Consider the integral ∫-1^1 sin(x^2) d x a. Find f^(4) for f(x)=sin(x^2) . (You may want to chec

Consider the integral ∫-1^1 sin(x^2) d x a. Find f^(4) for...

Key Concepts

Graphical Analysis for Derivative Bounds

Graphical methods can be used to estimate the maximum absolute value of higher-order derivatives over a given interval. By plotting the derivative function, one can visually identify its extreme values. This practice aids in establishing conservative yet practical bounds that ensure the error estimates in numerical integration are valid.

Higher-Order Differentiation Techniques

Computing higher-order derivatives, such as the fourth derivative, often requires the systematic application of differentiation techniques like the chain rule, product rule, and power rule. Mastery of these techniques is essential in obtaining error estimates for numerical methods, as they enable the determination of bounds on derivatives used in error formulas.

Mesh Size and Partitioning in Numerical Integration

Selecting an appropriate mesh size, or equivalently the number of subintervals, is critical in controlling the error of numerical integration methods such as Simpson's Rule. The mesh size directly influences the error bound; smaller ?x values lead to a more accurate approximation, and determining the minimum number of partitions to achieve a desired error tolerance is a key aspect of applying these numerical techniques effectively.

Simpson's Rule

Simpson's Rule is a numerical method for approximating definite integrals by partitioning the integration interval into even subintervals and approximating the function by quadratic polynomials over each pair of subintervals. It is particularly effective for smooth functions and provides high accuracy when the function is well-approximated by parabolic segments.

Error Analysis in Numerical Integration

The error in Simpson's Rule can be estimated using the fourth derivative of the integrand. The error bound typically takes the form involving (?x)^4 multiplied by a constant that depends on the maximum absolute value of the fourth derivative over the interval. This concept is crucial for understanding how the choice of step size affects the accuracy of the integral approximation.

Transcript

00:01 Here we consider the integral from negative 1 to 1 of sine of x square.

00:07 We will find the fourth derivative of f, where f is the integral function f of x equal sine of x squared.

00:16 In par p, we graph the fourth derivative of f in the interval of negative 1 -1.

00:21 In par c, we try to find a bound for the f -servative of the fourth derivative.

00:25 We show that that bound could be 30, where all the points in negative 1 -1.

00:33 Part d, we showed that the error in simpson's rule satisfy the inequality absolute value of the error in simpson's rule less than or equal to delta x, step size to the fourth over three.

00:46 In part e, we show that the absolute value of the error in simpson's rule is less than or equal to 0 .01 if the step size delta x is less than or equal to 0 .4.

00:58 And in part of which we are going to address the problem of finding how large and should be in order to have the inequality delta x less than or equal to 0 .4.

01:13 So we start in part a by calculating the fourth derivative of sine of x square.

01:25 So the first derivative is equal to a derivative of sine is cosine of x squared.

01:34 Times derivative of x square is 2x so this is already the first derivative now the second derivative of f is equal 2 2 is a constant so we get a product x and times cosine of x square and the derivative of x first is 1 times cosine of x square plus x times derivative of x square plus x times derivative of of x squared is negative sign of x squared times derivative x squared is 2x so the second derivative is equal to 2 cosine of x square minus 4x square sign of x square so this is the second derivative now the third derivative of f differentiate formula here so we get 2 is a constant here in this first term and derivative of cosine of x squared is negative sign of x squared times the derivative of x squares 2x minus 4 is a constant so we get a product x squared times sign of x square derivative of the first function x square is 2x that multiplied by sine of x square plus x squared times derivative of sine of x squared is cosine of x squared times derivative of x squares to x so this is negative 2 sign sorry rather it's negative 4 2 times 2 4 x sine of x square minus 4 multiplying 2x sine of x square and here we get plus 2x cube cosine of x square now we distribute this 4 here and we get negative 4 x sign of x square minus 8 x sine of x square minus 8 x cube cosine of x square so this is the third derivative and now finally the 4 derivative of f is equal to then this term to simplify a little bit more for doing this derivative here we can't simplify these two terms is one and this one which are similar and we get negative 12 x sine of x square minus 8 x cube cosine of x square.

05:07 And now we can find the derivative of this to have the fourth derivative of f, and that is negative 12.

05:17 It's a constant, and we get the product, x times sine of x square.

05:22 Derivative of x is 1 times sine of x squared plus x times derivative of sine of x square is cosine of x squared times derivative x squared is 2x.

05:34 So this is the first part.

05:37 Now, get negative 8 is a constant times and we have a product again and this is the first term.

05:47 The first factor is x cubed.

05:49 The derivative of that is 3x square times the other factor cosine of x square plus x cubed times the derivative of cosine of x square is negative sign of x square times the derivative of x squared is 2x.

06:06 And there is, so this is negative 12 times sine of x square plus 2x square, cosign of x square, minus a times 3x square, cosine of x square, minus 2x to the fourth, sine of x square.

06:36 Now we distribute the coefficients in front of the square brackets in this case, and we get negative 12 sine of x square minus 24 x squared cosine of x square, minus 24 x squared cosine of x square, minus 24 x square coside of x square, plus 16 x to the fourth sign of x square.

07:11 And as we can see here, these two terms are similar.

07:18 And then we get the default derivative of f is equal to 16 x to the fourth sine of x square.

07:34 These two terms here give us negative 48 x squared cosine of x square.

07:45 And the last term, this one minus 12.

07:51 Sine of x squared and this is the fourth derivative of f and this we are going to plot in the interval or over the integral negative 1 -1 we have done this over here we got the graph of this fourth derivative of f we can see this is a negative function overall the interval negative 1 -1 here we have the endpoints at the one here, one here.

08:35 And it's always negative...

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