Problem:
Using the TI-86 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:

1. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
2. If necessary, keep pressing the F4 key until only y1= appears on the screen.
3. Be sure that PLOTS 1, 2 and 3 are turned-off.
4. At the prompt \y1=, type in

x^4

and press the ENTER key.

5. The WINDOW settings for the graph above is  [-1.5, 1.5] × [-2, 4]:

Go to the WINDOW menu by pressing 2nd M2.

6. Press the EXIT key.
7. Press 2nd MATH.
8. Pick MISC from the menu by pressing F5 key.
9. Pick seq from the menu by pressing F3 key.
10. 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

11. 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.

12. Press STO key and type in A. This assigns the name A to the list generated above.
13. 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.

14. 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.

15. Press STO key and type in M. This assigns the name M to the list of slopes of the secant lines generated above.
16. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
17. Press the ENTER key to get the prompt y2=.
18. 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.
19. Press the EXIT key followed by the F5 key to pick GRAPH from the menu. You should see the graph