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balka75
02.01.2020 •
Mathematics
Consider the integral integral from 0 to 1 e superscript 6 x baseline dx with nequals 25 . a. find the trapezoid rule approximations to the integral using n and 2n subintervals. b. find the simpson's rule approximation to the integral using 2n subintervals. c. compute the absolute errors in the trapezoid rule and simpson's rule with 2n subintervals.
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Ответ:
a.
With n = 25,![\int_{0}^{1}e^{6 x}\ dx \approx 67.3930999748549](/tpl/images/0440/3164/a87ae.png)
With n = 50,![\int_{0}^{1}e^{6 x}\ dx \approx 67.1519320308594](/tpl/images/0440/3164/3c10d.png)
b.![\int_{0}^{1}e^{6 x}\ dx \approx 67.0715427161943](/tpl/images/0440/3164/0a861.png)
c.
The absolute error in the trapezoid rule is 0.08047
The absolute error in the Simpson's rule is 0.00008
Step-by-step explanation:
a. To approximate the integral
using n = 25 with the trapezoid rule you must:
The trapezoidal rule states that
where![\Delta{x}=\frac{b-a}{n}](/tpl/images/0440/3164/5eccf.png)
We have that a = 0, b = 1, n = 25.
Therefore,
We need to divide the interval [0,1] into n = 25 sub-intervals of length
, with the following endpoints:
Now, we just evaluate the function at these endpoints:
...
Applying the trapezoid rule formula we get
![\int_{0}^{1}e^{6 x}\ dx \approx \frac{1}{50}(1+2.54249830064281+3.23214880438579+...+634.696657835701+403.428793492735)\approx 67.3930999748549](/tpl/images/0440/3164/7eef8.png)
To approximate the integralWe have that a = 0, b = 1, n = 50.
Therefore,
We need to divide the interval [0,1] into n = 50 sub-intervals of length
, with the following endpoints:
Now, we just evaluate the function at these endpoints:
...
Applying the trapezoid rule formula we get
b. To approximate the integral
using 2n with the Simpson's rule you must:
The Simpson's rule states that
where![\Delta{x}=\frac{b-a}{n}](/tpl/images/0440/3164/5eccf.png)
We have that a = 0, b = 1, n = 50
Therefore,
We need to divide the interval [0,1] into n = 50 sub-intervals of length
, with the following endpoints:
Now, we just evaluate the function at these endpoints:
...
Applying the Simpson's rule formula we get
c. If B is our estimate of some quantity having an actual value of A, then the absolute error is given by![|A-B|](/tpl/images/0440/3164/25cf7.png)
The absolute error in the trapezoid rule is
The calculated value is
and our estimate is 67.1519320308594
Thus, the absolute error is given by
The absolute error in the Simpson's rule is
Ответ:
Step-by-step explanation:
The general form of a fraction is written as p/q, where p is called the numerator and q is called the denominator. In other words, a fraction is defined as two numbers written vertically which are separated by a horizontal line such that the top number is called numerator, and the bottom number is called denominator.
Here you can find a decimal to fraction chart and also write any decimal number as a fraction. Step 2: Multiply both top and bottom by 10 for every number after the decimal point: As we have 1 numbers after the decimal point, we multiply both numerator and denominator by 10. So, 2.5 1 = (2.5 × 10) (1 × 10) = 25 10.