Using the TI-86 graphing calculator, graphically find the tangent line to the graph of the function

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

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

- The
**WINDOW**settings for the graph above is**[-1.5, 1.5]**×**[-2, 4]**:Go to the

**WINDOW**menu by pressing 2nd M2. - Press the EXIT key.
- Press 2nd MATH.
- Pick
**MISC**from the menu by pressing F5 key. - Pick
**seq**from the menu by pressing F3 key. - 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**for^{-x}**x = 0, 1, 2, 3, 4, 5** - 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**. - Press STO key and type in
**A**. This assigns the name**A**to the list generated above. - 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. - 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 = x**whose^{4}**x**-coordinates are**0.5**and a number from the list**A**. - Press STO key and type in
**M**. This assigns the name**M**to the list of slopes of the secant lines generated above. - Press the GRAPH key and then pick
**y(x)=**by pressing the F1 key. - Press the ENTER key to get the prompt
**y2=**. - 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. - Press the EXIT key followed by the F5 key to pick
**GRAPH**from the menu. You should see the graph

This page was created with the help of Chris O'Brien.