Applications of Exponential Functions

Definition (Growth Functions) A function of the form where and is called a growth function. The domain is the set of all real numbers and the range is The asymptote is the -axis (negative half ) and the intercept is Growth functions are continuous functions on A growth function that is modeling a physical situation is called a growth model.

Definition (Decay Functions) A function of the form where and is called a decay function. The domain is the set of all real numbers and the range is The asymptote is the -axis (positive half ) and the intercept is Decay functions are continuous functions on A decay function that is modeling a physical situation is called a decay model.

Example (Applications of the Growth and Decay Functions)

(a) If is invested for years at compounded quarterly, the interest earned is What is the interest earned after 15 years.

    Solution. The interest earned is

(b) The percent concentration of a certain drug in the bloodstream at any time is given by the equation Graph this equation for

    Solution. There is no intercept in this domain, the horizontal asymptote is the line the intercept is and the graph is

applications of exponential functions _gr_33.gif]

(c) The demand function for a product is given by where is the number of units and is the price per unit. (i) At what price per unit will the quantity demanded equal 6 units? (ii) If the price is per unit, how many units will be demanded, to the nearest unit?

    Solution. The price per unit when the quantity demanded is 6 units is found by If the price is then we will solve   for First, dividing by 4000 and then converting to logarithmic form we obtain: Multiplying by we obtain, or units.

(d) Say the demand function for a product is given by



(i) What will be the price if 19 units are demanded? (ii) How many units, to the nearest unit, will be demanded if the price is $29.40?

    Solution. If 19 units are demanded then the price will be If the price is $29.40 then the number of units demanded will satisfy the equation Solving for we have,

(e) If the supply function for a product is given by where represents the number of hundreds of units, what will be the price when the producers are willing to supply 600 units?

    Solution. We have

(f) If the total cost function for a product is given by where is the number of items produced, what is the total cost of producing 30 units?

    Solution. We have,

(g) If the demand function for a product is given by where is the price per unit when units are demanded, what is the total revenue when 40 units are demanded and supplied?

    Solution. The price per unit is so when 40 units are supplied the price is per unit. So the total revenue is

(h) If is invested at compounded continuously, the future value at any time (in years) is given by (i) What is the amount after year? (ii) How long before the investment doubles?

    Solution. After one year, we have The investment doubled when and so we solve for in We have, or 7 years.

(i) By using data from the U.S. Bureau of Labor Statistics for the years 1968-2000, the purchasing power of a 1983 dollar can be modeled with the function where is the number of years past 1960. (i) Find and   and for each, write a sentence that interprets its meaning. (ii) How long before it will cost to purchase goods that cost in 1983?

    Solution. We have and As years go by the power of the $1 of 1983 loses its power. This can also be seen from the graph of :

applications of exponential functions _gr_91.gif]

We want to solve for in the equation to find out when the is to purchase goods that cost in 1983. We have, or 52.5 years. The year will be