Section 5.2The Natural Exponential Function
Objectives
- Understanding the Characteristics of the Natural Exponential Function
- Sketching the Graphs of Natural Exponential Functions Using Transformations
- Solving Natural Exponential Equations by Relating the Bases
- SolvingApplications of the Natural Exponential Function
Objective 1: Understanding the Characteristics of the Natural Exponential Function
We learned in the previous section that any positive number bwhere can be used as the base of anexponential function. However, there is one number that appears as the base in exponential applications more than any other number. This number is called the natural base and is symbolized using the letter e. The number e is an irrational number that is defined as the value of the expression as n approaches
infinity.
The table below shows the values of the expression for increasingly large values of n.
As the values of n get large, the value e(rounded to 6 decimal places) is . The function is called the natural exponential function. Since it follows that the graph of lies between the graph of and as seen in Figure 1.
The graph of the natural
exponential function
n /1 / 2
2 / 2.25
10 / 2.5937424601
100 / 2.7048138294
1000 / 2.7169239322
10,000 / 2.7181459268
100,000 / 2.7182682372
1,000,000 / 2.7182804693
10,000,000 / 2.7182816925
100,000,000 / 2.7182818149
Characteristics of the Natural Exponential Function
The Natural Exponential Function is the exponential function with base eand is defined as .
The domain of is and the range is . The graph of and some of its characteristics are stated below.
The graph of intersects the y-axis at .
The line is a horizontal asymptote.
The function is one-to-one.
It is important that you are able to use your calculator to evaluate various powers of e. Most calculators have an key.
Objective 2: Sketching the Graphs of Natural Exponential Functions
Again we can use the transformation techniques that were discussed in Section 3.4 to sketch variations of
the natural exponential function.
Objective 3: Solving Natural Exponential Equations by Relating the Bases
Recall the method of relating the bases for solving exponential equationsfrom Section 5.1. If we can write an exponential equation in the form of then . This method for solving exponential equations certainly holds true for the natural base as well.
Objective 4: Solving Applications of the Natural Exponential Function
Continuous Compound Interest
Recall the Periodic Compound Interest Formula that was introduced in Section 5.1. Some banks use continuous compounding, that is, they compound the interest every fraction of a second every day! If we start with the formula for periodic compound interest ,, and let n (the number of times the interest is compounded each year) approach infinity, we can derive the formula which is the formula for continuous compound interest.
Continuous Compound Interest Formula
Continuous compound interest can be calculated using the
formula
where
Total amount after t years
Principal
Interest rate per year
Number of years
Present Value
Recall that the present value Pis the amount of money to be invested now to obtain A dollars in the future. To find a formula for present value on money that is compounded continuously, we start with the formula for continuous compound interest and solve for P.
Continuous compound interest formula.
Divide both sides by.
Rewriteas
Present Value Formula
The present value of A dollars after t years of continuous compound
interest, with interest rater , is given by the formula
.
Exponential Growth Model
You have probably heard that some populations grow exponentially. Most populations grow at a rate proportional to the size of the population. In other words, the larger the population, the faster the population grows. With this in mind, it can be shown in a more advanced math course that the mathematical model that can describe population growth is given by the function.
Exponential Growth
A model that describes the population, P, after a certain time, t, is
where is the initial population and is a constant
called the relative growth rate. (Note: may be given as a percent.)
The graph of the exponential growth model is shown below. Notice that the graph has a y-intercept of .
Figure 3
The graph of the
exponential growth
model