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. $$ f(x)=\sin ^{2} x \quad \text { on } \quad[0, \pi] $$

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.
$$
f(x)=\sin ^{2} x \quad \text { on } \quad[0, \pi]
$$

Key Concepts

Solving Equations Involving Functions

Solving equations that set a function equal to a constant, such as the average value, is crucial in finding the input values (or points) where the function achieves a particular value. This process often involves algebraic manipulation or numerical methods, and is important for locating critical points and understanding the function’s behavior in relation to average or target values.

Average Value of a Function

The average value of a function on a closed interval is computed by summing the function’s values at selected points (often obtained through numerical methods) and dividing by the number of sample points or using an integral formulation. This concept gives an overall measure of the function’s magnitude over the given interval.

Interval Partitioning and Midpoint Evaluation

Interval partitioning involves dividing a given interval into smaller, equal subintervals. Evaluating a function at the midpoint of each subinterval is a common numerical technique used in approximating integrals. This method, known as the Midpoint Rule, can provide an estimate of the function's average behavior over the interval.

Function Plotting

Function plotting is the process of graphically representing a mathematical function over a specified interval. It helps in visualizing the behavior of functions, including identifying key features such as maximums, minimums, intercepts, and periodicity, which is essential for both analysis and interpretation of results.

Transcript

00:01 Okay, what we want to step through is it's all going to be through kind of technology.

00:07 We want to use a cas system to graph a function, and the function we want to graph is f of x is equal to sine squared of x on the interval from zero to pi.

00:25 And then we want to be able to use the midpoint rule to estimate the area for n equal to 100, 200, and a thousand of equal with partitions.

00:51 And then we're going to also kind of add onto that.

00:54 Okay, so when n is equal to 100, we're going to go ahead and i'm actually going to be using a desmos application.

01:05 Desmos is an online tool, scientific calculator, graphing calculator that you can get on.

01:14 There's also a lot of teachers, instructors, and other people have actually written desmos applications.

01:23 So i just kind of found a midpoint room on some application.

01:27 And this is a pretty cool one because it actually shows you what it's actually doing.

01:33 So, of course, here is our, here is basically our riemann sum, and then we have some additional things down there that kind of helps this program out.

01:45 And so you can actually go in here and type in any function you want.

01:51 So we are going to do sign squared here.

01:58 And so they're sign squared.

02:00 Couple things, though, we're doing a trick function, hit this toolbar.

02:05 We want to make sure that we are in radium mode.

02:08 Because we're on the interval from zero to pi, i wanted it kind of blown up so we could actually see what we're doing.

02:14 I could actually even change this if i wanted to, on the y -axis, go from negative 1 to maybe just positive 1, or even positive 1 .5 .5.

02:34 See, i have my thing in the way and i can't see what i'm doing.

02:37 So 1 .5 to kind of even give it a little bit bigger so you can kind of play around with that.

02:46 Then here, this is how many partitions, so i can either use the slider or i could type in if i wanted to the thing.

02:56 And then here is my a value, which is going to be your left most.

03:05 Start of your interval and then of course b is 3 .14 and they're both on a slider as well so you can slide it back and forth.

03:14 So we've got it set up for our first n equal to partitions and this is of course it's a midpoint rule so that what that's going to do and if i want to show you if i put that to 10 it's just going to place equal with rectangles where the middle of the rectangle actually touches the graph.

03:36 So the height is actually going to be in the very exact middle of it.

03:41 So i'm going to go ahead and change that back to 100.

03:43 And so here is actually the value of that midpoint rule.

03:49 So that area is 1 .5708.

03:55 So let's go ahead and do that.

04:00 So this area is, so the area is equal to 1 .57 .7.

04:09 See now i've already forgotten what it was.

04:14 08, 08, i guess.

04:16 And then we're going to do it for 200, and then we're going to do it for a thousand.

04:22 So we'll already get that set up.

04:24 So now i can just kind of, if i wanted to, if i wanted to go, i guess i could just do it by hand.

04:36 I mess that up, the slider.