- WE WANT TO WRITE THE EXPONENTIAL GROWTH FUNCTIONS
GIVEN THE INITIAL VALUE AND THE GROWTH RATE EXPRESSED
AS A PERCENTAGE.
WE'LL WRITE THE EXPONENTIAL GROWTH FUNCTIONS
IN THE FORM OF F OF X = "A" x B RAISED TO THE POWER OF X
WHERE "A" IS THE INITIAL AMOUNT AND B THE BASE
ALSO CALLED THE GROWTH FACTOR IS EQUAL TO 1 + R
WHERE R IS THE GROWTH RATE EXPRESSED AS A DECIMAL.
SO LOOKING AT THIS FIRST COLUMN
NOTICE HOW WE'RE GIVEN THE INITIAL VALUE
WHICH WOULD BE "A" IN THE EXPONENTIAL FUNCTION.
NOTICE HOW THE NEXT COLUMN IS THE GROWTH RATE
EXPRESSED AS A PERCENTAGE BUT TO FIND THE GROWTH FACTOR
WE NEED THE GROWTH RATE AS A DECIMAL.
SO FOR THE NEXT STEP,
LET'S CONVERT THESE PERCENTAGES TO DECIMALS.
REMEMBER A PERCENT IS A NUMBER COMPARED TO 100.
SO 18% IS EQUIVALENT TO 18 HUNDREDTHS OR 18 DIVIDED BY 100.
FROM HERE WE CAN PERFORM THIS DIVISION
OR RECOGNIZE THAT DIVIDING BY 100 IS EQUIVALENT
TO MOVING THE DECIMAL POINT TO THE LEFT TWO PLACES
GIVING US 0.18.
OF COURSE, WE COULD CHECK THIS ON TH CALCULATOR
18 DIVIDED BY 100 IS 0.18.
SO FOR 4.5% AS A DECIMAL WE'D HAVE 4.5 DIVIDED BY 100
WHICH WOULD BE EQUIVALENT TO 0.045
AND THEN FOR THE GROWTH RATE OF 115%
THIS IS GOING TO BE 115 DIVIDED BY 100 WHICH WOULD BE 1.15.
NOW THAT WE HAVE EACH GROWTH RATE AS A DECIMAL
THIS COLUMN NOW GIVES US R.
AGAIN WHERE THE GROWTH RATE R IS EXPRESSED AS A DECIMAL.
SO NOW WE HAVE R HERE, HERE, AND HERE.
SO NOW BEFORE WE WRITE THE EQUATION
WE JUST NEED TO FIND B WHERE B IS 1 + R.
SO FOR THIS FIRST ROW,
F OF X = "A" THE INITIAL VALUE OF 550 x 1
+ R RAISED TO THE POWER OF X
OR 1 + 0.18 RAISED TO THE POWER OF X
AND THEREFORE OUR EXPONENTIAL GROWTH FUNCTION IS F OF X = 550
x 1.18 RAISED TO THE POWER OF X.
FOR THE NEXT ROW, F OF X
IS EQUAL TO THE INITIAL VALUE OF 2,200 x 1 + 0.045
RAISED TO THE POWER OF X
AND THEREFORE F OF X = 2,200 x 1.045 RAISED TO THE POWER OF X.
AND THEN FOR THE LAST ROW,
WE'D HAVE F OF X EQUALS 192 x 1 + 1.15
RAISED TO THE POWER OF X AND THEREFORE THE EXPONENTIAL GROWTH
FUNCTION IS F OF X = 192 x 2.15 RAISED TO THE POWER OF X.
I DO WANT TO MAKE ONE COMMENT
ON HOW TO ENTER THESE FUNCTIONS INTO OUR ONLINE HOMEWORK.
THE HOMEWORK SYSTEM ALREADY CONTAINS F OF X EQUALS
SO FOR EXAMPLE, TO ENTER THIS FIRST EXPONENTIAL FUNCTION
WE CAN JUST USE THE MATH PALETTE OR FROM THE KEYBOARD
WE WOULD ENTER 550 (1.18),
THE EXPONENT KEY OR THE CARAT WHICH IS SHIFT 6, AND THEN X.
AND REMEMBER TO ALWAYS USE THE PREVIEW BUTTON TO SEE
HOW THE COMPUTER WILL INTERPRET YOUR ENTRY.
I HOPE YOU FOUND THIS HELPFUL.