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

In Exercises $23-26,$ use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into $n=100,200$, and 1000 subintervals of equal length and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation $f(x)=$ (average value) for $x$ using the average value calculated in part (c) for the $n=1000$ partitioning. $$ f(x)=\sin x \quad \text { on } \quad[0, \pi] $$

In Exercises $23-26,$ use a CAS to perform the following steps.
a. Plot the functions over the given interval.
b. Subdivide the interval into $n=100,200$, and 1000 subintervals of equal length and evaluate the function at the midpoint of each subinterval.
c. Compute the average value of the function values generated in part (b).
d. Solve the equation $f(x)=$ (average value) for $x$ using the average value calculated in part (c) for the $n=1000$ partitioning.
$$
f(x)=\sin x \quad \text { on } \quad[0, \pi]
$$

Key Concepts

Graphing Functions

Graphing functions involves creating a visual representation of a function's behavior over a specified interval. This is an essential step to understand the overall structure of the function, including any critical points, trends, or areas where the function behaves unusually. Visual aids help in verifying analytical work and in identifying regions of interest for further numerical analysis.

Interval Partitioning

Interval partitioning is the process of dividing a continuous interval into a series of smaller, equal subintervals. This subdivision is crucial in numerical methods, as it allows for the approximation of various properties of the function—such as integrals and average values—by working over finite segments instead of the entire interval at once.

Midpoint Rule

The midpoint rule is a numerical technique used to approximate the value of integrals or averages. In this method, the function is evaluated at the midpoint of each subinterval, and these values are used to estimate the overall integral or average. The midpoint rule is particularly useful because it often provides a better approximation than using endpoints, especially when the function exhibits non-linear behavior.

Average Value of a Function

The average value of a function over a given interval is defined as the integral of the function over that interval divided by the length of the interval. This concept offers a single representative value that summarizes the overall behavior of the function across the specified domain, making it an important tool in both theoretical and applied contexts.

Numerical Equation Solving

Numerical equation solving involves finding approximate solutions to equations when analytical solutions are difficult or impossible to obtain. Using computer algebra systems (CAS), iterative methods or other algorithms can be employed to solve equations with a high degree of precision, making this technique essential in the analysis of functions and in solving practical problems.

Transcript

00:01 Okay, what we want to walk through today is being able to use a cast system to go ahead and do some graphs to find the midpoint, remand some, or rule for finding the area under the curve.

00:24 And so what we're going to do is, first of all, we're actually going to graph f of x equal to, sign of x on the interval from zero to pi.

00:37 So that's the first thing where you do.

00:40 And then we're going to use the midpoint rule for n equal to 100, 200, and 1 ,000, and find out what that area is.

01:00 And where, of course, we're going to do equal subintervals of equal length.

01:05 And then we're also going to find average values and solve where the function equals that average value.

01:14 So the first thing i'm going to do is i actually am going to be using a desmos application.

01:25 Desmos is an online calculator system, graphing in just a straight scientific calculator, a graphing calculator, a graphing calculator.

01:35 And then there's also a lot of desmos applications out there that people have actually written that you can actually get onto.

01:45 And so i just found a midpoint room on some application through desmos.

01:51 And then it's going to let me, it will step you through the directions, it's going to let me go ahead and type in a function.

01:57 So you can actually type in any function you want.

02:00 So we're actually going to be doing sign of x.

02:02 A couple things on this, though, is to make sure that, one, we're in radio mode.

02:08 So you would hit this little toolbar.

02:11 We're going to be, make sure we're in radio modes.

02:14 You always want to graph trig functions in radium mode.

02:18 And we went from zero to pi for our window.

02:25 And so i've already got that set up.

02:27 And in this application, we know that we want.

02:33 The interval to go from zero.

02:36 So that's my a value to pi, which is my b value.

02:41 So that's what it's going to do.

02:43 And now we want equal sub -intervals for when n is equal to 100.

02:49 And so then right here, down here, is actually that calculation for that area under the curve using 100 rectangles with the midpoint rule.

03:01 So we're going to go ahead and get that copied over.

03:07 So that's 2.

03:08 And i have 4 zeros and an 8.

03:12 So we'll do that.

03:14 So we're going to go back to our whiteboard and we're going to say the area.

03:20 So this is for n equal to 100.

03:23 My area is equal to 2 .000008.

03:28 And then for n equal to 200, we're going to go back.

03:32 Back here and this is pretty nice because i can just use my slider or i can actually type in a hundred so i mean 200 so here's my 200 and this will automatically do it and i'm not changing my interval zero to pi and so now here is now my remon sums and you notice i've get more and more rectangles here in this little picture so this one now is i have four zero and a 19.

04:07 So that is my sum.

04:11 So i have an area of 2 .000 -19...

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