Problem:
For which x is the linear approximation to

f(x) = ln(x)

at x = 1 accurate to within 0.05?

Visualization:

Using the TI-85 graphing calculator:

[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. Press the GRAPH key and then pick y(x)= by pressing the F1 key.
3. If necessary, keep pressing F4 key until only y1= appears on the screen.
4. At the prompt y1=, type in

ln x

and press the ENTER key.

5. At the prompt y2=, type in

y1 + .05

and press the ENTER key.

6. At the prompt y3=, type in

y1 - .05

and press the ENTER key.

7. At the prompt y4=, type in

x - 1

and press the ENTER key.

8. The range settings for the graph above is  [0.5, 1.5] × [- 0.5, 0.5]:

9. Press the F5 key to choose GRAPH.

10. 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.
11. Press the F1 key to choose MATH.
12. Press the MORE key so that ISECT is one of the menu items.
13. 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.

14. ( * )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.
15. 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.
16. Press the ENTER key.
17. 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.
18. Press the ENTER key. After a few seconds you will see the coordinates of the point of intersection at the bottom of the screen:

19. Press the EXIT key followed by the F5 key to choose ISECT again.
20. 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.

21. 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.