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- NOW WE'LL TAKE A LOOK AT A BUSINESS APPLICATION
OF AN IMPROPER INTEGRAL.
THE ACCUMULATED PRESENT VALUE OF A CONTINUOUS MONEY FLOW
INTO AN INVESTMENT ACCOUNT
AT A RATE OF P DOLLARS PER YEAR PERPETUALLY,
THAT MEANS FOREVER, WAS GIVEN BY THIS IMPROPER INTEGRAL
WHERE K IS THE CONTINUOUS INTEREST RATE.
AND THERE IS A SHORTCUT
FOR EVALUATING THIS IMPROPER INTEGRAL
AND IT'S P DIVIDED BY K,
AGAIN, WHERE P IS THE MONEY FLOW PER YEAR
AND K IS THE CONTINUOUS INTEREST RATE.
THIS IS VERY SIMILAR
TO THE ACCUMULATED PRESENT VALUE FORMULA WE USED BEFORE.
HOWEVER, THIS FORMULA ASSUMES THE MONEY FLOW IS FOREVER
WHICH MAKES IT AN IMPROPER INTEGRAL
BECAUSE THE UPPER LIMIT OF INTEGRATION
IS POSITIVE INFINITY.
SO LET'S TAKE A LOOK AT OUR EXAMPLE.
WE WANT TO DETERMINE
THE ACCUMULATED PRESENT VALUE OF AN INVESTMENT
FOR WHICH THERE IS A PERPETUAL CONTINUOUS MONEY FLOW
OF $2,400 PER YEAR
AT AN INTEREST RATE OF 1% COMPOUNDED CONTINUOUSLY.
NOW, WE'LL USE THE SHORTCUT FIRST
AND THEN WE'LL DO IT THE LONG WAY.
SO THE ACCUMULATED PRESENT VALUE IS GOING TO BE EQUAL
TO THE MONEY FLOW PER YEAR WHICH IS $2,400
DIVIDED BY THE CONTINUOUS INTEREST RATE AS A DECIMAL
WHICH WOULD BE 0.01.
AND THIS IS EQUAL TO 240,000 OR $240,000.
SO THIS PUTS A VALUE ON RECEIVING $2,400
PER YEAR FOREVER
WHICH WOULD EARN 1% CONTINUOUS INTEREST.
AND THIS KIND OF REMINDS ME OF THE LOTTERY
WHERE YOU'RE GIVEN A CHOICE OF A LUMP SUM OR A MONTHLY PAYMENT.
THE ONLY DIFFERENCE IS THAT
MOST LOTTERIES DON'T GIVE YOU MONEY FOREVER
IT'S ONLY FOR A CERTAIN AMOUNT OF TIME.
BUT THIS DOES GIVE US A LUMP SUM VALUE
OF A CONTINUOUS MONEY FLOW EARNING INTEREST FOREVER.
NOW LET'S SEE HOW WE WOULD EVALUATE THIS INTEGRAL BY HAND
WITHOUT THE SHORTCUT.
WE WOULD HAVE THE ACCUMULATED PRESENT VALUE
EQUAL TO THE IMPROPER INTEGRAL FROM 0 TO INFINITY OF 2,400.
E TO THE -0.01T, DT.
AND THEN BECAUSE THIS IS IMPROPER INTEGRAL
WE'D WRITE THIS AS THE LIMIT
AS B APPROACHES INFINITY OF INTEGRAL
FROM 0 TO B OF 2,400 E TO THE -0.01T, DT.
AND NOW WE'D HAVE TO PERFORM U SUBSTITUTION
IN ORDER TO DETERMINE THIS ANTIDERIVATIVE.
SO WE'D HAVE U = -0.01T.
DU = -0.01DT.
SO WE WOULD DIVIDE BOTH SIDES BY -0.01.
SO DT = DU DIVIDED BY -0.01.
NOW LET'S GO AHEAD AND WRITE THIS IN TERMS OF U.
I'M GOING TO GO AHEAD AND PULL OUT THE 2,400
AND WE'LL HAVE THE LIMIT AS B APPROACHES INFINITY
WITH THE INTEGRAL OF-- THIS WOULD BE E TO THE U.
BUT THEN DT IS DU DIVIDED BY -0.01.
SO I'LL PUT THE DU HERE BUT WE CAN PULL OUT THIS
AND PUT IT IN THE FRONT.
-0.01.
LET'S GO AHEAD AND FINISH THIS ON THE NEXT SLIDE.
AND NOTICE HOW WE LEFT THE LIMITS OF INTEGRATION OFF HERE
BECAUSE THIS IS IN TERMS OF U.
AND THE ANTIDERIVATIVE OF E TO THE U
IS JUST GOING TO BE E TO THE U OR E TO THE -0.01T.
THE LIMITS OF INTEGRATION WERE FROM 0 TO B.
BEFORE WE REPLACE T WITH B AND T WITH 0 REMEMBER THIS IS /1
SO WE COULD MOVE THIS DOWN TO THE DENOMINATOR
WHICH WOULD CHANGE THE SIGN OF THE EXPONENT.
LET'S GO AHEAD AND DO THAT.
SO WE'D HAVE 1/E TO THE 0.01T.
NOW WE'RE GOING TO REPLACE T WITH B AND T WITH 0.
SO WE'LL HAVE 1/E TO THE 0.01B - 1/E TO THE 0.
LOOKING AT THIS FRACTION HERE AS B APPROACHES POSITIVE INFINITY
THIS EXPONENT INCREASED
THEREFORE THIS ENTIRE DENOMINATOR INCREASED
WITHOUT BOUND
AND THE NUMERATOR STAYS AT 1.
SO THIS FRACTION APPROACHES 0.
AND E TO THE 0 = 1 SO WE HAVE -1.
SO THIS LIMIT IS EQUAL TO -1.
SO WE HAVE 2,400 DIVIDED BY -0.1 x -1
WHICH IS 2,400 DIVIDED BY 0.01
WHICH PEAKING BACK AT THE PREVIOUS SLIDE
WAS THE SHORTCUT FORMULA WHERE THIS IS THE ANNUAL MONEY FLOW
AND THIS IS THE CONTINUOUS INTEREST RATE.
SO THIS WILL GIVE US THE SAME VALUE OF $240,000
AS THE ACCUMULATED PRESENT VALUE.
SO YOU CAN SEE WHY IN MOST CASES
YOU'RE PROBABLY GOING TO PREFER TO USE THE SHORTCUT.
BUT I DID WANT TO PROVIDE ONE EXAMPLE
OF WHY THIS FORMULA ACTUALLY WORKS.
I HOPE THIS WAS HELPFUL.