Chapter Five



Section 5.2 The Natural Exponential Function

Objectives

1. Understanding the Characteristics of the Natural Exponential Function

2. Sketching the Graphs of Natural Exponential Functions Using Transformations

3. Solving Natural Exponential Equations by Relating the Bases

4. Solving Applications 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 b where[pic] can be used as the base of an exponential 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 [pic]as n approaches

infinity.

The table below shows the values of the expression [pic] for increasingly large values of n.

As the values of n get large, the value e (rounded to 6 decimal places) is [pic]. The function [pic] is called the natural exponential function. Since [pic] it follows that the graph of [pic] lies between the graph of [pic]and [pic]as seen in Figure 1.

The graph of the natural

exponential function [pic]

|n |[pic] |

|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 e and is defined as [pic].

The domain of [pic]is [pic] and the range is [pic]. The graph of [pic]and some of its characteristics are stated below.

The graph of [pic] intersects the y-axis at [pic].

[pic]

[pic]

The line [pic] is a horizontal asymptote.

The function [pic]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 [pic] 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 equations from Section 5.1. If we can write an exponential equation in the form of [pic] then [pic]. 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 ,[pic], and let n (the number of times the interest is compounded each year) approach infinity, we can derive the formula [pic] which is the formula for continuous compound interest.

Continuous Compound Interest Formula

Continuous compound interest can be calculated using the

formula

[pic] where

[pic]Total amount after t years

[pic]Principal

[pic]Interest rate per year

[pic]Number of years

Present Value

Recall that the present value P is 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.

[pic] Continuous compound interest formula.

[pic] Divide both sides by [pic].

[pic] Rewrite[pic]as [pic]

Present Value Formula

The present value of A dollars after t years of continuous compound

interest, with interest rate r , is given by the formula

[pic].

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[pic].

Exponential Growth

A model that describes the population, P, after a certain time, t, is

[pic]

where [pic]is the initial population and [pic]is a constant

called the relative growth rate. (Note: [pic]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 [pic].

Figure 3

The graph of the

exponential growth

model [pic]

[pic]

[pic]

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[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

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