3 The Laws of Exponents If b and c are positive and x and y are any real numbers, then the following laws hold: Law Quick Examples
4 Exponential Functions and Models Law Quick Examples
5 Exponential Functions and Models Exponential Function An exponential function has the form f (x) = Ab x, where A and b are constants with A 0 and b positive and not equal to 1. We call b the base of the exponential function. Quick Example f (x) = 2 x f (1) = 2 1 = 2 f (–3) = 2 –3 = f (0) = 2 0 = 1 Technology: A*b^x A = 1, b = 2;Technology: 2^x 2^1 2^(–3) 2^0
6 Exponential Functions from the Numerical and Graphical Points of View
7 The following table shows values of f (x) = 3(2 x ) for some values of x (A = 3, b = 2): Its graph is shown in Figure 10. Notice that the y-intercept is A = 3 (obtained by setting x = 0). Figure 10
8 Exponential Functions from the Numerical and Graphical Points of View In general: In the graph of f (x) = Ab x, A is the y-intercept, or the value of y when x = 0. What about b? Notice from the table that the value of y is multiplied by b = 2 for every increase of 1 in x. If we decrease x by 1, the y coordinate gets divided by b = 2. The value of y is multiplied by b for every one-unit increase of x.
9 Exponential Functions from the Numerical and Graphical Points of View On the graph, if we move one unit to the right from any point on the curve, the y coordinate doubles. Thus, the curve becomes dramatically steeper as the value of x increases. This phenomenon is called exponential growth.
10 Exponential Functions from the Numerical and Graphical Points of View Exponential Function Numerically and Graphically For the exponential function f (x) = Ab x : Role of A f (0) = A, so A is the y-intercept of the graph of f. Role of b If x increases by 1, f (x) is multiplied by b. If x increases by 2, f (x) is multiplied by b 2. If x increases by x, f (x) is multiplied by b x.... If x increases by 1, y is multiplied by b.
11 Exponential Functions from the Numerical and Graphical Points of View Quick Example Technology: 2^x; 2^(-x)
12 Exponential Functions from the Numerical and Graphical Points of View When x increases by 1, f 2 (x) is multiplied by. The function f 1 (x) = 2 x illustrates exponential growth, while illustrates the opposite phenomenon: exponential decay.
13 Example 1 – Recognizing Exponential Data Numerically and Graphically Some of the values of two functions, f and g, are given in the following table: One of these functions is linear, and the other is exponential. Which is which?
14 Example 1 – Solution Remember that a linear function increases (or decreases) by the same amount every time x increases by 1. The values of f behave this way: Every time x increases by 1, the value of f (x) increases by 4. Therefore, f is a linear function with a slope of 4. Because f (0) = 1, we see that f (x) = 4x + 1 is a linear formula that fits the data.
15 Example 1 – Solution On the other hand, every time x increases by 1, the value of g(x) is multiplied by 3. Because g(0) = 2, we find that g(x) = 2(3 x ) is an exponential function fitting the data. We can visualize the two functions f and g by plotting the data points (Figure 11). contd Figure 11
16 Example 1 – Solution The data points for f (x) clearly lie along a straight line, whereas the points for g(x) lie along a curve. The y coordinate of each point for g(x) is 3 times the y coordinate of the preceding point, demonstrating that the curve is an exponential one. contd
18 Applications Recall some terminology we mentioned earlier: A quantity y experiences exponential growth if y = Ab t with b 1. (Here we use t for the independent variable, thinking of time.) It experiences exponential decay if y = Ab t with 0 b 1.
19 Example 3(a) – Exponential Growth and Decay Compound Interest If $2,000 is invested in a mutual fund with an annual yield of 12.6% and the earnings are reinvested each month, then the future value after t years is which can be written as 2,000(1.0105 12 ) t, so A = 2,000 and b = 1.0105 12. This is an example of exponential growth, because b 1.
