In Exercises 23-26, use a CAS to perform the following steps. a. Plot the functions over the giv

In Exercises 23-26, use a CAS to perform the following...

Key Concepts

Function Plotting

Plotting a function over a specified interval is a fundamental technique used to understand the behavior and key characteristics of the function, such as continuity, limits, and oscillatory patterns. It aids in visualizing how the function behaves across the domain and helps to identify points where analytic methods may need to be applied.

Partitioning of an Interval

Partitioning an interval into a number of equal subintervals is a crucial step in numerical methods. This concept underpins strategies such as Riemann sums and other integration techniques where the domain of a function is discretized, allowing for a systematic approximation of integrals or average values.

Midpoint Rule

The midpoint rule is a numerical integration method that approximates the integral of a function by evaluating it at the midpoint of each subinterval formed by partitioning the interval. This technique yields a good approximation of the overall integral, particularly when the function is reasonably smooth over the interval.

Average Value of a Function

The average value of a function on a given interval is determined by integrating the function over that interval and then dividing the result by the length of the interval. This provides a single representative value that summarizes the overall behavior or performance of the function across the domain.

Root-Finding in Numerical Analysis

Root-finding techniques are used to solve equations where a function equals a given value or zero. In the context of numerical analysis, these methods—such as the bisection method, Newton-Raphson method, or using built-in CAS solvers—are crucial for finding inputs for which the function attains specific values, such as its computed average.

Transcript

00:01 Okay, what we're going to do is walk through how to use a cast system to one, graph the function f of x is equal to x times sine squared of 1 over x on the interval from pi over 4 to pi.

00:23 And then to actually use that cast system and the midpoint rule to estimate the area.

00:39 And we're going to do it for 100 rectangles, 200 ,000.

00:49 So of equal width, we want each of those subpartitions, subintervals to be of equal with.

00:56 And so what i'm going to go ahead and do is i'm actually going to be using what is called desmos.

01:03 And desmos is an online calculator.

01:06 You can use it just as a calculator.

01:09 You can use it as a graphing calculator.

01:12 And also a lot of instructors and people have written programs to do certain applications.

01:21 And so this one actually is called a midpoint room on some application.

01:26 Through desmos, and so it kind of allows us to put in any function we want to.

01:32 The slider tells us the partitions, and then, of course, a and b is the width of the interval that we're looking at, and this is actually how it's being calculated.

01:44 And so we're just going to go ahead and put in this is x times sine, and then, of course, this is going to be a squared, so i'm going to insert that in there.

01:55 And then this is a slider, so i'm going to take him back down, and notice that when i do this, you notice i'm going to try to get it as small as possible, that those rectangles, the midpoint of that rectangle actually is what touches the graph.

02:17 That's why it's called a midpoint rule.

02:20 And so a couple things also is over here, this toolbar allows me to make sure i'm in, radio mode because we are doing a trick function and then it also allows me to be able to set up my window and i'm going to go ahead and zoom in into that interval of pi over four to pi and then of course i can make it hot taller if i wanted to by resetting my y axis okay and then of course here is a slider to set my end points of my interval so this is pi over four i've already got it set i can actually use the slider or i can type in that value.

03:01 And then, of course, this is pie.

03:02 So i already have it set up.

03:04 And what i'm going to do is go ahead and take this all the way out to 100.

03:08 And so for 100 rectangles, okay, 100 different partitions, we have in that interval from pi over 4 to pi, we have the area is 1 .1 .1 .1 .1.

03:32 159919.

03:34 And so i'm going to go ahead and just keep, since it's already set up, is to slide this down to, and you can sell that i'm getting more and more rectangles.

03:44 And so for, for 200 partitions, i have 1 .159914.

03:57 And then we're going to take it all the way out to a thousand, partitions and that area is 1 .159912 now and so now i'm going to actually go back to the whiteboard and so my area was equal to 1 .159919 the second area was 1 .15914 and my third area was 1 .159914 and my third area was 1 .15914 and my third area was 1 .5.

04:35 159912.

04:37 And now the next thing i want to do is actually to find the average value.

04:43 And so don't forget how i calculate an average value.

04:51 It is going to be equal to that area divided by the entire interval, length of the interval.

05:00 And so this is going to be the 1 .159919 divided by pi minus pi over four, which is going to give me three pi over four.

05:11 And then i'm going to keep doing it because i want to do it for each of the three.

05:21 And then i'm going to do it for the last one...

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