Use a computer algebra system and the error formulas to find n such that the error in the approx

Use a computer algebra system and the error formulas to find...

Key Concepts

Simpson's Rule

Simpson's Rule is a numerical integration technique that improves on the trapezoidal rule by approximating the integrand with quadratic polynomials over pairs of subintervals. It typically offers much higher accuracy by taking into account the function’s curvature more effectively, and its error bound involves the fourth derivative of the function.

Error Bound Analysis

Error Bound Analysis in numerical integration involves estimating the maximum possible error of an approximation method by using known formulas that depend on the function's higher-order derivatives. By ensuring the error is below a specified tolerance, one can determine the necessary number of subintervals. This analysis is crucial for balancing computational efficiency with accuracy requirements.

Numerical Integration

Numerical integration refers to a suite of methods used to approximate the value of definite integrals, especially when an antiderivative of the function is hard or impossible to find. These methods replace the continuous problem with a discrete approximation by summing values of the function at specific points over the interval.

Trapezoidal Rule

The Trapezoidal Rule approximates the area under a curve by dividing the interval into subintervals and approximating the integrand with a straight line (or trapezoid) in each subinterval. Its accuracy depends on the function's curvature, which is quantified by the second derivative, and the error bound is proportional to the maximum of the absolute value of the second derivative over the interval.

Transcript

00:01 Using a computer algebra system and the error formulas, we want to find a natural number n such that the error in the approximation of the definite integral is less than 0 .001, using in part a the trapezoidal rule and in part b, simpson's rule.

00:26 We are talking about the integral from 0 to 1 of tangent of x square.

00:31 So we define the function f of x equal tangent of x squared, and we define that function on the interval, the close interval from 0 to 1, which is the interval of integration.

01:00 So in part a, we know that the absolute value of the error in trapezoidal rule, which we call e.

01:15 Values less than or equal to b minus a cube, where a and b are the lower and upper limits of integration respectively, over 12 n squared times, let's say m, where the constant m is an upper pound for the second derivative of the function f over the interval of integration ab.

01:50 So if the the second derivative of the function is continuous over the interval ab.

01:59 And for that reason, the function attained its extreme values, its maximum and minimum values.

02:06 Then we know that instead of m, we can use the maximum value of the absolute value of the second derivative.

02:12 But at least if we have a bound for that second derivative in absolute value, we can put it here to form a bound of the error in trapsoid or bough.

02:22 In this case, the derivatives of this function start to be complicated formula, so it is better to plot the second derivative of f.

02:38 So if we plot the absolute value of the second derivative of f over the interval 01, it is easy to see that the absolute value is always less than or equal to in this case, we see is 50.

03:19 So i'll overbound for that second derivative over the interval 01 is 50.

03:28 And so we can say that the absolute value of the error intrapsos as a rule will be less and or equal to 1 minus 0 cube over 12 n square times 50.

03:43 That is 50 over 12 and square, which is the same of 25 over 6.

03:51 Square.

03:54 So now if we wanted the error in absolute value in terms of the rule be less than 10 to a negative 5, it is sufficient to bound here by that number.

04:06 That is if 25 over 6 in square is less than or equal to 10 to a negative 5, then by transitivity, the absolute value of the error in troposite the rule will also be less than or equal to 10 to negative 5.

04:26 10 to negative 5 is this accuracy we want to achieve in the absolute value of the error intrapezoidal rule.

04:37 So what we got to do is to find a natural number and for which the bound 25 over 6n square be less or equal than 10 to a negative 5.

04:57 So we start from this inequality and considering that all the quantity, and expressions in this inequality are positive, we get that n squared got to be created down or equal to 25 times 10 to the fifth over 6.

05:25 And so n got to be created down or equal to the square root of 25 times 10 to the 5th over 6.

05:33 And this value, this square root here, is about 645.

05:44 And because n got to be a positive integer, we can say that n got to be greater than or equal to 646.

05:57 So taking 646 or more sub -interables in the trapezoidal rule, we know we approximate the given integral with an error less than or equal to 1025.

06:14 It's true that we can achieve that accuracy may be way before these numbers of intervals.

06:20 That is, we can surely achieve the accuracy we want way before this number of intervals...

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