20 Example 3(b) – Exponential Growth and Decay Carbon Decay The amount of carbon 14 remaining in a sample that originally contained A grams is approximately C(t) = A(0.999879) t. This is an instance of exponential decay, because b 1. contd
22 The Number e and More Applications Suppose we invest $1 in the bank for 1 year at 100% interest, compounded m times per year. If m = 1, then 100% interest is added every year, and so our money doubles at the end of the year. In general, the accumulated capital at the end of the year is (1+1/m)^m
23 The Number e and More Applications Now, we are interested in what A becomes for large values of m. Below is an Excel sheet showing the quantity for larger and larger values of m.
24 The Number e and More Applications Something interesting does seem to be happening! The numbers appear to be getting closer and closer to a specific value. In mathematical terminology, we say that the numbers converge to a fixed number, 2.71828..., called the limiting value of the quantities. This number, called e, is one of the most important in mathematics. The number e is irrational, just as the more familiar number is, so we cannot write down its exact numerical value. To 20 decimal places, e = 2.71828182845904523536....
25 The Number e and More Applications We now say that, if $1 is invested for 1 year at 100% interest compounded continuously, the accumulated money at the end of that year will amount to $e = $2.72 (to the nearest cent). The Number e and Continuous Compounding The number e is the limiting value of the quantities as m gets larger and larger, and has the value 2.71828182845904523536... If $P is invested at an annual interest rate r compounded continuously, the accumulated amount after t years is A(t) = Pe r t.
26 The Number e and More Applications Quick Example If $100 is invested in an account that bears 15% interest compounded continuously, at the end of 10 years the investment will be worth A(10) = 100e (0.15)(10) = $448.17.
27 Example 5 – Continuous Compounding a. You invest $10,000 at Fastrack Savings & Loan, which pays 6% compounded continuously. Express the balance in your account as a function of the number of years t and calculate the amount of money you will have after 5 years. b. Your friend has just invested $20,000 in Constant Growth Funds, whose stocks are continuously declining at a rate of 6% per year. How much will her investment be worth in 5 years?
28 Example 5 – Continuous Compounding c. During which year will the value of your investment first exceed that of your friend?
29 Example 5(a) – Solution We use the continuous growth formula with P = 10,000, r = 0.06, and t variable, getting A(t) = Pe r t = 10,000e 0.06t. In five years, A(5) = 10,000e 0.06(5) = 10,000e 0.3 $13,498.59.
30 Example 5(b) – Solution Because the investment is depreciating, we use a negative value for r and take P = 20,000, r = –0.06, and t = 5, getting A(t) = Pe r t = 20,000e –0.06t A(5) = 20,000e –0.06(5) = 20,000e –0.3 $14,816.36. contd
31 Example 5(c) – Solution We can answer the question now using a graphing calculator, a spreadsheet, or the Function Evaluator and Grapher tool at the Website. Just enter the exponential models of parts (a) and (b) and create tables to compute he values at the end of several years: contd
33 Example 5(c) – Solution From the table, we see that the value of your investment overtakes that of your friend after t = 5 (the end of year 5) and before t = 6 (the end of year 6). Thus your investment first exceeds that of your friend sometime during year 6. contd
34 The Number e and More Applications Exponential Functions: Alternative Form We can write any exponential function in the following alternative form: f (x) = Ae rx where A and r are constants. If r is positive, f models exponential growth; if r is negative, f models exponential decay.
35 The Number e and More Applications Quick Examples 1. f (x) = 100e 0.15x 2. f (t) = Ae –0.000 121 01t Exponential growth A = 100, r = 0.15 Exponential decay of carbon 14; r = –0.000 121 01
37 Example 6 – Exponential Regression: Health Expenditures The following table shows annual expenditure on health in the United States from 1980 through 2009 (t = 0 represents 1980). a. Find the exponential regression model C(t) = Ab t for the annual expenditure. b. Use the regression model to estimate the expenditure in 2002 (t = 22; the actual expenditure was approximately $1,640 billion).
38 Example 6 – Solution a. We use technology to obtain the exponential regression curve (see Figure 13): C(t) 296(1.08) t b. Using the model C(t) 296(1.08) t we find that C(22) 296(1.08) 22 $1,609 billion which is close to the actual number of around $1,640 billion. Coefficients rounded Figure 13