Problem:
Using the TI-85 graphing calculator, graphically find the tangent line to the graph of the function

y = x4

at the point (0.5, 0.0625) as the limit of secant lines.

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^4

and press the ENTER key.

5. The range settings for the graph above is  [-1.5, 1.5] × [-2, 4]: 6. Press 2nd MATH.
7. Pick MISC from the menu by pressing F5 key.
8. Pick seq from the menu by pressing F3 key.
9. Type in 0.5+2^-x,x,0,5,1) and press the ENTER key. This command

seq(0.5+2^-x,x,0,5,1)

generates a list or collection of numbers of the form

0.5 + 2-x for x = 0, 1, 2, 3, 4, 5

10. You will see a part of this list on the screen:

{1.5 1 .75 .625 .562...

You can see the rest of the list by using the left and right arrows. This list is a collection of six numbers which is getting close to 0.5.

11. Press STO key and type in A. This assigns the name A to the list generated above.
12. Type in

(evalF(y1,x,A)-evalF(y1,x,0.5))/(A-0.5)

You can get the expression evalF by pressing 2nd CALC key followed by the F1 key. evalF(y1,x,0.5) evaluates the function stored in y1 (see step 4 above) by replacing x by 0.5. Similarly, evalF(y1,x,A) evaluates the function stored in y1 by replacing x by each number in the list A; the result of this evaluation is a new list.

13. Press the ENTER key and you will see a new list which consists of the numbers

(a - 0.5)4/(a - 0.5)

for each a in the list A. You will see a part of this list on the screen:

{5 1.875 1.015625 .7...

Again, you can see the rest of the list by using the left and right arrows. This list is the collection of the slopes of the secant lines drawn between the two points on the graph of y = x4 whose x-coordinates are 0.5 and a number from the list A.

14. Press STO key and type in M. This assigns the name M to the list of slopes of the secant lines generated above.
15. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
16. Press the ENTER key to get the prompt y2=.
17. Type in M*(x-0.5)-0.0625 and press the ENTER key. You are entering the equation of a list of lines where the slope is coming from the list M and which contains the point (0.5, 0.0625). Since the list M has six numbers in it, you will get six lines.
18. Press the EXIT key followed by the F5 key to pick GRAPH from the menu. You should see the graph which is shown above.
19. If want to view the list M of slopes then press 2nd LIST. Press F3 to choose NAMES from the menu.
20. One of the menu items should now be M. Depending upon how many lists you have entered in the calculator, you may have to press the MORE to see M on the menu. Pick M by pressing the appropriate F key.
21. You should see the list of slopes: 