## 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 […]

## Ex: Exponential Growth Function – Population

– A GROWING CITY HAD A POPULATION OF 500,000 IN 2005, IN 2010 THE POPULATION WAS 760,000. WE WANT TO ASSUME EXPONENTIAL GROWTH AND THEN EXPRESS THE POPULATION AFTER T YEARS AS A FUNCTION OF T, PREDICT THE POPULATION IN 2025, AND THEN DETERMINE IN WHICH YEAR THE POPULATION WILL REACH 1,000,000. SO WE WANT TO START BY […]

## Ex: Exponential Growth of Bacteria (Intro Question)

– THE NUMBER OF BACTERIA IN A CULTURE IS N OF T AFTER T HOURS. SO THE FUNCTION THAT MODELS THE NUMBER OF BACTERIA, OR THE BACTERIA POPULATION, IS N OF T=500 x E RAISED TO THE POWER OF 0.15T. WHEN WE HAVE EXPONENTIAL GROWTH, WE CAN GATHER A LOT OF INFORMATION FROM THE EXPONENTIAL GROWTH FUNCTION. IN […]

## Ex: Limited Growth Differential Equation

– WHEN A PRODUCT IS ADVERTISED HEAVILY SALES WILL TEND TO GROW VERY QUICKLY BUT EVENTUALLY THE MARKET WILL REACH SATURATION AND THE SALES WILL SLOW. THIS TYPE OF GROWTH IS CALLED LIMITED GROWTH. FOR THE LIMITED GROWTH DIFFERENTIAL EQUATION IF A QUANTITY GROWS AT A RATE PROPORTIONAL TO THE DISTANCE FROM THE MAXIMUM VALUE M IT CAN […]

## Ex: Logistic Growth Differential Equation

– IF A QUANTITY GROWS AT A RATE PROPORTIONAL TO IT’S SIZE AND TO THE DISTANCE FROM THE MAXIMUM VALUE M IT CAN BE MODELED WITH LOGISTIC GROWTH USING THE DIFFERENTIAL EQUATION Y PRIME=R x Y x THE QUANTITY 1 – Y DIVIDED BY M WHERE WHY IS THE GROWTH RATE ABSENT CONSTRAINTS AND M IS THE MAXIMUM […]

## Ex: Inverse Variation Application – Number of Workers and Job Time

– THE TIME, T, REQUIRED TO DO A JOB VARIES INVERSELY WITH THE NUMBER OF PEOPLE, P, WORKING ON THE JOB. IF IT TAKES SIX HOURS FOR EIGHT WORKERS TO COMPLETE A JOB, HOW LONG WOULD IT TAKE IF THERE WERE NINE WORKERS? SO, SOMETIMES WHEN WE HAVE AN INVERSE VARIATION APPLICATION LIKE THIS, IT’S A LITTLE MORE […]

## Ex: Exponential Growth Function – Bacterial Growth

– A BACTERIA CULTURE STARTING WITH 200 BACTERIA GROWS AT A RATE PROPORTIONAL TO ITS SIZE. AFTER THREE HOURS, THERE WILL BE 900 BACTERIA. THIS IS AN EXAMPLE OF AN EXPONENTIAL GROWTH. THIS IS AN EXPONENTIAL GROWTH MODEL. SO WE’LL BE USING THE FUNCTION P OF T=P SUB 0 x E RAISED TO THE POWER OF KT TO […]

## Ex: Find an Exponential Growth Function Given Two Points – Initial Value Given

– IN THIS QUESTION WE’RE ASKED TO DETERMINE AN EXPONENTIAL FUNCTION THAT PASSES THROUGH THE POINTS (0, 15) AND (5, 480). WE ALSO WANT TO KNOW WHAT THE EXPONENTIAL GROWTH OR DECAY RATE IS. WE ALSO WANT TO FIND THE FUNCTION VALUE WHEN X=2. IF AN EXPONENTIAL FUNCTION IS IN THIS FORM HERE THEN THE BASE B IS […]