Exponential Functions Exponential functions are functions that change at a constant percent rate. This percent rate will be called the growth rate when it results in an increase and the decay rate when it results in a decrease. Example. If you have a starting annual salary of $40000, and you get a 6% raise each year, what will your resulting salary be in subsequent years? Here, the growth rate is 6%. After 1 year, your salary is (1.06)(40000) = After 2 years, your salary is (1.06)(42400) = and so on... After t years, your salary is (1.06) t (40000).

Use of Maple to graph the salary from the previous slide > plot(40000*(1.06)^t,t=0..20,color=black);

Example. Suppose that you invest $10000 in the latest “dotcom” venture which allows two people anywhere to get married on the web. Unfortunately, you find that the value of your investment at the end of each year is 5% less than it was at the beginning of the year. Here, the decay rate is 5%. After 1 year, the value is (0.95)(10000) = 9500 After 2 years, the value is (0.95)(9500) = 9025 and so on... After t years, the value is (0.95) t (10000).

Use of Maple to graph the investment value of previous slide >plot(10000*(0.95)^t,t=0..80,color=black);

For an exponential function, we have New amount = (growth factor) (Old amount) where the old amount is present at the beginning of a period and the new amount is present at the end of a period. In the salary example, the growth factor is 1.06 per year. In the investment example, the growth factor is 0.95 per year. Note that we use the term “growth factor” even though the value of the investment is decreasing or decaying.

The formula for an exponential function Q = f(t) is given by where the parameter a is the initial value of Q (at t = 0) and the parameter b is the growth factor. If a > 0, then b > 1 gives exponential growth and 0 < b < 1 gives exponential decay. The growth factor or base is given by where r is the decimal representation of the percent rate of change. In the salary example, a = 40000, b = 1.06, and r = In the dotcom example, a = 10000, b = 0.95, and r = –0.05.

Comparing exponential and linear functions For a table that gives y as a function of x with x constant: If the difference of consecutive y-values is constant, the table could represent a linear function. If the ratio of consecutive y-values is constant, the table could represent an exponential function. Example. Which function is linear? exponential? x f(x) g(x)

Finding a formula for an exponential function To find a formula for the exponential function on the previous slide, we must determine the values of a and b in the formula g(x) = ab x. From the table, we have From the formula, we have Thus, Finally, we solve a( ) 20 = 1000 for a, which yields

Comparison of exponential and linear functions The 19th century clergyman and economist, Thomas Malthus, assumed that the food supply increases linearly and the population increases exponentially. This assumption combined with the following fact led to a gloomy prediction of the future. It can be shown that an exponentially increasing quantity will, in the long run, always outpace a linearly increasing quantity. For the example shown in the table below, the exponential outpaces the linear function at some t value between 4 and 5. t t t

Parameters for exponential and linear functions The general formulas for exponential and linear functions both have two parameters. Hence, two data points are sufficient to determine a particular function of either type. The two parameters for linear functions are the slope and the vertical intercept. The parameters a and b in the formula ab t determine the shape of the corresponding graph. In the current lecture, we assume that a > 0. A graph with a 0. If a = 0, the formula yields the zero function. As we did before, we assume that b > 0. Note that the value of an exponential function at 0 is ab 0 = a.

Graphs of exponential functions Q = f(t) = ab t with b > 1 These functions are both increasing since their graphs rise as t increases. Note that the greater the value of b, the more rapidly the graph rises. These graphs satisfy and we say that the graph has the t-axis as a horizontal asymptote. Alternatively, we write Q = 50(1.2) t Q = 50(1.4) t

Graphs of exponential functions Q = f(t) = ab t with 0 < b < 1 These functions are both decreasing since their graphs fall as t increases. Note that the smaller the value of b, the more rapidly the graph falls. These graphs satisfy and we again say that the graph has the t-axis as a horizontal asymptote. Alternatively, we write Q= 50(0.8) t Q= 50(0.6) t

Concavity and rates of change Again consider the salary example. Since the rate of change increases with time, the graph bends upward. We say such graphs are concave up. 4 years smaller increase larger increase

Concavity and rates of change, continued 1 hr Larger increase time (hours) distance (miles) Exponential functions Q = ab t with a > 0 are all concave up. We can consider the graph showing the distance traveled by a certain bicycle rider as a function of time. Since the bicycle rider’s rate of change decreases with time, the graph bends downward. We say such graphs are concave down. 1 hr Smaller increase

Motivating the number e using compound interest Suppose we invest $1.00 at 100% interest once a year. At the end of the year, we have $2.00. Suppose we invest $1.00 at 50% interest twice a year. At the end of the year, we have $2.25. Suppose we invest $1.00 at 25% interest four times a year. At the end of the year, we have $ As the frequency of computation increases, the balance at the end of the year approaches $ , and this limiting value is referred to as the number e.

The number e = is an irrational number which is often used as the base of an exponential function. Base e is often referred to as the “natural base”. Example. Solve graphically for x: e x = 100. Using a calculator to find the intersection, x ≈ e x x = ?

Continuous growth and the number e If Q = ab t, b > 0, this relation can be rewritten as: where e k = b. The value k is called the continuous growth rate. The value of the continuous percent growth rate may be given as a decimal or a percent. If t is in years, for example, then the units of k are per year; if t is in minutes, then k is per minute. Example. If you have a starting annual salary of $40000, and you get a 6% raise each year, what is the continuous growth rate of your salary, Q? We have Q = 40000(1.06) t, so a = and b = Using a calculator, we can solve e k = 1.06 to get:

Compound Interest What if your salary increase was not 6% compounded yearly, but 6% compounded over a shorter period? Then the formulas used to compute your salary might be:

Effective annual yield When the interest (or salary increase) of an account is compounded more frequently than once a year, the account effectively earns more than the nominal rate, which is 6% in the salary example. The effective annual rate gives the interest (or salary increase) actually earned by the account. Example. When your salary is compounded monthly (see previous slide), after one year you have: and the effective annual rate is 6.168%. When your salary is compounded continuously, after one year you have: and the effective annual rate is 6.184%.

General formulas for compounding If interest at an annual (nominal) rate of r is compounded n times a year, then r/n times the current balance is added n times a year. Therefore, with an initial deposit of $P, the balance t years later is: If interest on an initial deposit of $P is compounded continuously at an annual (nominal) rate of r, the balance t years later can be calculated using the formula:

Summary for exponential functions