## 2.2 Exponential Functions and Models

(This topic is also in Section 2.2 in Applied Calculus and Section 10.2 in Finite Mathematics and Applied Calculus)

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### Goodies: On-line Technology

Brush up on your Algebra For all of the tutorials on nonlinear functions and models, you should be familiar with the algebra of exponents and radicals.

Exponential functions are frequently used to model the growth or depreciation of financial investments, population growth, radioactive decay, and phenomena where a quantity is allowed to undergo unrestrained growth.

Exponential Function

An exponential function of the variable x is a function that can be written in the form

 f(x) = Abx Function form y = Abx Equation form
where A and b are constants with b positive. We call b the base of the exponential function.
Technology formula: A*b^x

Examples
 1. f(x) = 2x A = 1, b = 2 Technology: 2^x 2. g(x) = 1000(1.05x) A = 1000, b = 1.05 Technology: 1000*1.05^x 3. h(x) = 4(32x) Rewrite this as 4((32)x) = 4(9x) A = 4, b = 32 = 9 Technology: 4*9^x or 4*3^(2*x)) 4. k(x) = 100(2-x) A =   b =

In the following practice question, you are required to fill in some values of an exponential function and recognize its graph from the data. (See p. 95 of Applied Calculus or p. 573 of Finite Mathematics and Applied Calculus for a discussion of this example.)

Let f(x) = 3(2x). Fill in the missing values in the following table, and then click on the correct graph.

x -3-2-10123
f(x) = 3(2x)
 38

12 24

Q Multiple Choice: For every 1-unit increase in x, f(x)

 Increases by 3 units. Increases by 2 units. Increases by a factor of 3. Increases by a factor of 2. Is squared. Is cubed.

### Distinguishing Exponential Growth from Linear Growth

In the following table, decide which of the rows represent exponential functions of x, which represent linear functions, and which represent neither.

x -3-2-10123
 Select one Linear Exponential Neither HELP!
1197531-1
 Select one Linear Exponential Neither HELP!
1/31392781243
 Select one Linear Exponential Neither HELP!
84210.50.250.125
 Select one Linear Exponential Neither HELP!
9410149
 Select one Linear Exponential Neither HELP!
0123456

### Fitting an Exponential Function to Two Data Points

We want to find an exponential equation of the form y = Abx for the curve passing through (1, 6) and (3, 8.64).

Substituting (x, y) = (1, 6) in the equation   y = Abx   yields
Substituting (x, y) = (3, 8.64) in the equation   y = Abx   yields
Now divide the second equation above by the first, and cancel the A.
Now solve the equation you just obtained for b.
 b =
Next, substitute your value of b into either the first or second equation to obtain A.
 A =
Finally, since you now have A and b, you can write down the exponential equation relating y and x:

### The Number e and Exponential Growth

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, then the compound interest formula tells us that, by the end of the year, you will have

1+ 1m
m

dollars.
Compounding more times a year increases the value of m. For very large values of m, something seems to happen -- try experimenting by entering larger and larger values of m (positive whole numbers only, please!)
m 110100100010000
 1 + 1m m
22.593742462.704813832.716923932.71814593

What you might notice is, that as m gets larger and larger, the value of our investment seems to approach a fixed number:

e = 2.71828182...
(A more accurate value is e = 2.71828182845904523536....) Press here for an extremely accurate approximation (50,000 decimal places!) obtained from http://www.cs.arizona.edu/icon/oddsends/other.htm.

Q What is the point of this number e?
A This number is extremely important to mathematicians and physicists. In calculus, e is the best behaved base for exponential functions. Besides its theoretical usefulness, e can be used to model continuous "exponential" growth:

Continuous Compounding: If \$P is invested at an annual interest rate r compounded continuously, the accumulated amount after t years is

A(t) = Pert           Technology : P*e^(r*t) or P*EXP(r*t)
Example for you:

Pablo Pelogrande
Pablo Pelogrande invests \$10,000 in Continuity Continental, which offers 8% per year with continuous compounding on deposits. How much will Pelogrande's investment be worth after 5 years? (Round to the nearest cent.)
 A =

You now have several options

• Try some of the questions in the true/false quiz (warning: it covers the whole Chapter) by pressing the button on the sidebar.
• Try some of the on-line review exercises (press the "Review Exercises" button on the side.)
• Try some of the exercises from Section 2.2 in Applied Calculus or Section or Section 10.2 of Finite Mathematics and Applied Calculus.

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Last Updated: August, 2004