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Chapter 7 Exponential Functions

computer chip, Andre Kudyusov/Getty Images
In Chapter 1 we studied linear functions, which model quantities that have a constant rate of change. That is, they increase or decrease by the same amount in equal time periods (or equal increments of the input variable). In Example 1.14 we considered a bicycle rental program where the hourly rental fee of $3 was added each hour to the initial insurance fee of $5. We found a formula for the cost of renting a bike for \(t\) hours:
\begin{equation*} C(t) = 5 + 3t \end{equation*}
Recall that the general formula for linear functions is
\begin{equation*} f(t) = b + mt \end{equation*}
where \(m\) stands for the rate of change. Ponder the fact that multiplying \(m\) by \(t\) means repeatedly adding \(t\) copies of \(m\) to the initial amount \(b\text{.}\)
We next consider another important family of functions, called exponential functions. These functions describe growth by a constant factor in equal time periods. Just as multiplication means repeated addition, you also know that raising to an exponent means repeated multiplication. For example, the expression
\begin{equation*} a \cdot b^5 \end{equation*}
means to multiply \(a\) by \(5\) copies of \(b\text{,}\) so we will use exponents to describe the effect of constant growth factors.
Exponential functions model many familiar processes, including the growth of populations, compound interest, and radioactive decay. Here is an example.
In 1965, Gordon Moore, the cofounder of Intel, observed that the number of transistors on a computer chip had doubled every year since the integrated circuit was invented. Moore predicted that the pace would slow down a bit, but the number of transistors would continue to double every 2 years. More recently, data density has doubled approximately every 18 months, and this is the current definition of Moore’s law. Most experts, including Moore himself, expected Moore’s law to hold for at least another two decades.
Year Name of circuit Transistors
\(1971\) \(4004\) \(2300\)
\(1972\) \(8008\) \(3300\)
\(1974\) \(8080\) \(6000\)
\(1978\) \(8086\) \(29,000\)
\(1979\) \(8088\) \(30,000\)
\(1982\) \(80286\) \(134,000\)
\(1985\) \(80386\) \(275,000\)
\(1989\) \(90486\) \(1,200,000\)
\(1993\) Pentium \(3,000,000\)
\(1995\) Pentium Pro \(5,500,000\)
\(1997\) Pentium II \(7,500,000\)
\(1998\) Pentium II Xeon \(7,500,000\)
\(1999\) Pentium III \(9,500,000\)
The data shown are modeled by the exponential function \(N(t) = 2200(1.356)^t\text{,}\) where \(t\) is the number of years since \(1970\text{.}\)