Use two iterations of Newton’s method with the initial approximation
𝑥
0
=
−
3.5
to find the third approximation
𝑥
2
the equation
2
𝑥
3
+
7
𝑥
2
+
𝑥
+
8
=
0
. (Give your answer to
4
decimal places).

Hints: Newton's method
𝑥
𝑛
+
1
=
𝑥
𝑛
−
𝑓
(
𝑥
𝑛
)
𝑓
′
(
𝑥
𝑛
)
.

Question

Use two iterations of Newton’s method with the initial approximation 
𝑥
0
=
−
3.5
 to find the third approximation 
𝑥
2
 the equation 
2
𝑥
3
+
7
𝑥
2
+
𝑥
+
8
=
0
. (Give your answer to 
4
 decimal places).

Hints: Newton's method  
𝑥
𝑛
+
1
=
𝑥
𝑛
−
𝑓
(
𝑥
𝑛
)
𝑓
′
(
𝑥
𝑛
)
.

BlackTom AI · Accepted Answer

The question asks to use two iterations of Newton’s method with x0 = −3.5 to approximate the root of f(x) = 2x^3 + 7x^2 + x + 8, and to provide x2 to 4 decimal places.

First, set up Newton’s iteration formula: x_{n+1} = x_n − f(x_n)/f'(x_n), where f'(x) = 6x^2 + 14x + 1.

Step 1: Compute x1 from x0 = −3.5
- Evaluate f(x0): with x0 = −3.5, x0^2 = 12.25 and x0^3 = −42.875. Thus f(x0) = 2(−42.875) + 7(12.25) + (−3.5) + 8 = −85.75 + 85.75 − 3.5 + 8 = 4.5.
- Evaluate f'(x0): f'(−3.5) = 6(12.25) + 14(−3.5) + 1 = 73.5 − 49 + 1 = 25.5.
- Compute x1: x1 = x0 − f(x0)/f'(x0) = −3.5 − 4.5/25.5 = −3.5 − 0.1764706 ≈ −3.6764706.

Step 2: Compute x2 from x1
- Use x1 ≈ −3.6764706. Compute x1^2 ≈ 13.5158 and x1^3 ≈ −49.69 (approximate from multiplying x1^2 by x1).
- Evaluate f(x1): f(x1) = 2x1^3 + 7x1^2 + x1 + 8 ≈ 2(−49.69) + 7(13.5158) + (−3.6764706) + 8 ≈ −99.38 + 94.6106 − 3.6765 + 8 ≈ −0.4459.
- Evaluate f'(x1): f'(x1) = 6x1^2 + 14x1 + 1 ≈ 6(13.5158) + 14(−3.6764706) + 1 ≈ 81.0948 − 51.5706 + 1 ≈ 30.5242.
- Compute x2: x2 = x1 − f(x1)/f'(x1) ≈ −3.6764706 − (−0.4459)/30.5242 ≈ −3.6764706 + 0.01461 ≈ −3.66186.

Rounding to 4 decimal places, x2 ≈ −3.6619.

Note on the answer choices: The provided data shows an answer of −3.6619 as the single option, but the options list is empty in the input. If you were expected to choose from multiple options, you would select −3.6619 as the correct numeric approximation to four decimals based on the Newton iterations above.

SMAT011 Weekly Quiz 4 |LA004

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