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Published on March 11, 2014

Author: nkosi820401615



Exponential Functions More Mathematical Modeling

Internet Traffic  In 1994, a mere 3 million people were connected to the Internet.  By the end of 1997, more than 100 million were using it.  Traffic on the Internet has doubled every 100 days.  Source: The Emerging Digital Economy, April 1998 report of the United States Department of Commerce.

Derivatives of Exponential Functions

Exponential Functions  A function is called an exponential function if it has a constant growth factor.  This means that for a fixed change in x, y gets multiplied by a fixed amount.  Example: Money accumulating in a bank at a fixed rate of interest increases exponentially.

Exponential Function  An exponential equation is an equation in which the variable appears in an exponent.  Exponential functions are functions where f(x) = ax + B, where a is any real constant and B is any expression. For example, f(x) = e-x - 1 is an exponential function.  Exponential Function: f(x) = bx or y = bx, where b > 0 and b ≠ 1 and x is in R For example, f(x) = 2x g(x) = 10x h(x) = 5x+1

Exponential Equations with Like Bases  Example #1 - One exponential expression.  Example #2 - Two exponential expressions. Evaluating Exponential Function 32x 1 5 4 32x 1 9 32x 1 32 2x 1 2 2x 1 x 1 2 1. Isolate the exponential expression and rewrite the constant in terms of the same base. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation. 3x 1 9x 2 3x 1 32 x 2 3x 1 32x 4 x 1 2x 4 x 5

Exponential Equations with Different Bases  The Exponential Equations below contain exponential expressions whose bases cannot be rewritten as the same rational number.  The solutions are irrational numbers, we will need to use a log function to evaluate them.  Example #1 - One exponential expression. 32x 1 5 11 or 3x 1 4x 2 32 x 1 5 11 32 x 1 16 ln 32x 1 ln 16 (2x 1)ln3 ln16 1. Isolate the exponential expression. 3. Use the log rule that lets you rewrite the exponent as a multiplier. 2. Take the log (log or ln) of both sides of the equation.

Exponential Functions  Consider the following example, is this exponential? x y 5 0.5 10 1.5 15 4.5 20 13.5

Exponential Functions  For a fixed change in x, y gets multiplied by a fixed amount. If the column is constant, then the relationship is exponential. x y 5 0.5 10 1.5 1.5 / 0.5 3 15 4.5 4.5 / 1.5 3 20 13.5 13.5 / 4.5 3

This says that if we have exponential functions in equations and we can write both sides of the equation using the same base, we know the exponents are equal. If au = av, then u = v 82 43x The left hand side is 2 to the something. Can we re-write the right hand side as 2 to the something? 343 22 x Now we use the property above. The bases are both 2 so the exponents must be equal. 343x We did not cancel the 2’s, We just used the property and equated the exponents. You could solve this for x now.

Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent. x xf 2 Let’s look at the graph of this function by plotting some points. x 2x 3 8 2 4 1 2 0 1 -1 1/2 -2 1/4 -3 1/8 2-7 -6 -5 -4 -3 -2 -1 1 5 730 4 6 8 7 1 2 3 4 5 6 8 -2 -3 -4 -5 -6 -72 1 21 1 f Recall what a negative exponent means: BASE

x xf 2 x xf 3 Compare the graphs 2x, 3x , and 4x Characteristics about the Graph of an Exponential Function where a > 1 x axf What is the domain of an exponential function? 1. Domain is all real numbers x xf 4 What is the range of an exponential function? 2. Range is positive real numbers What is the x intercept of these exponential functions? 3. There are no x intercepts because there is no x value that you can put in the function to make it = 0 What is the y intercept of these exponential functions? 4. The y intercept is always (0,1) because a 0 = 1 5. The graph is always increasing Are these exponential functions increasing or decreasing? 6. The x-axis (where y = 0) is a horizontal asymptote for x - Can you see the horizontal asymptote for these functions?

The Rule of 72  If a quantity is growing at rate r% per year (or month, etc.) then the doubling time is approximately (72 ÷ r) years (or months, etc.)  For example, if a quantity grows at 8% per month, its doubling time will be about 72 ÷ 8 = 9 months.

Ex: All of the properties of rational exponents apply to real exponents as well. Lucky you! Simplify: 3232 555 Recall the product of powers property, am an = am+n

Ex: All of the properties of rational exponents apply to real exponents as well. Lucky you! Simplify: 10 2525 6 6)6( Recall the power of a power property, (am)n= amn

Application: Compound Interest  Suppose: - A: amount to be received P: principal r: annual interest (in decimal) n: number of compounding periods per year t: years n n r ptA 1)( Example  What would be the yield for the following investment? P = 8000, r = 7%, n = 12, t = 6 years 612 12 07.0 18000A ≈ $12,160.84

References  functions-24925841?qid=833b5856-9eca-411e-ab90- c8e1a4352e6b&v=default&b=&from_search=3  -functions-4772163?qid=833b5856-9eca-411e-ab90- c8e1a4352e6b&v=default&b=&from_search=8  exponential-functions?qid=833b5856-9eca-411e-ab90- c8e1a4352e6b&v=default&b=&from_search=15  exponential-functions?qid=3a63ce8a-d910-43eb-b5b1- 8aac6231e021&v=qf1&b=&from_search=2  equations-and-graphing?qid=3a63ce8a-d910-43eb-b5b1- 8aac6231e021&v=qf1&b=&from_search=7 Date accessed:03 March 2014

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