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Chapter 4 Exponential Functions

computer chip, Andre Kudyusov/Getty Images

We next consider another important family of functions, called exponential functions. These functions describe growth by a constant factor in equal time periods. 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

\begin{equation*} N(t) = 2200(1.356)^t\text{,} \end{equation*}

where \(t\) is the number of years since \(1970\text{.}\)

growth
Investigation 4.1. Population Growth.

  1. In a laboratory experiment, researchers establish a colony of \(100\) bacteria and monitor its growth. The colony triples in population every day.

    1. Fill in the table showing the population P(t) of bacteria t days later.

    2. Plot the data points from the table and connect them with a smooth curve.

    3. Write a function that gives the population of the colony at any time \(t\text{,}\) in days. Hint: Express the values you calculated in part (1) using powers of \(3\text{.}\) Do you see a connection between the value of \(t\) and the exponent on \(3\text{?}\)

    4. Graph your function from part (3) using a calculator. (Use the table to choose an appropriate domain and range.) The graph should resemble your hand-drawn graph from part (2).

    5. Evaluate your function to find the number of bacteria present after \(8\) days. How many bacteria are present after \(36\) hours?

    \(t\) \(P(t)\)
    \(0\) \(100\) \(P(0)=100\)
    \(1\) \(\) \(P(1)=100\cdot 3=\)
    \(2\) \(\) \(P(2)=[100\cdot 3]\cdot 3=\)
    \(3\) \(\) \(P(3)=\)
    \(4\) \(\) \(P(4)=\)
    \(5\) \(\) \(P(5)=\)

    grid for graph of bacteria population

  2. Under ideal conditions, the number of rabbits in a certain area can double every \(3\) months. A rancher estimates that \(60\) rabbits live on his land.

    1. Fill in the table showing the population \(P(t)\) of rabbits \(t\) months later.

    2. Plot the data points and connect them with a smooth curve.

    3. Write a function that gives the population of rabbits at any time \(t\text{,}\) in months. Hint: Express the values you calculated in part (1) using powers of \(2\text{.}\) Note that the population of rabbits is multiplied by \(2\) every \(3\) months. If you know the value of \(t\text{,}\) how do you find the corresponding exponent in \(P(t)\text{?}\)

    4. Graph your function from part (3) using a calculator. (Use the table to choose an appropriate domain and range.) The graph should resemble your hand-drawn graph from part (2).

    5. Evaluate your function to find the number of rabbits present after \(2\) years. How many rabbits are present after \(8\) months?

    \(t\) \(P(t)\)
    \(0\) \(60\) \(P(0)=60\)
    \(3\) \(\) \(P(3)=60\cdot 2=\)
    \(6\) \(\) \(P(6)=[60\cdot 3]\cdot 2=\)
    \(9\) \(\) \(P(9)=\)
    \(12\) \(\) \(P(12)=\)
    \(15\) \(\) \(P(15)=\)

    grid for graph of rabbit population