第1章関数とグラフ

For any base $b>0,b\ne 1,$ the exponential function $f(x)=b^x$ is defined for all real numbers $x$ and $b^x>0.$ Therefore, the domain of $f\left(x\right)=b^x$ is $\left(\text{−}\infty ,\infty \right)$ and the range is $\left(0,\infty \right).$ To graph $b^x,$ we note that for $b>1,b^x$ is increasing on $(\text{−}\infty ,\infty )$ and $b^x\to \infty$ as $x\to \infty ,$ whereas $b^x\to 0$ as $x\to \text{−}\infty .$ On the other hand, if $0<b<1,f\left(x\right)=b^x$ is decreasing on $(\text{−}\infty ,\infty )$ and $b^x\to 0$ as $x\to \infty$ whereas $b^x\to \infty$ as $x\to \text{−}\infty$ (Figure 1.44).

Note that exponential functions satisfy the general laws of exponents. To remind you of these laws, we state them as rules.

A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests $P$ dollars in a savings account with an annual interest rate $r,$ compounded annually. The amount of money after 1 year is

The amount of money after $2$ years is

More generally, the amount after $t$ years is

If the money is compounded 2 times per year, the amount of money after half a year is

The amount of money after $1$ year is

After $t$ years, the amount of money in the account is

More generally, if the money is compounded $n$ times per year, the amount of money in the account after $t$ years is given by the function

What happens as $n\to \infty ?$ To answer this question, we let $m=n\text{/}r$ and write

and examine the behavior of ${\left(1+1\text{/}m\right)}^m$ as $m\to \infty ,$ using a table of values (Table 1.12).

Looking at this table, it appears that ${\left(1+1\text{/}m\right)}^m$ is approaching a number between $2.7$ and $2.8$ as $m\to \infty .$ In fact, ${\left(1+1\text{/}m\right)}^m$ does approach some number as $m\to \infty .$ We call this number $e$. To six decimal places of accuracy,

The letter $e$ was first used to represent this number by the Swiss mathematician Leonhard Euler during the 1720s. Although Euler did not discover the number, he showed many important connections between $e$ and logarithmic functions. We still use the notation $e$ today to honor Euler’s work because it appears in many areas of mathematics and because we can use it in many practical applications.

Returning to our savings account example, we can conclude that if a person puts $P$ dollars in an account at an annual interest rate $r,$ compounded continuously, then $A\left(t\right)=P{e}^{rt}.$ This function may be familiar. Since functions involving base $e$ arise often in applications, we call the function $f\left(x\right)=e^x$ the natural exponential function. Not only is this function interesting because of the definition of the number $e,$ but also, as discussed next, its graph has an important property.

Since $e>1,$ we know $e^x$ is increasing on $(\text{−}\infty ,\infty ).$ In Figure 1.45, we show a graph of $f\left(x\right)=e^x$ along with a tangent line to the graph of at $x=0.$ We give a precise definition of tangent line in the next chapter; but, informally, we say a tangent line to a graph of $f$ at $x=a$ is a line that passes through the point $\left(a,f\left(a\right)\right)$ and has the same “slope” as $f$ at that point $.$ The function $f\left(x\right)=e^x$ is the only exponential function $b^x$ with tangent line at $x=0$ that has a slope of 1. As we see later in the text, having this property makes the natural exponential function the most simple exponential function to use in many instances.