Answer :
By using Simpson's rule the correct answer is 151.
To find out how many points we need to take in order to achieve an error ≤ 1 × 10^(-6) using Simpson's rule, we need to use the error formula for Simpson's rule.
The error formula for Simpson's rule is given by:
Error ≤ (1/180) * (b - a) * h^(4) * max|f''''(x)|,
where:
- a and b are the limits of integration (in this case, a = 0 and b = 1),
- h is the step size (h = (b - a) / n, where n is the number of points),
- and max|f''''(x)| is the maximum value of the fourth derivative of the function f(x) within the interval [a, b].
In this case, f(x) = 2x^(5) + 3x^(4).
To find the fourth derivative of f(x), we need to take the derivative of f''(x) twice.
First, let's find f''(x):
f''(x) = (d^(2)/dx^(2))(2x^(5) + 3x^(4))
= (10x^(3) + 12x^(2)).
Now, let's find the fourth derivative of f(x):
f''''(x) = (d^(4)/dx^(4))(10x^(3) + 12x^(2))
= (60x + 24).
To find the maximum value of f''''(x) within the interval [0, 1], we need to evaluate it at the critical points and endpoints.
The critical point occurs when f''''(x) = 0:
60x + 24 = 0
x = -4/5.
Since x = -4/5 is not within the interval [0, 1], we only need to evaluate f''''(x) at the endpoints of the interval.
When x = 0, f''''(0) = 24.
When x = 1, f''''(1) = 84.
So, the maximum value of f''''(x) within the interval [0, 1] is 84.
Now, let's substitute the values into the error formula and solve for n:
Error ≤ (1/180) * (1 - 0) * (1/n)^(4) * 84 ≤ 1 × 10^(-6).
Simplifying the equation:
(1/180) * (1/n)^(4) * 84 ≤ 1 × 10^(-6),
(1/n)^(4) * 84 ≤ 1 × 10^(-6) * 180,
(1/n)^(4) ≤ 1 × 10^(-6) * 180 / 84,
(1/n)^(4) ≤ 0.002142857,
1/n ≤ (0.002142857)^(1/4),
n ≥ 1 / (0.002142857)^(1/4).
Evaluating the expression:
n ≥ 150.8.
Since n must be a whole number, we round up to the nearest whole number, which is 151.
Therefore, we must take at least 151 points in order to achieve an error ≤ 1 × 10^(-6) using Simpson's rule.
So, the correct answer is 151.
Learn more about Simpson's rule on
https://brainly.com/question/34267272
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