For which

at **x = 1** accurate to within **0.05**?

Using the TI-85 graphing calculator:

[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. - At the prompt
**y1=**, type in**ln x**and press the ENTER key.

- At the prompt
**y2=**, type in**y1 + .05**and press the ENTER key.

- At the prompt
**y3=**, type in**y1 - .05**and press the ENTER key.

- At the prompt
**y4=**, type in**x - 1**and press the ENTER key.

- The range settings for the graph above is
**[0.5, 1.5]**×**[- 0.5, 0.5]**: - Press the F5 key to choose
**GRAPH**. - Note that the tangent line meets the top curve which is the graph of
**y2**. We want to find the intersection of the tangent line with the graph of**y2**. Press the MORE key so that**MATH**is one of the menu items. - Press the F1 key to choose
**MATH**. - Press the MORE key so that
**ISECT**is one of the menu items. - Press the F5 key to choose
**ISECT**. You will see a blinking cursor in the center of the screen:Note that coordinates of the cursor at the bottom of the screen and note the number 1 in the upper right-hand corner of the screen. This tells us that the cursor is on the graph of the function

**y1**. - ( * )Since we want the top graph
**y2**, we use the up or down arrow keys to move the cursor to the graph of**y2**. The number 2 will appear in the upper right-hand corner of the screen when this happens. - Now using the left arrow key, move the cursor close to the intersection of
the intersection of the graph of
**y2**with the graph of the tangent line. - Press the ENTER key.
- The cursor moves to one of the other graphs. Note the number in the upper right-hand corner of the screen. Use the up or down arrow keys to move the cursor to the graph of the tangent line. The number 4 will appear in the upper right-hand corner of the screen when this happens.
- Press the ENTER key. After a few seconds you will
see the coordinates of the point of intersection at the bottom of the screen:
- Press the EXIT key followed by the F5 key to choose
**ISECT**again. - Find the other point of intersection by repeating the steps above starting
with ( * ). This time, instead of moving the cursor
to the left, move the cursor to the right.
- Hence, for
**x**in the interval**(0.71618945517, 1.350403256)**the linear approximation to**ln(x)**at**x = 1**is accurate to within**0.05**.