Precal Blog

Given that a sequence is an ordered list of values, our job is to find a way to generalize the patterns we find. We can either do that recursively where we define the next term in “terms” of the previous ones, or we can do it expicitly where any term is determined simply by its place in the sequence. There are advantages and disadvantages to both – what are they?

We will be concerned with two special sequences – arithmetic and geometric. You should be able to write both recursive and explicit formulas for both. You should also be aware of the general numerical patterns that hold for both. We also talked today about how to use sequence mode on your calculator to plot discrete graphs of sequences.

Lastly then be warned that sequences other than arithmetic and geometric will not be as easy to find generalizations for – especially explicit formulas. For example, you should have no trouble writing a recursive formula for the Fibonacci sequence, but spend a little while trying to write an explicit formula – then give up and look it up.

It’s nice to have the calculator do most of the work for you when it comes to residual plots but don’t forget that I will expect you to be able to create a residual plot by hand. Since you already know how to calculate the coefficient of determination by hand this shouldn’t be an issue – just create a graph of the the residuals versus the x – values.

We also mentioned today that whenever you run regression on your calculator, the calculator automatically computes the residuals and stores them in a new list called RESID. You can access that list by going to 2nd STAT, then scrolling down until you see it.

Another cool time saver is accessing the actual regression equation which your calculator stores automatically also. You can find it by going to VARS, then down to STATISTICS, then over to EQ, then down to RegEQ. You could use this to store the regression equation in Y= if you forgot to do it automatically when you ran the regression.

The oak tree data from today is not clearly modeled by any particular function. We have to do the best we can to choose the most appropriate model. Is the coefficient of determination acceptable? Is the end behavior appropriate? If we extrapolate, do we get a reasonable value? You may not find a function that is perfect in all areas but taking the context of the data into consideration should help you weigh the pros and cons of each model you consider.

By the way, did anyone try squaring the y-values…?

Today we looked at an exponential function, and like yesterday it was linearized by taking the log of the y-values. Working through the algebra is really as simple as using the definition of logarithm to rewrite the log y = ax + b as a power of 10 exponential equation and simplifying.

The second problem was a power function that was linearized by taking the log of both x and y. This gave us a messy equation: log(y) = a⋅log(x) + b. Our (my) first thought was to raise both sides to the power of 10. This left us with logs on both sides of the equation – sort of like leaving yourself with x’s on both sides of an equation. With that said, our next attempt should probably be to get both logs together on one side and go from there. Can anyone do it for tomorrow?

We came to some important conclusions today. First, we discovered that for anything other than linear regression your calculator computes an R² value instead of the normal r². The difference is that R² is calculated using the line of least squares that fits the transformed, “linearized” data – not the actual curve that best fits the original data. If you want the r² value for the original data using the curve of best fit you must calculate by hand.

So, how do you find the best fit curve for non-linear data? You must transform one or both of the variables by squaring, or taking the the log, or the square root, or….? The goal is to plot the transformed points and see a line. Then running linear regression we can find the slope and y-intercept of the best fit line for the transformed data. Then re-write your y = mx + b form substiuting log(y) or x² (or whatever transformation you made) in for x and y. Solve for y and we have the curve.

It is not a random process deciding on the transformation to use. Today we looked at exponential data and wound up taking the log of the y-values. That is not a coincidence that we “un-did” exponents with logs. Work with the problems for tomorrow and you will see some other techniques and where to use them.

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

-Mathematical Intelligencer, v. 4, no. 4.

Let’s not forget that the main goal of our current unit is to choose an appropriate function to model whatever set of data we are looking at. Using that model will allow us to make predictions via interpolation and/or extrapolation.

So far we have one tool in our box and that is the coefficient of determination. We should never try and choose a model based only on this one value. As we will see tomorrow, end behavior is also an important consideration to make when choosing a function to model our data. We will also discuss techniques for finding functions when the data is not linear.

What does it determine? What it determines should help you remember how to calculate it. The coefficient of determination (r²) determines how much (what percent) of the original deviation from the mean line we have accounted for with our line of best fit. Subtracting the SS_res from the SS_dev tells you how much square area you have reduced the error by. Then by dividing by SS_dev will give your result as a percentage of the orignal deviation.

We did not discuss correlation today but you can handle that on your own in the reading. Suffice it to say that the (positive or negative) square root of the coefficient of determination is called the correlation coefficient and tells us how strongly the variables in the problem are associated.

Interesting problem today, converting a vertically stretched base-10 log into a base-n log with no vertical stretch. Does it work in general? What does it look like if you try to convert a dilated base-3 log to a non-dilated base-n log?

We also talked about the two forms of the logistic function and showed them equivalent although a few people thought it was magic.

I decided on the Cramer’s rule homework because I think it is interesting (the rule I mean) and something you will definitely see again in your college studies. It also extends the matrix concepts we talked about. I’d like that by next Friday.