\documentclass{article}
\usepackage{graphicx}
\usepackage{url}
\usepackage{amssymb}
\usepackage{amsmath}

\newcommand{\R}{{\sf R}}
\newcommand{\code}[1]{\texttt{#1}}
\title{Lab 1: solutions}
\author{\copyright 2005 Ben Bolker}

\newcounter{exercise}
\numberwithin{exercise}{section}
\newcommand{\exnumber}{\addtocounter{exercise}{1} \theexercise \thinspace}


\begin{document}
\maketitle
\addtocounter{section}{2}
\textbf{Exercise\exnumber}:

\begin{enumerate}
\item
<<>>=
2^7/(2^7-1)
(1-1/2^7)^-1
@ 
\item 
<<>>=
1+0.2
1+0.2+0.2^2/2
1+0.2+0.2^2/2+0.2^3/6
exp(0.2)
@ 
\item
<<>>=
x=1
1/sqrt(2*pi)*exp(-x^2/2)
dnorm(1)
x=2
1/sqrt(2*pi)*exp(-x^2/2)
dnorm(2)
@
\end{enumerate}

\addtocounter{section}{1}
\setcounter{exercise}{0}

\textbf{Exercise\exnumber}:
(nothing to write down!)

\addtocounter{section}{2}
\setcounter{exercise}{0}
\textbf{Exercise\exnumber}:
<<>>=
X=read.table("ChlorellaGrowth.txt",header=TRUE) 
Light=X[,1]; rmax=X[,2] 
logLight=log(Light)
@ 

<<fig=TRUE>>=
op <- par(cex=1.5,cex.main=0.9)
plot(logLight,rmax,
     xlab="Log light intensity (uE/m^2/s)",
     ## expression(paste("Log light intensity ",(mu*E/m^2/s))), ylab=
     ylab="Maximum growth rate rmax (1/d)",pch=16) 
title(main="Data from Fussmann et al. (2000) system")
fit=lm(rmax~logLight)
abline(fit) 
rcoef=round(coef(fit),digits=3)
text(3.7,3.5,paste("rmax=",rcoef[1],"+",rcoef[2],"log(Light)"))
par(op)
@ 

\textbf{Exercise\exnumber}:
explained in text

\textbf{Exercise\exnumber}:
<<fig=TRUE>>=
plot(Light,rmax,xlim=c(0,120),ylim=c(1,4))
@ 

\textbf{Exercise\exnumber}:
<<>>=
X=read.table("ChlorellaGrowth.txt",header=TRUE) 
Light=X[,1]; rmax=X[,2]
logLight=log(Light)
logrmax = log(rmax)
@ 

<<fig=TRUE>>=
op <- par(mfcol=c(2,1))
plot(Light,rmax,xlab="Light intensity (uE/m2/sa)", ylab="Maximum growth rate rmax (1/d)",pch=16) 
title(main="Data from Fussmann et al. (2000) system")
plot(logLight,logrmax,xlab="Log light intensity", ylab="Log max growth rate",pch=16) 
title(main="Data from Fussmann et al. (2000) system")
par(op)
@


\textbf{Exercise\exnumber}:

<<>>=
x = 3:8
y=5*x+3
@ 

<<fig=TRUE>>=
op = par(mfrow=c(2,2))
plot(x,y,lty=1,col=1,type="l")
plot(x,y,lty=2,col=2,type="l")
plot(x,y,lty=3,col=3,type="l")
plot(x,y,lty=4,col=4,type="l")
par(op)
@ 

\textbf{Exercise\exnumber}:
(nothing to say)

\addtocounter{section}{3}
\setcounter{exercise}{0}

\textbf{Exercise\exnumber}:
<<keep.source=TRUE>>=
seq(1,13,by=4)
## or
seq(1,by=4,length=4)
seq(1,5,by=0.2)
@ 

\textbf{Exercise\exnumber}:
<<keep.source=TRUE>>=
z=c(1,3,5,7,9,11) ## set value of z
z[c(2,1,3)]
@ 

\textbf{Exercise\exnumber}:
<<>>=
x=1:10
y = (x-1)/(x+1)
@ 

<<fig=TRUE>>=
plot(x,y,type="b")
@ 

\textbf{Exercise\exnumber}:
<<>>=
r=0.5
n=10
G = r^(0:n)
sum(G)
1/(1-r)
n=50
sum(r^(0:n))
2-sum(r^(0:n))
@ 

\textbf{Exercise\exnumber}:
<<>>=
set.seed(273)
x = runif(20)
x[x<mean(x)]
@ 

\textbf{Exercise\exnumber}:
<<>>=
which(x<mean(x))
@ 
or
<<>>=
p = 1:length(x)
p[x<mean(x)]
@ 

\textbf{Exercise\exnumber*}:
<<>>=
x = 1:40
n = length(x)
x[seq(1,n,by=2)]
x[-seq(2,n,by=2)]
@ 

\addtocounter{section}{1}
\setcounter{exercise}{0}

\textbf{Exercise\exnumber*}:
<<>>=
v = c(1,2,1,2,1,2,1,2)
X=matrix(v,nrow=2,ncol=4)
@ 
or
<<>>=
v = rep(1:2,4)
X=matrix(v,nrow=2,ncol=4)
@ 
or
<<>>=
v = rep(1:2,each=4)
X=matrix(v,nrow=2,ncol=4,byrow=TRUE)
@ 

\textbf{Exercise\exnumber*}:
<<>>=
set.seed(273)
v = rnorm(35,mean=1,sd=2)
matrix(v,nrow=5,ncol=7)
@

\textbf{Exercise\exnumber}:
nothing to do

\end{document}