Ex: Find the Initial Value and Exponential Growth or Decay Rate Given an Exponential Function

Ex: Find the Initial Value and Exponential Growth or Decay Rate Given an Exponential Function


– IN THIS PROBLEM WE’RE GIVEN
SIX EXPONENTIAL FUNCTIONS. THREE OF THEM ARE EXPONENTIAL
GROWTH AND THREE OF THEM ARE EXPONENTIAL DECAY. FOR EACH FUNCTION WE WANT
TO FIND THE INITIAL VALUE AND ALSO THE GROWTH OR DECAY
RATE AS A PERCENTAGE. TO BEGIN, LETS IDENTIFY WHICH
FUNCTIONS ARE EXPONENTIAL GROWTH,
AND WHICH ARE EXPONENTIAL DECAY. FOR EXPONENTIAL GROWTH
IN THIS FORM HERE, THE BASE B IS CALLED THE GROWTH
FACTOR, WHICH IS EQUAL TO 1 + R, WHERE R IS THE GROWTH RATE
OF THE DECIMAL. SO BECAUSE B IS EQUAL TO 1 + R, THE BASE IS ALWAYS
GREATER THAN 1 AND, THEREFORE,
WE HAVE AN INCREASING FUNCTION. FOR EXPONENTIAL DECAY, THE BASE
B IS CALLED THE DECAY FACTOR, WHICH IS EQUAL TO 1 – R, WHERE R
IS THE DECAY RATE AS A DECIMAL. SO BECAUSE WE’RE SUBTRACTING
R HERE FOR EXPONENTIAL DECAY, THE BASE IS ALWAYS
LESS THAN ONE AND, THEREFORE,
WE HAVE A DECREASING FUNCTION. SO LOOKING AT OUR SIX FUNCTIONS
NOTICE HOW HERE THE BASE IS 1.2, WHICH IS GREATER THAN 1, SO THIS
WOULD BE EXPONENTIAL GROWTH. NEXT WE HAVE A BASE OF 0.86,
WHICH IS LESS THAN 1, THIS WOULD EXPONENTIAL DECAY. NEXT WE HAVE A BASE OF 0.4,
THIS IS ALSO EXPONENTIAL DECAY. NEXT WE HAVE A BASE OF 1.38,
WHICH IS GREATER THAN 1, SO THIS WOULD BE EXPONENTIAL
GROWTH. NEXT WE HAVE A BASE OF 3,
WHICH IS GREATER THAN 1, THIS IS EXPONENTIAL GROWTH. AND, FINALLY,
WE HAVE A BASE OF 0.92, SO WE HAVE EXPONENTIAL DECAY. NOTICE IN ALL OF THESE
EXPONENTIAL FUNCTIONS “A” IS THE INITIAL AMOUNT
OR INITIAL VALUE, WHICH WE CAN EASILY IDENTIFY. SO FOR THIS FIRST FUNCTION, NOTICE HOW THE INITIAL VALUE OR
THE INITIAL AMOUNT WOULD BE 310. AND NOW FOR THE GROWTH RATE WE CAN FIND THAT USING
THE EQUATION B=1 + R, SINCE WE KNOW B IS EQUAL TO 1.2. WE CAN PROBABLY DETERMINE THIS
VALUE MENTALLY, BUT LET’S GO AHEAD AND SHOW SOME
WORK ON THE NEXT SLIDE. AGAIN, FOR EXPONENTIAL GROWTH WE
KNOW THAT THE BASE B OR THE GROWTH FACTOR
IS EQUAL TO 1 + R, WHERE FOR OUR FIRST EQUATION
B IS EQUAL TO 1.2. SO WE’D HAVE 1.2=1 + R, WHICH, AGAIN, WE CAN PROBABLY
TELL R MUST BE 0.2 OR WE CAN SUBTRACT 1 ON BOTH
SIDES, GIVING US R=0.2. SO THIS IS THE DECIMAL FORM
OF THE GROWTH RATE, BUT WE WANT THIS AS A
PERCENTAGE, NOT AS A DECIMAL. SO TO CONVERT A DECIMAL
TO A PERCENTAGE WE MULTIPLY BY 100
