Goals:

This activity will cover the materials in section 1.7 of Stewart's Calculus and Concepts textbook. The skills you should acquire are:

• The ability to determine from a table of data for f(x), whether or not a linear formula, f(x) = m*x + b will produce a decent model of the tabular data.
• The ability to determine from a table of data for f(x), whether or not an exponential formula, f(x) = k*b^x will produce a decent model of the tabular data.
• The ability to determine (by hand) a linear or exponential formula which fits tabular data from a linear or exponential function.
• The ability to run various regressions (fit a formula to tabular data) for a specified function type on the TI-83 calculator.

Shown below are two plots of data of the form (x_k, y_k). From a graphical perspective, data is linear if you can draw a straight line which comes close to "going through" all the data points. If no straight line does a good job of "going through" all the data, the data is not linear.

1. Label one plot as linear and the other as not-linear.






Here are the two data sets above in tabular format:



Linear functions are characterized by the property that they have a constant slope. Slopes can be quickly calculated from a table of values, since both X and Y values appear in the table. Normally, the X values in a table are "equally spaced", so D X, the change in X, is always the same (for consecutive data points in the table). In the table above, as we move from row to row, the value of X increases by D X = 0.3 each time. This means that the denominators in slope calculations, D Y / D X, will always be 0.3 (again, as long as we compute slopes for consecutive data pairs in the table). What we might call the "first slope" for Y1 above, is thus (2.41 - 2.98) / 0.3 = -1.9. The "second slope" is (1.76 - 2.41) / 0.3 = -2.17 (which isn't too different from the first slope).

2. Calculate the remaining slopes for Y1 and Y2 and fill in the table below (I've done several for you):

 
 




3. Are the slopes for Y1 all about the same? If so, what single value of m are they all close to?

 

 

4. Are the slopes for Y2 all about the same? If so, what single value of m are they all close to?

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

The test for determining whether or not a tabular function will satisfy a linear formula is simply to compute several (usually all "consecutive") slopes. If the slopes are all about the same, the data is roughly linear. If the slopes are not all about the same, the data is not linear. When the slopes are all about the same, select some reasonable value (maybe the average slope, perhaps the slope from the first data point to the last data point, etc.) for this constant slope and call it m. You now know f(x) = m*x + b, and you know the value of m, just plug in any data point (or several and take an average) and solve for b. You will then have a formula, and you should use the table features of your calculator to check that your formula does a decent job of reproducing the values in the table. Don't expect perfection, you simply want a formula that "comes close".

5. Which of the functions Y1, Y2 and Y3, below are linear? Select one that is linear and find a formula.




























We've already seen that exponential functions of the form f(x) = k*b^X have "magic ratios". That is, . Normally, X values in a table all differ by the same amount (so D X, for consecutive entries, is usually the same). If D X is constant, b^DX will also be constant. Thus, instead of constant slopes, which indicates linear data, we check for "constant ratios".

An Example:



The "first ratio" is Y1(2) / Y1(1.3) = 22.50 / 16.94 = 1.328. The next is 29.88 / 22.50 = 1.328. The last consecutive ratio is 39.69 / 29.88 = 1.328. They're all the same! We will be able to find an exponential formula that fits this data well. The key is to compute these ratios algebraically using the general exponential formula f(x) = k*b^X. All of the ratios will simplify to b^0.7. For example, = 1.328. Raising both sides to the 1 / 0.7 power gives b = 1.328^(1/0.7) = 1.5. Now, pick any of the data points, plug it into f(x) = k*1.5^X, and solve for k.

6. What value do you get for k?

 

 
 
 
 
 
 

7. Find formulas (linear or exponential) that come close to fitting the data below:

The calculator can also find formulas (of many different kinds). This is called regression, and in most cases the calculator uses statistics and calculus (which we'll learn later) to find these "least squares" formulas. The basic scheme is this (details will be given below):

1. Put the independent variable data (often X values) into the list L₁.
2. Put the dependent variable data (often Y values) into the list L₂.
3. Select and run the appropriate regression command from the [STAT] [CALC] menu.
4. Check to see how well the formula fits the data.

