between x=0 and x=1 which is accurate to within 0.00001 using the TI-85 graphing calculator.

[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 F4 key until only y1= appears on the screen.
4. Type in cos x - x^2

   and press the ENTER key.
   

5. The range settings for the graph above is  [0, 1] × [-1, 1]:
   

6. Pick ZOOM by pressing the F3 key.
7. Press the MORE key twice so that ZFACT appears on the menu.
8. Press the F1 key to pick ZFACT.
9. Since we want the answer to be accurate to within 0.00001, we want to zoom in quickly. We enter 10 at each of the prompts xFact and yFact followed by the ENTER key. With these settings, the zooming in will be at a factor of 1/10. Since we started with an interval of length 1, the length of the interval after one zooming in is 0.1; after two zooming in's, the length is 0.01, etc.
10. Pick ZOOM by pressing the F3 key again.
11. This time pick ZIN by pressing the F2 key.
12. You will now see a flashing tick mark in the middle of the screen and the coordinates of the tick mark at the bottom of the screen:
    

13. Using the arrow keys, move the tick mark as close to the intersection of the graph with the horizontal line (which is the x-axis) as possible and press the ENTER key.
14. Repeat the latter step at least six times.
15. Press the EXIT key followed by the F2 key to pick RANGE. The following screen was obtained:
    

    after seven zooming in's. This indicates that the root is between 0.824132251587 and 0.824132351587; 0.824132 satsifies the desired accuracy.

    A more efficient way to find the root with the TI-85 graphing calculator is to use the SOLVER key.