AND ADD A PERCENT SIGN, WHICH IS EQUIVALENT TO MOVING THE DECIMAL POINT TO THE RIGHT
TWO PLACES, GIVING US 20%. SO FOR OUR FIRST EQUATION
WE HAVE A GROWTH RATE OF 20%. OUR NEXT FUNCTION IS EXPONENTIAL
DECAY. THE INITIAL AMOUNT OR INITIAL
VALUE IS STILL 15, AND NOW WE’LL SHOW SOME WORK
TO FIND THE DECAY RATE. FOR EXPONENTIAL DECAY THE BASE B
OF THE DECAY FACTOR IS EQUAL TO 1 – R,
WHERE B IS EQUAL TO 0.86, SO WE HAVE 0.86=1 – R. AND, AGAIN, WE PROBABLY
CAN SOLVE THIS MENTALLY AND RECOGNIZE THAT R
IS EQUAL TO 0.14, BUT IF NOT, WE CAN ALWAYS SOLVE
THIS AS AN EQUATION. SUBTRACT 1 ON BOTH SIDES. THIS WOULD GIVE US -R=-0.14, DIVIDE BY -1
AND WE HAVE R=0.14, BUT THIS IS THE DECIMAL FORM AND
WE WANT THE PERCENT. SO 0.14 x 100 WOULD GIVE US 14,
ADD THE PERCENT SIGN. SO WE HAVE A DECAY RATE OF 14%. NEXT WE HAVE ANOTHER EXPONENTIAL
DECAY FUNCTION WHERE THE DECAY FACTOR IS 0.4. SO WE’D HAVE 0.4 MUST EQUAL
1 – R. AGAIN, WE PROBABLY RECOGNIZE
THAT R IS GOING TO BE 0.6, OR 60%, BUT IF WE DON’T WE CAN
SHOW THE WORK. -R=-0.6, DIVIDE BY -1. WE HAVE R=0.6,
WHICH WOULD BE 0.6 x 100. AS A PERCENTAGE THAT’D BE 60%. NOTICE HOW THIS IS THE SAME
AS MOVING THE DECIMAL POINT TO THE RIGHT TWO PLACES. SO OUR DECAY RATE,
AS A PERCENTAGE, IS 60% AND THE INITIAL VALUE
OR INITIAL AMOUNT IS 112. OUR NEXT FUNCTION
IS EXPONENTIAL GROWTH WHERE THE INITIAL VALUE IS 310, SO IF B OR THE GROWTH FACTOR
IS 1.38, WE SHOULD BE ABLE TO RECOGNIZE
THAT R WOULD BE 0.38, OR 38%. BUT, AGAIN, I’LL GO AHEAD
AND SHOW THE WORK. WE WOULD HAVE 1.38=1 + R, SUBTRACTING 1 ON BOTH SIDES
GIVES US 0.38=R AND 0.38 x 100 WOULD BE 38
OR 38%. WE HAVE ANOTHER EXPONENTIAL
GROWTH FUNCTION HERE WHERE THE INITIAL VALUE IS 528, AND LET’S GO AHEAD AND SHOW SOME
WORK FOR THE GROWTH RATE. NOTICE HOW IF THE GROWTH FACTOR
OR THE BASE IS 3, WE’D HAVE 3=1 + R, SUBTRACTING
1 ON BOTH SIDES WE HAVE R=2. SO THE GROWTH RATE AS A DECIMAL
IS 2. WELL, 2 x 100=200 OR 200%,
WHICH IS OUR GROWTH RATE AS A PERCENTAGE. OUR LAST FUNCTION IS EXPONENTIAL
DECAY WITH AN INITIAL VALUE OF 90, AND THE DECAY FACTOR OR THE BASE
B IS EQUAL TO 0.92. SO IF WE RECOGNIZE THE DECAY
RATE IS GOING TO BE .08 OR 8%, THAT’S PERFECT. IF NOT, WE CAN SHOW THE WORK.
WHERE WE’D HAVE B=0.92=1 – R, SUBTRACT 1 ON BOTH SIDES, -R=- 0.08 AND DIVIDE BY -1. SO WE HAVE R=0.08
AS A DECIMAL. AS A PERCENTAGE,
MULTIPLY BY 100, THIS WOULD GIVE US 8%. SO OUR DECAY RATE EQUALS 8%. I HOPE YOU FOUND THIS HELPFUL.  

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