MATH 133 – Calculus with Fundamentals 1 Discussion Day on Exponential Functions September 18, 2015

Background

Moore’s law is the observation that the number of transistors in the densest integrated circuits (like those used in computer hardware) doubles approximately every two years. The observation is named after Gordon E. Moore, the co-founder of Intel and Fairchild Semiconductor. Moore wrote a paper in 1965 describing a doubling every year in the number of components per integrated circuit and projected this rate of growth would continue for at least another decade. In 1975, looking forward to the next decade, he revised the forecast to doubling every two years. This prediction proved very accurate for several decades, and the law was used in the semiconductor industry to guide long-term planning and to set targets for research and development. Advancements in digital electronics are strongly linked to Moore’s law: quality-adjusted microprocessor prices, memory capacity, sensors and even the number and size of pixels in digital cameras and cell phones. (In other words, your “smart phones” would not be possible without it!)

Today we want to see how Moore’s Law leads to something closely related to one of the exponential functions studied in Section 1.5 of the text and today’s video. Recall that the basic exponential function graphs have the form: y = f(x) = b^{x}. For the purposes of this discussion, let’s consider a slightly more general form:

y = f(x) = c · b^{x},  (1)

where c is a constant.

Question

What we’ll do first in this question is to take two actual data points and find a function of the form (1) that agrees with them.

(a) In 1990, the Intel 80486 CPU chip (one industry standard at the time) contained about 1,000,000 transistors. In 2005, the AMD K8 CPU chip contained about 100,000,000 transistors. To simplify things, let’s say x represents the number of years after 1990, so 1990 ↔ x = 0 and 2005 ↔ x = 15. There is exactly one function of the form (1) whose graph contains the points (0, 1000000) and (15, 100000000). To find the c and b that work, first substitute these x and y values to get

1000000 = c · b^{0}  (2)
100000000 = c · b^{15}.  (3)

To finish this off, use the first equation to solve for c, then substitute your value for c into the second equation and solve for b. Use your values for b, c to write a function that represents the number of transistors in the densest circuits as a function of time.

(b) Now, we want to know how close this is to saying “the number of transistors doubled approximately every two years.” If we started from the known value f(0) = 1000000 (the number of transistors in the 80486 chip from 1990), then the number after 2 years should be 2×1000000, then after 4 years the number should be 4 × 1000000, the number after 6 years should be 8 × 1000000, and so forth. Explain why

g(x) = 1000000 · 2^{x/2}

matches this pattern, so we’ll take that as the formula for all x ≥ 0.

(c) Rewrite 2^{x/2} in the form b^{x} for some b (you’ll need to recall and use a rule for exponents to do this). How does this value compare with the b you found in part (a) of the question?

(d) To be clear, “Moore’s law” is an just observation or projection of what has happened in the development of computer technology; it is obviously not a physical or natural law, and it cannot continue to hold true indefinitely into the future. Why not? Explain in your own words by thinking about the formula you developed earlier.