Use the error formulas to find n such that the error in the approximation of the definite integr

Use the error formulas to find n such that the error in the...

Key Concepts

Numerical Integration

Numerical integration involves estimating the value of a definite integral when an exact analytical solution is either difficult or impossible to obtain. This method substitutes the integral with a weighted sum of function values at specified points within the interval, making it useful in applied contexts where functions are too complex for direct integration.

Trapezoidal Rule

The trapezoidal rule is a simple numerical integration technique that approximates the area under a curve by dividing it into trapezoids rather than precise shapes of the function. The accuracy of this method depends on the smoothness of the function and the number of subdivisions, and it uses the values of the function at the endpoints of each subinterval to form the trapezoids.

Error Bound for the Trapezoidal Rule

The error bound for the trapezoidal rule gives an estimate of the worst-case error of the approximation. This bound is directly related to the maximum absolute value of the second derivative of the function over the interval of integration, indicating that functions with high curvature may require more subdivisions to achieve a desired accuracy.

Simpson's Rule

Simpson's rule is a numerical integration method that improves on the trapezoidal rule by approximating the integrand with quadratic polynomials over pairs of subintervals. By fitting parabolic segments, Simpson's rule generally offers higher accuracy especially when the function is smooth, making it very effective for approximating definite integrals.

Error Bound for Simpson's Rule

The error bound for Simpson's rule provides an estimate of the maximum error in its approximation and depends on the fourth derivative of the function over the integration interval. As a result, functions that are well approximated by polynomials of degree up to three will typically yield very small errors when Simpson's rule is applied with a sufficient number of subintervals.

Transcript

00:01 All right, for this problem, we are asked to use the error formulas to find n, so that the error in the approximation of the integral from 1 to 3 of e to the power of 2x is less than 0 .001, using first the trapezoidal rule, then simpson's rule.

00:19 So the way that we can begin here is essentially we want to set this upper bound equal to the desired value and then rearrange for n.

00:30 So we would have b minus a cubed over 12 n squared times the maximum value of the second derivative equals d, which we can rearrange without too much effort to find that n is going to equal the square root of b minus a cubed times i'm just going to write m for max divided by that'll be 12d.

00:56 Or d is our desired value.

01:00 Then we will need to figure out m to actually calculate this.

01:04 So m is going to be the maximum value of the second derivative of our function.

01:11 And since our function is e to the power of 2x, the first derivative would be 2e to the power of 2x.

01:17 Second derivative will be 4e to the power of 2x.

01:20 And on the interval from 1 to 3, its maximum value is going to be 4e to the power of 6.

01:26 So we have m is 4e to the power of 6.

01:29 E minus a is going to be 3 minus 1.

01:32 So we'll have 3 minus 1 cubed, 2 cubed, we'll have 8.

01:37 8 times 4e to the power of 6, divided by 12 times 0 .001.

01:43 I'm going to pause and calculate that.

01:48 The result is going to be a whopping 3 ,279 .95...