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1 Exponential Functions The greatest shortcoming of the human race is our inability to understand the exponential function. - Albert A. Bartlett The function f(x) = 2 x, where the power is a variable x, is an example of an exponential function. Exponential functions take the general form of A(t) = P b t where b is positive and P is some constant. We call b the base of this exponential function. You may remember the exponential function as P e rt, we ll discuss this ambiguity later. Review the rules of exponents on page 138 of the text. Please make sure you re completely comfortable with algebraically manipulating exponents as we will use exponential functions regularly. We will commonly use the following exponent rules. 1 a n = a n a n a m = a n+m a 0 = 1 a n a m = an m a nm = a nm a = a 1/2 where a > 0, n, and m are real numbers. Given a general exponential function of the form P b t the constant P represents the constant b represents In particular, for: b > 1 we have growth/decay(circle one), and for b < 1 we have growth/decay(circle one). Plot some examples of functions to understand how values of b effect exponential growth and decay. 8

2 1. (a) The figure below gives the plot of 5 x and e x. Label the two functions. Note e (b) The figure below gives the plot of 0.7 x and 0.9 x. Label the two functions. Exponential functions change by a constant percent (or relative amount) unlike linear functions which change by a constant amount. 2. Consider the following table. Up to 2 decimal places, does this growth appear to be linear? If not, what type of function fits this data? Justify your answer(s) mathematically. Year Population(millions)

3 Fitting an Exponential Function Example 1 The U.S. population was 180 million in 1960 and 309 million in Use the data to give an exponential growth model showing the U.S. population A as a function of time t in years since Use your model to predict the population in Solution 3. The amount of a hormone in the body can change rapidly. Suppose the initial amount is 20 mg. Find a formula for H, the amount in mg, at a time t minutes later if H is (a) Increasing by 0.4 mg per minute. (b) Increasing by 3% per minute. (c) Decreasing by 0.4 mg per minute. (d) Decreasing by 3% per minute. As we ve now seen, an exponential function s behavior is completely determined by the two parameters, A and b. 10

4 Compound and Continuous Exponential Growth Exponential functions also characterize many financial applications. For example we may have an account which earns interest at a certain percentage at the end of some period of time(e.g. once every month). For these situations it helps to generalize our formula Ab t to so we can modify the terms for interest rate and number of times we are compounding. The standard equation that describes this type of exponential growth is given by ( A(t) = P 1 + r ) mt m where A(t) represents the future amount given a principal amount P, compounded m times per year at a rate r given as decimal. Note: If we choose m = 1 we get our original form P b t. Thus we can think of our original formula as compounding yearly, or annually. 4. Shari invests $5000 into an account that pays an interest rate of 2.15% compounded quarterly. What is the value of the account after 5 years? Continuously Compounded Growth In many cases, such as in biological organisms, it is unatural to describe growth( or interest ) that compounds at fixed times. Certain types of growth occur not at fixed times but at every moment in time. This type of growth is known as continuously compounding growth. Continuous compounding can be described by the formula where A(t) is the future amount given a principal amount P, compounded continuously at a rate r given as decimal. A(t) = P e rt 11

5 Notice that this form looks different than our original general form P b t however using exponent rules we have that A(t) = P e rt = P (e r ) t so we see e r is equivalent to b in our original formula P b t. We chose to use either form depending on the problem we want to solve. Some problems may ask for you to convert an annual percent growth to a continuous percent growth rate. In that case simply set the two formulas equal and solve for the unknown rate. Example 2 What continuous percent decay rate is equivalent to an annual percent decay rate of 11%? Solution Exponential Formula Summary At this point we ve discussed several different formulas for exponential functions so it can be confusing to know what to use in any particular situation. Below you will find some general tips for determining which form is most appropriate. General Exponential(Annual Growth) A(t) = P b t Use this general form for problems that specify exponential growth but give no explicit details about the rate of growth. e.g. Example 1 Discrete Compounding Growth ( A(t) = P 1 + r ) mt m Use this form for problems that specify a discrete time that an amount is compounded. e.g....an account compounds monthly... Continuously Compounding Growth A(t) = P e rt Use this form for problems that specify the exponential growth is continuous or provide a context where growth should be happening all the time. e.g....the exponential growth of a disease... 12

6 5. Suppose that the population of starfish in a pond decreased exponentially from 573 in 1980 to 300 in 1995 and continued to decrease at the same percentage rate between 1995 and 2000, calculate what the starfish population would be in The population of a city is decaying according to the formula P (t) = 43, 021e 0.2t where t is years after Determine the annual decay rate. Ideas From Today... exponent rules, exponential function, base, initial value, exponential decay, exponential growth, types of compounding For Next Time... review the logarithm and the natural logarithm 13

Daily Outcomes: I can evaluate, analyze, and graph exponential functions. Why might plotting the data on a graph be helpful in analyzing the data?

3 1 Exponential Functions Daily Outcomes: I can evaluate, analyze, and graph exponential functions Would the increase in water usage mirror the increase in population? Explain. Why might plotting the data