Problem:
Using the TI-85 graphing calculator, graphically determine which of the following functions

y = x7 and y = 4x

has larger values for x large.


Visualization:


[Press here to see animation again!]

  1. If you are not familiar with the graphing of functions on the TI-85, then first read the Initial Setup page from Little's Basic Guide to the TI-85.
  2. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
  3. If necessary, keep pressing the F4 key until only y1= appears on the screen.
  4. At the prompt y1=, type in

    x^7

    and press the ENTER key.

  5. At the prompt y2=, type in

    4^x

    and press the ENTER key.

  6. The range settings for the graph above is  [0, 10] × [0, 106]:

  7. Pick GRAPH from the menu by pressing the F5. Since y = x7 was entered first, it is graphed first:

    followed by the graph of y = 4x:

  8. It appears that the function y = x7 has larger values.
  9. However, this does not continue to happen for x larger. Choose range settings  [10, 15] × [0, 5 × 108]; the graph of y = x7 is

    and is followed by the graph of y = 4x:

  10. Now it appears that the function y = 4x has larger values. This, in fact, can be verified by using l'Hospital's Rule.

Alternative methods for doing this type of exploration:

  1. Consider the graph of x7 - 4x:

    with viewing window [0, 50] × [-10, 10].

  2. Consider the graph of x7 / 4x:

    with viewing window [0, 50] × [0, 2].