Using Newton's method to approximate the zero of
𝑓
(
𝑥
)
=
3
𝑥
2
−
3
𝑥
+
1
, determine the second approximation if the initial one is
𝑥
1
=
2
.

Question

Using Newton's method to approximate the zero of 
𝑓
(
𝑥
)
=
3
𝑥
2
−
3
𝑥
+
1
, determine the second approximation if the initial one is 
𝑥
1
=
2
.

BlackTom AI · Accepted Answer

The problem asks us to use Newton's method to approximate the zero of f(x) = 3x^2 − 3x + 1, given the initial approximation x1 = 2.
First, recall Newton's iteration formula: x_{n+1} = x_n − f(x_n)/f'(x_n).
We compute the derivative: f'(x) = 6x − 3.
Evaluating at x1 = 2:
f(2) = 3*(2)^2 − 3*(2) + 1 = 12 − 6 + 1 = 7.
f'(2) = 6*(2) − 3 = 12 − 3 = 9.
Thus the second approximation is x2 = x1 − f(x1)/f'(x1) = 2 − 7/9 = (18 − 7)/9 = 11/9.
Note: The provided data for answer options is missing in the prompt, so there are no explicit distractors listed to evaluate. The given answer entry '11
9' corresponds to the fraction 11/9, which matches the calculated x2 value. If you were choosing from multiple options, you would select the option equivalent to 11/9.
In summary, the second Newton iterate for f(x) with x1 = 2 is x2 = 11/9. The numerical result 11/9 is the same as the representation shown as '11' on one line and '9' on the next in the provided format.

MATH-112-301-001 Unproctored Midcourse Exam 3 Practice Exam 2

Using Newton's method to approximate the zero of 𝑓 ( 𝑥 ) = 3 𝑥 2 − 3 𝑥 + 1 , determine the second approximation if the initial one is 𝑥 1 = 2 .

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