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

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

Consider the integral $\int_{-1}^{1} \sin \left(x^{2}\right) d x$. (a) Find $f^{(4)}$ for $f(x)=\sin \left(x^{2}\right) .$ (You may want to check your work with a CAS if you have one available.) (b) Graph $y=f^{(4)}(x)$ in the viewing window [-1,1] by [-30,10] (c) Explain why the graph in part (b) suggests that $\left|f^{(4)}(x)\right| \leq 30$ for $-1 \leq x \leq 1$ (d) Show that the error estimate for Simpson's Rule in this case becomes $\left|E_{s}\right| \leq \frac{h^{4}}{3}$ (e) Show that the Simpson's Rule error will be less than or equal to 0.01 if $h \leq 0.4$ (f) How large must $n$ be for $h \leq 0.4 ?$

Consider the integral $\int_{-1}^{1} \sin \left(x^{2}\right) d x$.
(a) Find $f^{(4)}$ for $f(x)=\sin \left(x^{2}\right) .$ (You may want to check your work with a CAS if you have one available.)
(b) Graph $y=f^{(4)}(x)$ in the viewing window [-1,1] by [-30,10]
(c) Explain why the graph in part (b) suggests that $\left|f^{(4)}(x)\right| \leq 30$ for $-1 \leq x \leq 1$
(d) Show that the error estimate for Simpson's Rule in this case becomes $\left|E_{s}\right| \leq \frac{h^{4}}{3}$
(e) Show that the Simpson's Rule error will be less than or equal to 0.01 if $h \leq 0.4$
(f) How large must $n$ be for $h \leq 0.4 ?$

Key Concepts

Step Size and Subinterval Calculation

The step size, denoted by h, is the width of each subinterval in numerical integration. It is directly related to the number of subintervals, n, in the integration interval. Determining the appropriate value of h is essential to ensuring that the error stays within a desired tolerance, which in turn dictates how many subintervals are required.

Simpson's Rule and Its Error Bound

Simpson's Rule is a numerical integration method that approximates the integral of a function using parabolic arcs. The error bound for Simpson's Rule involves the fourth derivative of the function, indicating that the error decreases with smaller step sizes and is directly affected by the maximum value of the fourth derivative on the interval.

Error Analysis in Numerical Integration

Error analysis is the process of quantifying the difference between the true value of an integral and its numerical approximation. It involves deriving error bounds that depend on the function's derivatives, the step size, and the number of subintervals used in the numerical method.

Differentiation and Higher Order Derivatives

This concept involves computing successive derivatives of a function, in this case up to the fourth derivative. Understanding how to differentiate composite functions, such as sin(x^2), is crucial when analyzing their behavior and in estimating errors in numerical integration techniques.

Graphical Analysis of Functions

Graphing a function provides visual insight into its behavior over a specific interval. By plotting the fourth derivative, one can estimate its maximum absolute value, which is essential for accurately bounding the error in numerical methods.

Transcript

00:01 We want to estimate the integral from negative 1 to 1 of sine of x square using simpson's rule.

00:08 We have several parts here, part a.

00:11 We find the fourth derivative of f equal sine of x square.

00:16 We graph that fourth derivative on the interval negative 1 -1.

00:22 We explain why that graph tell us that the absolute value of the fourth derivative of f is less than or equal to 30.

00:34 4x between eaddy 1 and 1.

00:37 Part e, we show that the error estimate for simpson's rule.

00:40 In this case, becomes less than or equal to h to the fourth over 3.

00:47 Part e, we show that simpson's rule error will be less than or equal to 0 .01.

00:53 If h is less than or equal to 0 .4, and in part f, we discuss about how large must then be for h be less than or equal to 0 .4.

01:06 So we start with part a defining f equals sine of x square on the interval negative 1.

01:21 And we calculate the derivatives of f.

01:27 The first derivative is derivative of sine of x square is cosine of x square, times the derivative of x square is 2x.

01:39 Second derivative is two times, which time 2 is a constant.

01:44 And we have a product x times cosine of x squared.

01:48 So the derivative of x is 1 times cosine of x square is cosine of x squared plus x times derivative of cosine of x square is negative sign of x squared times 2x.

02:04 So the second derivative of f is 2 cosine of x square, putting this 2 again inside the square bracket, minus and 2 times 2 is 4 x times x x square sign of x square now the third derivative is the derivative of this first term is 2 times negative sine of x squared times 2x minus 4 times and product again so we have 2x sine of x square plus x squared times derivative of sine of x square is the same function we have.

02:55 So we calculated that here.

02:57 It's 2x cosine of x square.

03:03 And that becomes negative 4x sine of x square minus 8x sine of x square minus 8x cube 8x cube.

03:33 8x cube.

03:34 Cosine of x square and we can simplify these two terms so the third derivative is these two terms are similar so we have a negative 12 x sine of x squared minus 8 x cube cosine of x squared and the fourth derivative will be then derivative of this is a product we have negative 12 times derivative of x is 1 times sine of x square plus x times derivative of sine of x square is cosine of x squared times 2x and that's the derivative of the first term minus 8 times the product x cubed times cosine of x squared so derivative x cube is 3x square times cosine of x square plus x cubed times the derivative of cosine of x squared is negative sign of x square times 2x.

04:49 And we distribute the constant inside of square brackets.

04:54 We got negative 12 sine of x square minus 24 x square, cosine of x square, minus 24 x square cosine of x square minus 8 minus 16 okay let's revisit that a little bit here we have a negative negative plus 8 times 2 is 16 x to the 4th times sine of x square now i think is correct and finally simple final this we get this term is 16 x to the 4th sine of x square minus 48 x square cosine of x square which is the simplification of these two terms minus 12 sine of x square so we can use a cast to verify this result and doing that right now and is correct so this is our fourth derivative of the integral function sine of x square.

06:28 And now we want to graph that over the interval negative 1 -1.

06:33 I have done so previously and i got this graph.

06:38 I'm going to show here.

06:42 And we can see the window we have been asked to do is negative 1 -1, which i did.

06:52 Here is negative 1, here is 1.

06:56 But in the vertical it is net to 30, but not up to 10, but up to 3 or 4.

07:03 That's because the graphs is much more clearer that, and it's not necessary to go up to 10 because the maximum value of this function is 0 at 0.

07:18 So it's not necessary to go up to 10.

07:21 We can see here what all this happened.

07:24 So in part c, now we can see that the function is always negative or zero...

Please give Ace some feedback