1. Exponential Functions An exponential function is a function of the form the real constant a is called the base, and the independent variable x may assume any real value. The graph of y = 2 x is shown below. It is increasing as are all exponential functions with base > 1.

2. More about Exponential Functions The graph of y = is shown next. It is decreasing as are all exponential functions with 0 < base < 1. Since exponential functions are increasing or decreasing, it follows that they are one-to-one. Why? By examining the graph we conclude that the range of an exponential function is the set of positive real numbers.

3. More about Exponential Functions The graph of always passes through the points (0, 1) and (1, a). The graph of is the reflection about the y-axis of the graph of The graph of has the horizontal asymptote y = 0. If there are two exponential functions and a < b, then

4. Solving Exponential Equations If   Example. Solve 3 10 = 3 5x. By the previous bullet points, Example. Solve. 2 7 = (x–1) 7. By the previous bullet points,

5. The Number e As the real number m gets larger and larger, The limiting value 2.71828... is an irrational number known as e. In order to simplify certain formulas, exponential functions are often written with base e. For x > 0, 2 x < e x < 3 x.

6. Compound Interest When the money in an account receives compound interest, each interest payment includes interest on the previously accrued interest. Example. $100 compounded annually at 10% interest for 3 years, and P dollars compounded annually at r% interest for 3 years YearStarting Amount Ending AmountStarting Amount Ending Amount 1 100100(1+0.1) = 110PP(1+r) 2 110110(1+0.1) = 121P(1+r)P(1+r) 2 3 121121(1+0.1) = 133.10P(1+r) 2 P(1+r) 3

7. Compound Interest Formula If the interest on an account at r% annually is compounded k times per year, the interest rate applied to each accounting period is r/k. When k = 2, we say that interest is compounded semiannually, when k = 4, we say that interest is compounded quarterly, and when k = 12, we say that interest is compounded monthly. In general, if P dollars are invested at an annual interest rate r (expressed in decimal form) compounded k times annually, then the amount A available at the end of t years is

8. Compound Interest Example Suppose that $6000 is invested at an annual rate of 8%. What will be the value of the investment after 3 years if (a) interest is compounded quarterly? (b) interest is compounded semiannually? In which case, (a) or (b), is the total amount of interest greater? Why?

9. Continuous Compounding Suppose we let the number of compounding periods k increase without bound. (imagine compounding every second, then every millisecond, etc.). The amount of the investment of P dollars after t years approaches a limit: When the above situation pertains, we say that we are compounding continuously. In general, if P dollars are invested at an annual interest rate r (expressed in decimal form) compounded continuously, then the amount A available at the end of t years is

10. Compound Interest Examples Example. Suppose that $6000 is invested at an annual rate of 8%. What will be the value of the investment after 3 years if interest is compounded continuously? Note that the amount of the investment after 3 years is greater than it was when compounding was done semiannually or quarterly? Why? Example. Suppose that a principal P is to be invested at continuous compound interest of 8% per year to yield $10,000 in 5 years. How much should be invested?

11. Exponential Growth Model--World Population A model which predicts the quantity Q, which is number or biomass, for a population at time t is the following exponential growth model: Both q 0 and k are constants specific to the particular population in question, and k is called the growth constant. For the world population, k = 0.019 and q 0 = 6 billion when t = 0 corresponds to the year 2000. The model is: In the year 2010, the model predicts a world population of

12. Exponential Decay Model A model which predicts the quantity Q, which is mass, for a particular radioactive element at time t is the following exponential decay model: Both q 0 and k are constants specific to the particular radioactive sample in question, and k is called the decay constant. We use the term half-life to describe the time it takes for half of the atoms of a radioactive element to break down. A radioactive substance has a decay rate of 5% per hour. If 500 grams are present initially, how much remains after 4 hours?

13. Summary of Exponential Functions; We discussed Definition of an exponential function and its base Fact that exponential functions are increasing or decreasing and therefore they are one-to-one Range of an exponential function Horizontal asymptote of an exponential function Solving exponential equations The number e The formula for compound interest The formula for continuous compounding Exponential growth and decay