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Home Limits, Analytic Geometry, and Approximations Newton's Method Examples Example 3: Approximating an Intersection of Two Graphs
 
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Example 3: Approximating an Intersection of Two Graphs

Approximate the point x where sin x = ln x.

As one can see from the graphs of sin x and In x in Figure 5.9.6, sin x and ln x cross at one point x, which is somewhere between x = 1 (where ln x crosses the x-axis going up) and x = π (where sin x crosses the x-axis going down). To apply Newton's method, we let f(x) be the function

f(x) = sin x - ln x

shown in Figure 5.9.7. We wish to approximate the zero of f(x).

05_limits_g_approx-448.gif

Figure 5.9.6

05_limits_g_approx-449.gif

Figure 5.9.7

Step 1

Choose xl = 2 (since the zero of f(x) is between 1 and π).

Step 2

f'(x) = cos x - 1/x

Step 3

05_limits_g_approx-450.gif

Step 4

Repeat Step 3. The values of xn, f(xn), and f'(xn) are shown in the table.

 

n

xn

f(xn)

f'(xn)

1

2.000000000

0.216150246

-0.916146836

2

2.235934064

-0.017827280

-1.064407894

3

2.219185522

-0.000082645

-1.054519059

4

2.219107150

-0.000000001

-1.054472505

The answer is

x ~ 2.219107150.

On a calculator we find that

sin (2.219107150) = 0.797104929

ln (2.219107150) = 0.797104930.
Home Limits, Analytic Geometry, and Approximations Newton's Method Examples Example 3: Approximating an Intersection of Two Graphs

Last Update: 2006-11-15