11.
Explanation
The minimum number of intervals required to evaluate the integral ∫011+x1dx with an accuracy of 10−i using Simpson's 1/3 Rule, we can use the error formula for Simpson's Rule. The error in Simpson's 1/3 Rule is proportional to (b−a)5/n4, where a and b are the limits of integration and n is the number of intervals.
The error formula can be written as:
E≈−180(b−a)5⋅max∣f(4)(x)∣
where f(4)(x) represents the fourth derivative of the function f(x).
In our case, a=0 and b=1. We want the error to be less than or equal to 10−i, so we set:
(1−0)5180⋅∣≤10−180(1−0)5⋅max∣f(4)(x)∣≤10−i
This simplifies to:
Max f(4)(x)∣≤10−i180
max∣f(4)(x)∣≤180⋅10i
Now, to find the minimum number of intervals (n), we can use the fact that f(4)(x) represents the fourth derivative of the function f(x)=1+x1. Let's calculate the fourth derivative:
2f′(x)=−(1+x)21
3f′′(x)=(1+x)32
4f′′′(x)=−(1+x)46
5f(4)(x)=(1+x)524
Now, we need to find the maximum value of f(4)(x) within the interval [0, 1]. Since the function f(4)(x) is decreasing, its maximum value occurs at the lower limit (x = 0):
max∣f(4)(x)∣=(1+0)524=24
Now, we can use this value to find the minimum number of intervals n by rearranging the inequality:
24≤180⋅10i
18024≤10i
log10(18024)≤i
Using logarithms, we can find the minimum value of i required to satisfy the accuracy condition:
i≥log10(18024)
Now, calculate the value:
i≥log10(152)≈−0.8239
Since i must be an integer (representing the number of decimal places of accuracy), we round up to the next higher integer, which is 1.
Therefore, the minimum number of intervals required to evaluate the integral with an accuracy of 10−i using Simpson's 1/3 Rule is 1 interval.