Introduction: It is easy to graph lines and find their x-intercepts. You will be guided through the basic ideas of Newton's Method, which uses x-intercepts of appropriate lines to approximate x-intercepts of more difficult functions. Note: We need zeroes of a function y to find its x-intercepts; zeroes of y' to find stationary points of y; and zeroes of y'' to find possible points of inflection of y. Sometimes we just need to find where two functions cross. Many calculators use Newton's Method with y=x²-a and an initial guess of 1 to find the square root of a.

Elements of this lab were adapted from Solow's "Learning by Discovery", Edwards & Penney's "Single Variable Calculus", and Harvey & Kenelly's "Explorations with the TI-85". More information can be found in the annotated Bibliography .

A Let y = f(x) be a function. On the figure below, graph the tangent line to f(x) at x₀. Label the point (x₀, f(x₀)), the graph y=f(x), the tangent line T1(x), the root r of y=f(x), and the x-intercept x₁ of the tangent line.

What is the equation of the line T1(x) tangent to the graph of f at (x_0,f(x₀))?

Show that the x-intercept of T1(x), x₁, is given by x₁= x₀-f(x₀)/f'(x₀) .

We repeat the process, using (x₁, f(x₁)) as our new point at which to draw the tangent line. The x-intercept of the new line is x_2. On the figure above, sketch the tangent lines T1 and T2. Show x₁ , and x₂. Show x₃, if possible.

Write a formula for x₂ in terms of x₁.

Write a formula for x_n+1 in terms of x_n.

B Let's work with a function with a nice alegraic definition and some graphs.

Let y1=x³ - 4 x² - 1 in the Graph menu of the calculator.

1. View the graph with a ZoomStd. Use Trace to approximate the value of the root

r=___________.

2. Exit from the graph screen to the home screen. Access "Catalog" and type "t" (no alpha key needed here). Scroll down until you see "TanLn". Press "Enter". You should see "TanLn(" in the home screen. Complete the command by typing "y1,5)" (note the lower case y.) You will see the tangent line drawn on the previous graph. (To clear this at any time that you wish, choose "CLRDRW" from the "DRAW" sub-menu of "Graph".) Make a sketch of your graph.

3. Write the derivative of f(x) = x³ - 4 x² - 1.

In the "Graph y=" menu, set up y2=f'(x). Exit to the home screen.

4. Let's use Newton's Method.  First, one guesses x₀. Then, for n=1,2,3,..., x_n+1= x_n-f(x_n)/f'(x_n) . The method is repeated until the value doesn't change much. Here are instructions to perform it on the TI-85. Skip any steps you have already done.

From the home screen, set the number of decimals you wish to use with "MODE", "FLOAT" and the number of decimals; let's use 7.

In the "Graph y=" menu, define

y1 = original function y2 = its derivative y3 = x-y1/y2.

Deselect all functions but y1.

Graph in an appropriate window. Zoom in to find an initial guess for a root.

Exit to the home screen.

Store your initial guess, (we'll use 5 now), to x:

5 STO x (remember to use lower case x).

Since the value of the new x will be calculated in y3 and re-used as x, our next command is

y3 STO x (remember to use lower case y and x).

We can see the value on the screen.

"Entry" "Enter" .

Repeat the "Entry" and "Enter". Stop when the value no longer changes, or you have reached the number of iterations specified, or something wierd seems to be happening.

4a. Use Newton's Method on y = x³ - 4 x² - 1 with x₀ = 5. Record your values in the table. Once the value repeats, you may quit. Notice that the Method converges to the root. Usually, when Newton's method does converge.
 

n	xn	n	xn
0	5.0000000	5
1		6
2		7
3		8
4		9

4b. Newton's Method does not always work well. It is sensitive to your initial guess. Use Newton's Method on y = x³ - 4 x² - 1 with x₀ = 2. Record your values here. Notice that the Method does not converge to the root. What seems to be happening?

 
 

n	xn	n	xn
0	2.0000000	5
1		6
2		7
3		8
4		9

 

C. Deselect all previous functions. Plot y4=3sinx and y5=lnx with xmin=-5, xmax=30, ymin=-5, and ymax=5. Note that they intersect several times. To find these intersections, perform Newton's method with y1=y4-y5. Calculate the derivative, y2 and record here.

1. Begin with x₀=3.
 

n	xn	n	xn
0	3.0000000	5
1		6
2		7
3		8
4		9

2. Choose appropriate values of x₀ to find the other zeroes.
 

n	xn	n	xn
0		5
1		6
2		7
3		8
4		9

 

n	xn	n	xn
0		5
1		6
2		7
3		8
4		9

 

n	xn	n	xn
0		5
1		6
2		7
3		8
4		9

 

 
 

n	xn	n	xn
0		5
1		6
2		7
3		8
4		9

 

n	xn	n	xn
0		5
1		6
2		7
3		8
4		9