004 TO 2016, WHICH WOULD BE A LOT OF WORK.
BUT USING OUR EXPLICIT FORMULA,
WE CAN JUST SUBSTITUTE THE VALUE OF N FOR THE YEAR 2016.
THEREFORE, WE'LL USE THE EXPLICIT EQUATION
TO ANSWER THIS QUESTION.
SO AGAIN, WE KNOW THAT P SUB N = 200 + 22N.
BUT WE STILL HAVE TO DETERMINE THE VALUE OF N,
WHERE AGAIN N IS THE NUMBER OF YEARS AFTER 2004.
SO IF WE'RE TRYING TO MAKE A PREDICTION FOR THE YEAR 2016,
N = 2016 - 2004 = 12.
2016 IS 12 YEARS AFTER 2004,
SO THE POPULATION AFTER 12 YEARS OR P SUB 12,
IS EQUAL TO 200 + 22 x 12, WHICH WOULD BE 200 + 264 = 464.
SO IN 2016 THE PREDICTED POPULATION IS 464 JAVELINA.
AND NOW FOR PART THREE WE WANT TO PREDICT
WHEN THE POPULATION WILL REACH 500.
SO AGAIN, WE'LL USE THE EXPLICIT EQUATION.
WE'LL SUBSTITUTE 500 FOR PS SUB N AND SOLVE FOR N.
SO WE WANT TO SOLVE THE EQUATION 500 = 200 + 22N,
WE'LL SUBTRACT 200 ON BOTH SIDES,
THAT WOULD GIVE US 300 = 22N DIVIDE BOTH SIDES BY 22.
300 DIVIDED BY 22 IS APPROXIMATELY 13.64.
REMEMBER, N IS THE NUMBER OF YEARS AFTER 2004.
SO 2004 + 13.64 WOULD BE 2017.64.
ROUNDING TO THE NEAREST YEAR,
WE CAN SAY THE POPULATION WILL REACH 500 IN 2018.
I HOPE YOU FOUND THIS HELPFUL.