Problem:
Find and graph the tangent line to the curve represented by the following pair of parametric equations:

x = t2 + 1,   y = t3 - t

at t = 1.5 using the TI-85 graphing calculator.

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. In order to graph a system of parametric equations, the calculator must be in the PARAM graphing mode. To change into this mode, press 2nd MODE. You should see a screen similar to the following:

3. Using the arrow keys, move the blinking cursor to the PARAM item of the fifth row and press the ENTER key.
4. Press the EXIT key.
5. Press the GRAPH key and then pick E(t)= by pressing the F1 key.
6. If necessary, keep pressing F4 key until only the two items xt1= and yt1= appear on the screen.
7. At the prompt xt1=, type in

t ^ 2 + 1

and press the ENTER key.

8. At the prompt yt1=, type in

t ^ 3 - t

and press the ENTER key.

9. The range settings for the graph above are

10. Pick the item GRAPH from the menu by pressing the F5 key.

11. Press the MORE key so that MATH appears on the menu. Pick MATH by pressing the F1 key.
12. Press the MORE key again so that TANLN appears on the menu. Pick TANLN by pressing the F1 key.
13. Press the right arrow key until the value t = 1.5 appears in the left lower corner of the screen:

A blinking cursor indicates the point on the curve corresponding to t = 1.5. The coordinates of the point, x = 3.25 and y = 1.875 appear at the bottom of the screen.

14. Press the ENTER key to draw the tangent line and to calculate its slope which appears at the bottom of the screen, dy/dx = 1.9166666667.

Hence, the equation of the tangent line is

y - 1.875 = 1.9166...(x - 3.25).