Lab Overview

The purpose of this lab is for you to find the best linear ( ), exponential ( ), power ( ), and logarithmic ( ) function that fits a given set of data. In the process of finding these four functions, you should see that - in most instances - one of these four functions is better than the other three.

 This should provide you with some practice applying the properties of exponentials and logarithms in addition to being helpful in other courses that you are taking (or will take).

After a short review of exponential and power functions, we will look at some sample data and determine the best power or exponential fit for the data. You will then repeat this analysis on several additional sets of data. It is not necessary for you to complete the analysis of all of the data sets. Do only as much work as is needed to answer the questions. Note that some data sets will need to be adjusted prior to performing some of the analysis.

Note that this analysis is related to the uses of (semi-)log and log-log plots, but this topic will not be explored in detail here.

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Auxiliary Procedures (execute but do not change)

>	slope := proc( F )   if type( F, equation ) then     return lcoeff( rhs(F) )   else     return lcoeff( F )   end if end proc:

>	intercept := proc( F )   if type(F,equation) then     return tcoeff( rhs(F) )   else     return tcoeff( F )   end if end proc:

>	bestlinearfit := (X,Y)->fit[leastsquare[[x,y]]]([evalf(X),evalf(Y)]):

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dataplot := proc( X, Y )   local bestlinfit, opts, p1, p2, xlo, xhi;   if nargs>2 then opts:= args[3..-1] else opts := NULL end if;   p1 := plot( zip( (x,y)->[x,y], X, Y ), style=point );   bestlinfit := rhs(bestlinearfit(X,Y));   xlo := min(op(X),0);   xhi := max(op(X),0);   p2 := plot( bestlinfit, x=xlo..xhi, color=blue );   return display( [p1,p2], opts ) end proc:

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datafitplot := proc( X, Y, F )   local bestlinfit, f, opts, p1, p2, xlo, xhi;   if nargs>3 then opts:= args[4..-1] else opts := NULL end if;   if type(F,equation) then f := rhs(F) else f := F end if;   p1 := plot( zip( (x,y)->[x,y], X, Y ), style=point );   xlo := min(op(X),0);   xhi := max(op(X),0);   p2 := plot( f, x=xlo..xhi, color=blue );   return display( [p1,p2], opts ) end proc:

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Review of Exponential and Power Functions

We want to develop a simple (oftentimes visual) way to determine if a collection of data points is better approximated with an exponential function or a power function. Let's consider a general exponential function:

>	e1 := y=a*exp(b*x): e1;

If the natural logarithm is applied to both sides of this equation, and the right-hand side is simplified:

>	e2 := map( ln, e1 ): e2; simplify( e2 ) assuming real;

The assumption is necessary because of some technical concerns that are not relevant to us. (Be sure you can do this calculation by hand!)

The conclusion that we should draw is that if the plot of the -values vs. the (natural) logarithm of the -values is a straight line, then the data corresponds to a power function. Moreover, the slope of the line is the growth (or decay) rate of the exponential function and the y-intercept of the line is the (natural) logarithm of the coefficient of the exponential function.

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Next, consider a general power function:

>	p1 := y=a*x^b: p1;

Taking the logarithm of both sides of this equation, and simplifying the right-hand side leads to:

>	p2 := map( ln, p1 ): p2; simplify( p2 ) assuming positive;

Again, be sure you can do these calculations by hand.

The conclusion to be drawn from this result is that if the plot of the logarithm of the -values vs. the logarithm of the -values is a straight line, then the data appears to follow a power function. The slope of the line is the power of the power function and the y-intercept of the line is  the logarithm of the coefficient of the power function.

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Examples

General Data Processing

These commands can be used to visualize any set of data points stored in two vectors, X  and Y . Note that all work necessary to ensure the data values are all positive and are appropriately scaled or shifted are assumed to have been completed prior to executing these commands. In some cases it might be necessary to add a view=  option to some of the plot commands.

It is recommended that you copy this section to where you want to use it. The reason for this recommendation is that you will always have a good copy of these commands should you want to revert to the original form.

>	'X' = X; 'Y' = Y;

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Required Data Sets

Optional Data Sets

Lab Questions

1. Determine the best fitting power and logarithmic functions for the Regional Drainage Area data.

2. Determine the best fitting power and exponential functions for the Radium Emissions from a Watch Hand data.

3. Find one of the Optional Data Sets that is best fit with a power function; give the best fitting power function.

4. Find one of the Optional Data Sets that is best fit with an exponential function; give the best fitting power function.

5. Which of the four best fitting functions provides the best fit to the U.S. Census data?

Essay Question [2 points]: Explain how the behavior of the exponential, power, and logarithmic functions differ at the origin. How do these differences help select the most appropriate fit for the Regional Drainage Area and Radium Emissions data sets.

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