#intro.R
#Population Biology ,  Fall 2011 BIOL 763 / SCCC 411B / Math 523

#OBJECTIVES:
#
#  - Enter mathematical expressions 
#  - Plot functions and extract information from graphs
#  - Learn the solve and other commands for algebra and calculus
#
#WHAT IS R AND HOW CAN IT BE USED? 
#
#It is probably a good idea to begin EVERY worksheet that you create with the following command. 
rm(list=ls())             #this clears the system memory

#BASIC FACTS
#
#R is a computer language; it cannot read your mind. You need to learn how to communicate with Maple.
#
#The basic symbols:

# - assignments are made with  <-  (plain = has a different meaning) -- think of this as giving the value of the right hand side to the name that appears on the left hand side
# - commands can be terminated with a semicolon ( ; ) 
# - unlike Maple, there does not appear to be a command for "most recent result"
# - R is case sensitive -- that is, the names x and X are different, pi and Pi are not the same thing

# - []  -- selection operators for extracting subsets from vectors or matrices
# - :  --  as in a:b -- this is how R indicates a sequence of numbers from a to b
# - ?topic or help(topic)  -- a request to R for information

#Mathematical functions have their standard names, or ones that are easily guessed:
#  + (plus) , - (minus) , * (times) , / (divided by) , ^ (raised to the power) , sin, cos, tan, abs, sqrt, ...

#In the following lines, use paper and pencil to first predict what you think R will do. 
#Then execute the command and see what actually happens!  

N<- 4 *  6 + 12 /  6 - 1 
N

power<- (-3)^3 
power

pi   # note that in R the numerical value is called "pi" and in Maple it is "Pi"

v<- sin( pi / 4 )
v

tan(  -Pi / 2 )

3 / ( 5 - sqrt( N ) )

factorial(4)  # note the difference from Maple, which uses the ! symbol to indicate factorial

factorial(50)

#computer algebra calculations can be done in R using the Ryacas package
#yacas is "yet another computer algebra system"
#as of 28-Aug-2011 I have not tried yacas or Ryacas.  More information later.

#PLOTTING
t<- seq(-3,6,0.1)  # create a sequence of numbers from -3 to 6 in increments of 0.1
plot(t,3*t-2)   # the plot command is plot(x,y) where y is the vertical axis and x is the horizontal

x<-seq(-2*pi,2*pi,0.1)
plot(x,sin(3*x))

t<-seq(0,10,0.1)
y<-3.5*exp(0.2*t)
plot(t,y)

L<-ln(y)
plot(t,L)

#We illustrate how to plot data points, and the usefulness of the logarithm function.
pop<- c( 508, 711, 912, 1131, 1590, 1811, 2015, 2249, 2509, 3008, 3610, 3967)
yr<-  c( 1650, 1750, 1800, 1850, 1900, 1920, 1930, 1940, 1950, 1960, 1970, 1975)

log10pop<-log10(pop)
lnpop<- log(pop)

adjustedyr<-yr - 1650

plot(yr,pop,main="Population vs time")
plot(adjustedyr,pop,main="Population vs time",lwd=2, col="green")
plot(adjustedyr,log10pop main="Log10 of Pop vs time")
plot(adjustedyr,lnpop,main="Ln of pop vs time")

#We have two ways to make logarithmic plots. 
#We can either take log of the data values, as we did above, or we can use the built-in R command.

plot(adjustedyr,pop,log="y",main="Semilog plot of Pop vs time")

#Here are a few more functions and their graphs
x<-seq(-3,3,0.1)
f<-sin(x); g<-2*x^2; p<-f*g;
h<-cos((1/4)*x); s<-0.2*sin(2*x)+h;
plot(x,p)

#R can display the values on graphs via the locator() command.
#Position the cursor on a point on the graph and click the left button. 
#When you are done, press the right button and select "stop".
#The numbers are the coordinates of the current location of the cursor. 
#Use this technique to identify the maximum and minimum values of p(x) on the interval [-3, 3], and the x-values at which these are found.   
#

locator()

#To plot more than one set of data on a graph, use the points() or lines() command:
x<-seq(-6*pi,6*pi, 0.1)
f<-sin(x); g<-2*x^2; p<-f*g;
h<-cos((1/4)*x); s<-0.2*sin(2*x)+h;
plot(x,h)
lines(s,h)

x<-seq(-16,16,0.1)
 f<-sin(x); g<-2*x^2; p<-f*g;
   ymax<-max(g)  #find the largest value that we are planning to plot
   ymin<-min(-g) #find the smallest value
 plot(x,g,ylim=c(ymin,ymax),type="l",col="red")
 lines(x,-g,col="blue")
 lines(x,p,col="black")

#