Let
𝑓
(
𝑥
)
=
3
𝑥
2
−
2
𝑥
−
9
. Using Newton's method, and an initial guess of
𝑥
1
=
2
, find the next approximation,
𝑥
2
, for a root of this function.

Question

Let 
𝑓
(
𝑥
)
=
3
𝑥
2
−
2
𝑥
−
9
. Using Newton's method, and an initial guess of 
𝑥
1
=
2
, find the next approximation, 
𝑥
2
, for a root of this function.

BlackTom AI · Accepted Answer

The problem asks us to apply Newton's method to the function f(x) = 3x^2 − 2x − 9 with an initial guess x1 = 2, and to find the next approximation x2.
First, determine the derivative: f'(x) = 6x − 2.
Next, evaluate the function and its derivative at x1 = 2:
f(2) = 3*(2^2) − 2*(2) − 9 = 3*4 − 4 − 9 = 12 − 4 − 9 = −1.
f'(2) = 6*(2) − 2 = 12 − 2 = 10.
Now apply Newton's update formula: x2 = x1 − f(x1)/f'(x1).
Plugging in the values gives x2 = 2 − (−1)/10 = 2 + 0.1 = 2.1.
Note: the provided answer options are missing in this problem statement, so this calculation yields the next Newton iterates value directly from the formula.

MATH-112-301-001 Unproctored Midcourse Exam 3 Practice Exam 1

Let 𝑓 ( 𝑥 ) = 3 𝑥 2 − 2 𝑥 − 9 . Using Newton's method, and an initial guess of 𝑥 1 = 2 , find the next approximation, 𝑥 2 , for a root of this function.

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