Calculating Average Rates (TI 82/83)

Goal: We want to calculate values of [f(x+h)-f(x)]/h for a variety of values of h and we want to be able to easily change the function f and then repeat the calculations.

General method: Set up the function, f, as Y1 and the expression for the average rate in Y2. We then can compute values for Y1 for particular values of h and x and display the calculated values.

Since we will want to compute with many values of h for each choice of x, it works best to replace the symbol x by the symbol t and the symbol h by the symbol x in the expression for Y2. In other words, calculate values of [f(t+x)-f(t)]/x. Then, once these symbol changes are made and a particular value of t is set, you can compute Y2 values by displaying Y2(x) for values of x.

The expression Y2(x) can be typed directly on the home screen.

Step by step example: Consider f(t)=x² ö x and the interval [2, 2.4]. Then we will want to calculate the average rate when x is 2 and h is 0.4.

• On the Y= screen, enter Y1=X^2-X
• On the Y= screen, enter Y2=(Y1(T+X)-Y1(T))/X
  (Caution! Y1 in the above comes from the VARS Y-VARS FUNCTION screen)
• On the "Home" screen, enter 2 T and then Y2(0.4) to display the desired result.
  ( Y2 comes from the VARS Y-VARS FUNCTION screen)

In the case for which you would like to compare values of Y2 for many values of h, the Table program on your calculator is useful. To use this, after the above setup:

• On the TBLSET screen, set Indpnt: to Ask
• Go to the TABLE screen and enter 0.4 and your other values of h in the X column.
• Read the results in the Y2 column.

CAUTION! If you can just follow the above steps to obtain answers but do not understand how and why this procedure works you would be very wise to NOT use this method!

An alternate (and conceptually simpler) method is to set up Y1 as above and then enter (Y1(2+0.4)-Y1(2))/0.4 directly on the home screen. Other values of h can then be used by recalling that line after execution (use the ENTRY command to do this) and editing in a replacement for 0.4 in its two locations.

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