- WE WANT TO FIND THE FUTURE VALUE OF A CONTINUOUS MONEY FLOW
OF $2,400 PER YEAR
BEING INVESTED AT 4% CONTINUOUS INTEREST FOR 4 YEARS.
SO BECAUSE WE HAVE A CONTINUOUS MONEY FLOW
RATHER THAN A 1 TIME INVESTMENT,
WE'RE GOING TO HAVE TO USE CALCULUS TO SOLVE THIS PROBLEM.
WE WANT TO ACCUMULATE THE MONEY FLOW
AND ALL OF THE INTEREST EARNED OVER 4 YEARS.
SO WE'LL HAVE TO DETERMINE THE VALUE
OF THIS DEFINITE INTEGRAL HERE
WHERE T IS THE TIME IN YEARS.
R OF T REPRESENTS THE MONEY FLOW PER YEAR.
K IS THE INTEREST RATE EXPRESSED AT A DECIMAL.
AND T IS OUR VARIABLE OF INTEGRATION.
LOOKING AT THIS INTEGRAND IT SHOULD REMIND YOU
OF THE PRESENT VALUE FORMULA FOR A 1 TIME INVESTMENT.
THE ONLY DIFFERENCE IS THAT WE HAVE THIS FUNCTION R OF T
FOR THE MONEY
RATHER THAN P SUB 0 AS THE INITIAL AMOUNT.
SO THE FUTURE VALUE OF THIS INVESTMENT
IS GOING TO BE EQUAL TO THE INTEGRAL FROM 0 TO 4
BECAUSE IT'S 4 YEARS OF THE MONEY FLOW
WHICH IS CONSTANT AT $2400.
SO 2,400 x E RAISE TO THE POWER OF K x T
WHERE K AS THE INTEREST RATE EXPRESSES A DECIMAL.
SO IT'S GOING TO BE 0.04.
AND THEN WE LEAVE "T" AS THE VARIABLE OF INTEGRATION.
THIS WILL GIVE US THE FUTURE VALUE
BUT NOTICE HOW THIS WILL REQUIRE U SUBSTITUTION.
WE'LL HAVE U EQUAL TO 0.04T,
THEREFORE, DIFFERENTIAL U IS GOING TO BE 0.04DT.
SO YOU WANT TO WRITE THIS IN TERMS OF U.
LET'S GO AHEAD AND DIVIDE BOTH SIDES BY 0.04.
THIS TELLS US THAT DT IS GOING TO BE EQUAL TO DU
DIVIDED BY 0.04.
SO WE'LL NEED TO USE THIS AND THIS
TO REWRITE THIS INTEGRAL IN TERMS OF U.
AND BECAUSE WE'RE GOING TO WRITE THIS IN TERMS OF U,
WE'LL LEAVE OFF THE LIMITS OF INTEGRATION TEMPORARILY
BECAUSE THESE ARE IN TERMS OF "T".
LET ME GO AHEAD AND PULL OUT THE 2,400
x THE INTEGRAL OF E TO THE U.
THEN AGAIN DT IS EQUAL TO DU DIVIDED BY 0.04.
SO WE'LL PUT THE DU HERE AND THEN WE'LL DIVIDE THIS BY 0.04.
SO NOW WE'RE GOING TO HAVE 2,400 DIVIDED BY 0.04.
AND THE ANTIDERIVATIVE OF E TO U IS JUST E TO U
WHICH IS E TO THE 0.04T.
AND NOW WE NEED TO EVALUATE THIS AT 4 AND 0.
SO WE'LL HAVE 2,400 DIVIDED BY 0.04
AND THEN WE'LL HAVE "E" TO THE 0.04 x 4 - E TO THE 0
AND THIS WILL GIVE US OUR FUTURE VALUE
OF THIS CONTINUOUS CASH FLOW UNDER THESE CONDITIONS.
SO NOW LET'S GO TO THE CALCULATOR.
SECOND NATURAL LOG BRINGS UP E TO THE POWER.
SO WE HAVE 0.04 x 4 - E TO THE 0 WHICH IS GOING TO BE 1.
SO THE FUTURE VALUE OF THIS ACCOUNT AFTER 4 YEARS
IS GOING TO BE $10,410.65 ROUNDED TO THE NEAREST PENNY.
I HOPE YOU FOUND THIS HELPFUL.