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Home Limits, Analytic Geometry, and Approximations Newton's Method Examples Example 2: Approximating a Fifth Root
 
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Example 2: Approximating a Fifth Root

Approximate the fifth root of 6 by Newton's method.

Step 1

We must find the zero of f(x) = x5 - 6. The graph is shown in Figure 5.9.5. Choose x1 = 1.5.

Step 2

f'(x) = 5x4

Step 3

05_limits_g_approx-445.gif

Step 4

05_limits_g_approx-446.gif

As a check we compute

(x3)5 ~ 6.001

In this example more iterations would be necessary if our first approximation had not been chosen as well. For instance, starting with x1 = 1 we would not reach the approximation 1.431 until x6, obtaining the successive approximations

x1 = 1,

x2 = 2,

x3 = 1.675,

x4 = 1.49245,

x5 = 1.43583,

x6 = 1.43100.

05_limits_g_approx-447.gif

Figure 5.9.5
Home Limits, Analytic Geometry, and Approximations Newton's Method Examples Example 2: Approximating a Fifth Root

Last Update: 2006-11-15