- Pick any value x
_{1}> 0 as a first estimate. - To get a better second estimate x
_{2}, divide the average of x_{1}^{2}and N by x_{1}. - To get an even better estimate x
_{3}, repeat the process using x_{2}in place of x_{1}. - Likewise, get x
_{4}, x_{5}and so on.

- Pick your N: use N = 1.5XY where XY are the fourth and fifth digits of your student ID (i.e., the numbers between the hyphens, as in 123-XY-0012): N = 1.599 (say)
- Express x
_{2}in terms of x_{1}(use correct mathematical notation): x_{2}= ((x_{1}^{2}+ N)/2)/x_{1} - Likewise, express x
_{3}in terms of x_{2}and then x_{n+1}in terms of x_{n}:- x
_{3}= ((x_{2}^{2}+ N)/2)/x_{2} - x
_{n+1}= ((x_{n}^{2}+ N)/2)/x_{n}

- x
- Using N from above and 3 for your initial estimate
x
_{1}, indicate what you get for x_{2}through x_{6}(if x_{6}is not very close to N^{1/2}, you've made a mistake somewhere):- x
_{2}= 1.76650000000 - x
_{3}= 1.33583986697 - x
_{4}= 1.26641981342 - x
_{5}= 1.26451714901 - x
_{6}= 1.26451571758 (note: N^{1/2}= 1.26451571757729)

- x
- On the accompanying graph (BELOW) of f(x) = x
^{2}, very carefully and accurately draw in and label the line y = N. What is the significance of the x-coordinate where the graphs of y=f(x) and y=N intersect? (Use a complete sentence in your answer.)

The x-coordinate of the point where the two graphs intersect is N^{1/2}. - Again on the accompanying graph of f(x) = x
^{2}, plot and label the points P_{1}= (x_{1}, f(x_{1})) and Q_{1}= (x_{2}, N) and draw in the line through these points P_{1}and Q_{1}. Do likewise for P_{2}= (x_{2}, f(x_{2})) and Q_{2}= (x_{3}, N). (If your points and lines are not very carefully placed, you will not see what you need to see.) What is special about the line through P_{1}and Q_{1}among all lines through P_{1}? Is this special behavior true for the line through P_{2}and Q_{2}? (Use complete sentences in your answer.)

Both lines are tangent to the graph of f(x) = x^{2}. - Based on your answers above, describe a procedure
for how you could graph the points (x
_{2}, N), (x_{3}, N), (x_{4}, N), and so on, just using a ruler and your graphs of f(x) = x^{2}and y= N, without ever resorting to a calculator to compute any values of x_{n}. Explain briefly, perhaps referencing your accompanying graph, how your procedure shows that x_{n}gets closer and closer to N^{1/2}as n gets bigger and bigger. (Use good English and complete sentences.)

To plot the point (x_{2}, N), draw in the line tangent to the graph of f(x) = x^{2}at the point P_{1}. This line crosses the line y = N at the point Q_{1}= (x_{2}, N). The point P_{2}is the point where the vertical line through Q_{1}crosses the graph of f(x) = x^{2}. Now draw in the line tangent to the graph of f(x) = x^{2}at the point P_{2}. This line crosses the line y = N at the point (x_{3}, N). In general, given the point Q_{n}= (x_{n+1}, N), P_{n+1}is the point where the vertical line through Q_{n}crosses the graph of f(x) = x^{2}, and the line tangent to the graph of f(x) = x^{2}at the point P_{n+1}crosses the line y = N at the point Q_{n+1}= (x_{n+2}, N). By repeatedly drawing in these tangents, we can obtain the points (x_{2}, N), (x_{3}, N), (x_{4}, N), and so on.

Note that the tangent line at P_{n}intersects the line y = N to the left of P_{n}. Thus x_{n+1}is always smaller than x_{n}. But the tangent lines are always below the graph of f(x), so the points Q_{1}, Q_{2}, etc., although proceeding to the left, are always to the right of the point where y = f(x) and y = N intersect. I.e., the values x_{1}, x_{2}, etc., are decreasing but are all bigger than N^{1/2}. Thus the values of x_{n}are getting closer and closer to N^{1/2}.