• Press [2^nd][CATALOG] (the [0] key), then [ALPHA][D] to jump to the commands which begin with the letter D.
• Use the down cursor to scroll down to DiagnosticsOn and press [ENTER] twice.
• Your calculator should display "Done" to indicate that diagnostics are now on.

I'll walk through problem 3, for the fourth-degree polynomial fit, to illustrate running a regression. The only thing that changes is the command you use from the [STAT] [CALC] menu, but each command has similar syntax, namely commandname Y₁. You should try to reproduce all my steps (so you can do the other problems that require a regression). In general, linear and exponential curve fits, can (and should, since it is usually faster) be done "by hand", as detailed above. For other regressions, you will normally be told what type of formula to look for, and using the calculator to find this regression formula is the suggested manner to proceed.

1. Set up the statistical editor with the lists L₁ and L₂ (these will be the only lists displayed and the first command clears the contents of these lists) by pressing
[STAT] [4:ClrList] [2^nd] [L₁] [ , ] [2^nd] [L₂] [ENTER] and then
[STAT] [5:SetUpEditor] [2^nd] [L₁] [ , ] [2^nd] [L₂] [ENTER].

2. Access the statistical editor by pressing [STAT] [ENTER].
3. Use the cursor arrow keys ([ENTER] and [downarrow] both work for moving down) to enter the "years from 1970" in the first column, and the emissions data in the second column.
4. When your data is entered, press [Y=] and delete the contents (if any) of Y₁. Then press [2^nd] [QUIT] to return to the home screen and run a quartic regression, storing the regression formula in Y₁ by pressing:

If you press the down cursor, you'll see an R^2 value very close to 1, indicating an excellent fit. The answer to part (b) is simply Y₁(2) = 183.97 (for 1972) and Y₁(12) = 43.52 (for 1982). To set up a stat-plot of the data and the regression formula, press [2^nd][STAT PLOT] [ENTER] to select the options for stat-plot 1. Using the cursor keys to move and the [ENTER] key to select options, get your screen to look like the one below on the left. Set your window to that of the middle screen below (normally, you set the x range from the data values in L₁, and then press [ZOOM][9: ZoomStat] ). A graph like the one shown on the right should appear by pressing [GRAPH] (turn off all functions other than Y₁ first).

You can compare the regression function values, namely Y₁(L₁) to the data values (i.e. L₂) by "attaching a formula" to L₃ as follows:

Press [STAT] [ENTER] to load the statistical editor. Move the cursor into the top "name row" and move over to the third column (it should be blank). At the name prompt type [2^nd] [L₃] [ENTER]. If you had data in L₃, it will appear, so press [CLEAR]. If L₃ was empty, you do not need to press [CLEAR]. In either case, the TI-83 should now be waiting for a definition of L₃.

See the bottom line of the screen below. The regression function values should appear in the third column (see the screen below). The Lock symbol also appears in the third column (indicating that L₃ has a formula attached to it). If you never modify L₃, this attached formula will remain set in your calculator. You can change Y₁ or L₁, and L₃ will automatically update itself. Thus, you can start future regressions by adding L₃ to the end of the SetUpEditor command (actually you need "comma L₃") , and the regression function values will automatically be displayed in the statistical editor. To remove this attached formula (leaving it hurts nothing however), either clear the values of L₃ with [STAT] [4:ClrList] [2^nd] [L₃], or store a new value to some element of L₃ from the home screen, like 0 [STO>] L₃(1). Again, the regression function values (in L₃) are very close to the actual data displayed in L₂. Like the table of the TI-83, you see more digits of the data by moving the cursor over a data entry and viewing the "full version" at the bottom of the editor's window.

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