Example 4 from Algebra and Trigonometry, 6.7 Exponential and Logarithmic Models

Finding a Function That Describes Exponential Growth

According to Moore’s Law, the doubling time for the number of transistors that can be put on a computer chip is approximately two years. Give a function that describes this behavior.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The formula is derived as follows:
$\begin{array}{l} t = \frac{\ln 2}{k} & The doubling time formula . \\ 2 = \frac{\ln 2}{k} & Use a doubling time of two years . \\ k = \frac{\ln 2}{2} & Multiply by k and divide by 2 . \\ A = A_{0} e^{\frac{\ln 2}{2} t} \begin{matrix} \end{matrix} & Substitute k into the continuous growth formula . \end{array}$
The function is $A_{0} e^{\frac{\ln 2}{2} t} .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Recent data suggests that, as of 2013, the rate of growth predicted by Moore’s Law no longer holds. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account.

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solution

Your turn

Solution

Your turn