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8.5 and 8.6 Exponential Growth and Decay Functions. Internet Traffic. In 1994 , a mere 3 million people were connected to the Internet. By the end of 1997 , more than 100 million were using it. Traffic on the Internet has doubled every 100 days .
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Internet Traffic • In 1994, a mere 3 million people were connected to the Internet. • By the end of 1997, more than 100 million were using it. • Traffic on the Internet has doubled every 100 days. • Source: The Emerging Digital Economy, April 1998 report of the United States Department of Commerce.
Internet Technology • The Internet is growing faster than all other technologies that have preceded it. • Radio existed for 38 years before it had 50 million listeners. • Television took 13 years to reach that mark. • The Internet crossed the line in just four years.
Exponential Functions • A function is called an exponential function if it has a constantgrowth factor. • This means that for a fixed change in x, y gets multiplied by a fixed amount. • Example: Money accumulating in a bank at a fixed rate of interest increases exponentially. In some cases.
Exponential Functions • Consider the following example, is this exponential?
Exponential Functions • For a fixed change in x, y gets multiplied by a fixed amount.If the column is constant, then the relationship is exponential.
Exponential Functions • Consider another example, is this exponential?
Exponential Functions • For a fixed change in x, y gets multiplied by a fixed amount.If the column is constant, then the relationship is exponential.
Other Examples of Exponential Functions • Populations tend to grow exponentially not linearly. • When an object cools (e.g., a pot of soup on the dinner table), the temperature decreases exponentially toward the ambient temperature. • Radioactive substances decay exponentially. • Viruses and even rumors tend to spread exponentially through a population (at first).
Exponential Growth • Exponential growth occurs when some quantity regularly increases by a fixed percentage. • The equation for an exponential relationship is given by • where A is the initial value of y, and ris that rate of growth. • An example of the equation of the last relationship above is simply y = 500 (1+.03)t.
Example: In 2000, the U.S. population was 282 million. The U.S. population has been growing by about 0.8% each year. In this case, population A is growing byr % each year. After one year, population A will become
Exponential Functions • If a quantity grows by a fixed percentage change, it grows exponentially. • Example: Bank Account • Suppose you deposit $100 into an account that earns 5% annual interest. • Interest is paid once at the end of year. • You do not make additional deposits or withdrawals. • What is the amount in the bank account after eight years?
Economists refer to inflation as increases in the average cost of purchases. The formula C = c(1 + r)n can be used to predict the cost of consumer items at some projected time. In this formula C represents the projected cost of the item at the given annual inflation rate, c the present cost of the item and r is the rate of inflation (in decimal form), and n is the number of years for the projection. Suppose a gallon of milk costs $2.69 now. How much would the price increase in 6 months with an inflation rate of 5.3%?
Exponential Decay • Exponential Decay occurs whenever the size of a quantity is decreasing by the same percentage each unit of time. • The best-known examples of exponential decay involves radioactive materials such as uranium or plutonium. • Another example, if inflation is making prices rise by 3% per year, then the value of a $1 bill is falling, or exponentially decaying, by 3% per year.
Exponential Decay: Example • China’s one-child policy was implemented in 1978 with a goal of reducing China’s population to 700 million by 2050. China’s 2000 population is about 1.2 billion. Suppose that China’s population declines at a rate of 0.5% per year. Will this rate be sufficient to meet the original goal?
Exponential Decay: Solution The declining rate = 0.5%/100 = 0.005 Using year 2000 as t = 0, the initial value of the population is 1.2 billion. We want to find the population in 2050, therefore, t = 50 New value = 1.2 billion × (1 – 0.005)50 New Value = 0.93 billion ≈ 930 million
Exponential Decay • The fixed amount of time that it takes a quantity to halve is called its half-life. • Suppose Putonium has a half life of 50 years and you start with 100 grams of plutonium. How what is the rate of decay?
Exponential Review • If the factor b is greater than 1, then we call the relationshipexponential growth. • If the factor b is less than 1, we call the relationshipexponential decay.
