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Example: Find the value of (75)1/4 to five decimal places.
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Recall Strategy:
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1. |
Write down the
expression for f(x). In this
example, f(x) =
x4 ˗ 75
= 0
and f '(x)
= 4 x3 .
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2. |
Set up a table with
columns for xn ,
f(xn) , f ' (xn)
, f(xn)
/ f ' (xn) , and xn+1 .
See below.
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3. |
Identify the starting
value, x1. In this example 24 =
16 and 34 = 81
So pick the value
of 3 for the starting value
of x.
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4. |
Once the value for the
root, xn+1 is found, it represents the new
value for xn .
Repeat the iteration until reaching the desired accuracy.
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n xn (xn)4
˗ 75
4 (xn)3 [(xn)4
˗ 75] / 4 (xn)3 xn+1
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1 |
3.0 |
6.0 |
108.0 |
0.055555555 |
2.94444444 |
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2 |
2.94444444 |
0.16461857 |
102.110425 |
0.001612162 |
2.94283228 |
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3 |
2.94283228 |
0.00013515 |
101.942792 |
0.000001326 |
2.94283095 |
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Result: The value of (75)1/4 is approximately 2.94283
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Click here for another example.
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