Using the TI-85 graphing calculator, find a

Let

[Press here to see animation again!]

- 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.
- Press the GRAPH key and then pick
**y(x)=**by pressing the F1 key. - If necessary, keep pressing F4 key until only
**y1=**appears on the screen. - Type in the function
**abs(x^3 + 5 x + 1 - 19) - 0.1**and press the ENTER key. You can get the function

**abs**by pressing 2nd CATALOGUE and the ENTER keys. - Press EXIT and the F2 keys
to pick
**RANGE**from the menu. - Let us begin with the viewing window
**[ 1, 3] × [-1, 1]**: - Press the F5 key to graph the function:
- To get a clearer picture of what is happening, we shall zoom in. Press
F3 key to pick
**ZOOM**from the menu. - Now press F2 key to pick
**ZIN**, the command for zooming in. - Press the ENTER key several times to get a good
graph:
- We shall now find the two roots of our functions which are shown in the
above graph. Press the EXIT key twice folowed by the
MORE key so that
**MATH**appears on the menu. - Now press F1 key to pick
**MATH**. - Press F3 key to pick
**ROOT**. - Use the arrow keys to move the blinking cursor close to one of the
roots and press the ENTER key. The following
shows the left-hand root:
which is

**1.9941053957...**. - Repeat the preceding step to find the other root:
which is

**2.005870179...**. - We see that the interval
**[1.9945, 2.0055]**lies between the two roots and, hence, if we choose**d = 0.0055**then we have that**0 < |x - 2| < 0.0055**implies that**|f(x) - 19| < 0.1**. - We can visualize this by graphing the function
**f(x) = x**using the window^{3}+ 5 x + 1**[1.9945, 2.0055] × [18.9, 19.1]**:and get

**Acknowledgment:** The first place which the author saw this way to approach
this problem is in Jerry Johnson and Benny Evans' **Discovering Calculus with
Derive**.