\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\usepackage{latexsym,amsmath,amssymb,epsfig}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.0in}
\begin{document}
\centerline{MATH 135 -- Calculus 1}
\centerline{Discussion Day on Exponential Functions}
\centerline{September 12, 2016}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
{\it Moore's law} is the observation that the number of transistors in the densest integrated 
circuits (like those used in computer hardware) {\it 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 {\it 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 {\it 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:
\begin{equation}
\label{genexp}
y = f(x) = c b^x,
\end{equation}
where $c$ is a constant.  

\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
\begin{enumerate}
\item[(a)] What we'll do first today is to take two actual data points and find a function of 
the form \eqref{genexp} that agrees with them.  
 In 1990, the Intel 80486 CPU chip (one industry standard at the time -- we had lots
of PC's at Holy Cross with those CPUs!) 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 $1990 \leftrightarrow x = 0$
and $2005 \leftrightarrow x = 15$.   
 There is exactly one function of the form \eqref{genexp}
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
\begin{eqnarray}
1000000 &=& c b^0\\
100000000 &=& c b^{15}.
\end{eqnarray}
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.  
\item[(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 $2 \times 1000000$, then
after 4 years the number should be 
$$2 \times 2 \times 1000000 = 4 \times 1000000,$$ 
the number after 6 years should be 
$$2 \times 2 \times 2 \times 1000000 = 8 \times 1000000,$$ 
and so forth.  In general, suppose $x$ is the number of years after 1990, then $x/2$ would give
the number of 2-year periods between 1990 and $1990 + x$, and hence the \emph{number of doubling factors}.  
Find a formula that would give
the predicted number of transistors as a function of $x$.  (Hint:  The number would have
doubled $x/2$ times to 
match this pattern).  
\item[(c)] Explain how to 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?  
\item[(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.  
\end{enumerate}
\end{document}
