A computer takes 2 seconds to compute a particular definite integral accurate to 4 decimal places. Approximately how long does it take the computer to get 12 decimal places of accuracy using each of the LEFT, MID, and SIMP rules?
Round your answers to one decimal place.
(a) LEFT ¡Ö years
(b) MID ¡Ö hours
(c) SIMP ¡Ö minutes
(a) LEFT
For left hand or right hand rule, every extra digit requires 10 times of work. 4 digit needs 2 second. From 4 to 12, it has 8 digits. 2 * 10^(8) seconds / 60 /60 /24 /365 = 6.3 years
(b) MID
For midpoint rule, every 2 digit requires 10 times of work.
2 * 10^(8/2) /60 /60 = 5.6 hours
(c) SIMP
Each 4 digits requires 10 times of work.
2 * 10^(8/4) /60 = 3.3 mins.
To estimate the time it takes for the computer to compute a definite integral with different rules, we need to consider the convergence rate of each rule.
For the LEFT rule, the convergence rate is O(1/n), meaning that by doubling the number of intervals (n), we can expect the precision to increase by a factor of 2.
For the MID rule, the convergence rate is O(1/n^2), meaning that by doubling the number of intervals (n), we can expect the precision to increase by a factor of 4.
For the SIMP rule, the convergence rate is O(1/n^4), meaning that by doubling the number of intervals (n), we can expect the precision to increase by a factor of 16.
Let's assume that the computer can compute the definite integral using each rule in 2 seconds accurately to 4 decimal places. To find out approximately how long it takes for each rule to compute 12 decimal places of accuracy, we need to determine how many times we need to double the number of intervals to increase the precision by a factor of 10^8 (16^2).
For the LEFT rule:
(1/n)^(10^8) = (1/2)^(4)
1/n = (1/2)^(4/(10^8))
n = (2^(4/(10^8)))
For the MID rule:
(1/n^2)^(10^8) = (1/2)^(4)
1/n^2 = (1/2)^(4/(10^8))
n = (2^(2/(10^8)))
For the SIMP rule:
(1/n^4)^(10^8) = (1/2)^(4)
1/n^4 = (1/2)^(4/(10^8))
n = (2^(1/(10^8)))
Using these values, we can estimate the time it takes for each rule to compute 12 decimal places of accuracy:
(a) For the LEFT rule:
time = 2 seconds * (2^(4/(10^8))) seconds
(b) For the MID rule:
time = 2 seconds * (2^(2/(10^8))) seconds
(c) For the SIMP rule:
time = 2 seconds * (2^(1/(10^8))) seconds
Let's calculate each approximation:
(a) LEFT ≈ 2 * (2^(4/(10^8))) ≈ 2 seconds
(b) MID ≈ 2 * (2^(2/(10^8)))
≈ 2 seconds
(c) SIMP ≈ 2 * (2^(1/(10^8)))
≈ 2 seconds
Therefore, for all three rules, it would take approximately 2 seconds to compute a definite integral accurate to 12 decimal places.
To estimate the time taken by the computer to achieve higher decimal places of accuracy using different rules, we need to consider the relationship between the number of decimal places and the computational effort required.
The general formula to estimate the number of decimal places of accuracy using the midpoint (MID) and Simpson's (SIMP) rules is given by:
n = -log10(d) + k
where n is the number of decimal places of accuracy, d is the desired accuracy (in this case, 12 decimal places), and k is a constant that depends on the rule used.
For the LEFT rule, the number of decimal places of accuracy can be estimated using the formula:
n = -log10(d) + k'
where k' is a different constant compared to the MID and SIMP rules.
Let's calculate the values for each rule:
(a) LEFT:
Using a similar approach, we can use the given information that the computer takes 2 seconds to compute a definite integral to 4 decimal places of accuracy. We are trying to estimate the time it takes to achieve 12 decimal places of accuracy using the LEFT rule. By equating the two formulas, we get:
-4 + k' = -12 + k
k' - k = -12 + 4
k' - k = -8
(b) MID:
Using the formula mentioned earlier, we can estimate the time in hours using the MID rule. By equating the formulas, we have:
-12 + k = -log10(12) + k
k = -log10(12)
(c) SIMP:
Similar to the MID rule, we can estimate the time in minutes using the SIMP rule. By equating the formulas, we have:
-12 + k = -log10(12) + k
k = -log10(12)
Now, let's calculate the values for each rule.
(a) LEFT:
Since we are given that the computer takes 2 seconds for 4 decimal places of accuracy, we can assume that the constant k' is 0 (since it will cancel out when calculating the difference):
k' - k = -8
0 - k = -8
k = 8
To estimate the approximation, we can equate the values of 12 decimal places and 8 constant value into the formula:
n = -log10(d) + k
n = -log10(12) + 8
n ≈ 7.4
Therefore, it would take approximately 7.4 years for the computer to achieve 12 decimal places of accuracy using the LEFT rule.
(b) MID:
Using the constant value we calculated earlier:
k = -log10(12)
k ≈ 2.08
To estimate the time in hours, we can equate the values of 12 decimal places and 2.08 constant value into the formula:
n = -log10(d) + k
n = -log10(12) + 2.08
n ≈ -9.9
Therefore, it would take approximately 9.9 hours for the computer to achieve 12 decimal places of accuracy using the MID rule.
(c) SIMP:
Using the constant value we calculated earlier:
k = -log10(12)
k ≈ 2.08
To estimate the time in minutes, we can equate the values of 12 decimal places and 2.08 constant value into the formula:
n = -log10(d) + k
n = -log10(12) + 2.08
n ≈ -9.9
Therefore, it would take approximately 9.9 minutes for the computer to achieve 12 decimal places of accuracy using the SIMP rule.
In summary:
(a) LEFT ≈ 7.4 years
(b) MID ≈ 9.9 hours
(c) SIMP ≈ 9.9 minutes