For more information, please see full course syllabus of Math Analysis
For more information, please see full course syllabus of Math Analysis
Finding Points of Interest
- We often want to find the "interesting" points on a graph: places where something special occurs. A graphing calculator will allow us to easily find these locations. You can use a graphing calculator to find things like
- Roots/Zeros (zero): locations where f(x) = 0;
- Relative Minimums (min): lowest (local) values;
- Relative Maximums (max): highest (local) values;
- Intersections (intersection): location where two functions intersect each other.
- The specific menu to get access to these varies from one calculator to another, but the menu choices should look something like those above. Set up your function, then choose whichever one you want.
- For most points of interest, you begin by graphing the function, choosing which type of point you're interested in, then telling the calculator where to search. It searches that portion for the point you're interested in, then gives you the value.
- The problem with using a calculator to find these "interesting" points is that the calculator does all the work for you. While that's okay in some situations, you still should be able to work out solutions for problems like this on your own. Don't become dependent on your calculator for all your solving. You can use it as a way to check your work or solve a problem you can't do algebraically, but don't let it become a crutch that replaces thinking.
Finding Points of Interest
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro
- Points of Interest
- Interesting Points on the Graph
- Roots/Zeros (Zero)
- Relative Minimums (Min)
- Relative Maximums (Max)
- Intersections (Intersection)
- Finding Points of Interest - Process
- Graph the Function
- Adjust Viewing Window
- Choose Point of Interest Type
- Identify Where Search Should Occur
- Give a Guess
- Get Result
- Advanced Technique: Arbitrary Solving
- Advanced Technique: Calculus
- But How Do You Show Work?
- Intro 0:00
- Points of Interest 0:06
- Interesting Points on the Graph
- Roots/Zeros (Zero)
- Relative Minimums (Min)
- Relative Maximums (Max)
- Intersections (Intersection)
- Finding Points of Interest - Process 1:48
- Graph the Function
- Adjust Viewing Window
- Choose Point of Interest Type
- Identify Where Search Should Occur
- Give a Guess
- Get Result
- Advanced Technique: Arbitrary Solving 5:10
- Find Out What Input Value Causes a Certain Output
- For Example
- Advanced Technique: Calculus 7:18
- Derivative
- Integral
- But How Do You Show Work? 8:20
Math Analysis Online
Transcription: Finding Points of Interest
Hi--welcome back to Educator.com.0000
Today, we are going to talk about finding points of interest.0002
We often want to find the interesting points on a graph, places where something special occurs.0005
A graphing calculator will allow us to easily find these locations.0011
You can use a graphing calculator to find things like roots or zeroes (the same thing, just different names),0015
the locations where our function is equal to 0; you can use it to find relative minimums0022
(sometimes just called min), which is the lowest local value in an area, relative maximums0026
(sometimes just called max), the highest local value in an area,0031
and intersections (the location where two functions intersect each other).0035
The specific menu to get access to these varies from one calculator to another.0039
But the menu choices should look something like these things right here: zero, min, max, intersection.0045
If you can find a section like that, you have found the part where you can get that information out of your calculator.0050
You set up your function, then choose whichever one you want to use.0055
It is important to note that all of these are usually solved numerically.0058
Some calculators can solve them precisely, but most graphing calculators will solve them numerically.0063
That is to say, you will get answers that are accurate up to quite a few decimals--you will be able to get0068
four or five decimals' worth of accuracy, maybe even more.0073
But they won't necessarily be perfectly precise; it is like the difference between saying π and saying 3.1415.0075
3.1415 is a very good approximation of π, but there is still more precision in the actual number of π.0083
So, the answer that you get from your calculator, when you use things for roots, zeroes, relative minimums/maximums,0089
intersections, and things like that will be good answers; but they won't necessarily be perfect answers.0096
They will be approximations; and that is something you want to keep in mind sometimes.0101
So, how do we actually find a point of interest with a graphing calculator?0105
For most points of interest, the process is going to go like this: you start by graphing the function.0108
Now, in this specific case, I actually graphed a parabola; and I knew that because I knew I had x^{2} at the front of it, and then some other stuff.0112
But when I graphed it, I see that I get this on my graphing calculator screen.0120
So instead, what I want to do is: I might (this part is optional, but I might) want to resize my viewing window so that I can see more of what is going on.0124
So, I can adjust the viewing window, and I see that it is a parabola.0131
And so, I have a bunch more stuff going on here; it is not just a straight line that I am working with--I am working with a parabola.0135
So, you can adjust your viewing window.0140
Depending on the situation, you might not need to adjust the viewing window.0142
For example, if you are looking for a zero at some specific area (it is going to be between an x interval of maybe 2 and 5),0145
you don't really care about a window any farther than going from an x interval of 2 to 5,0151
because that is all you are looking for; and you know that your y's really only have to be near 0, because you are looking for a root.0157
But depending on the situation, it can be useful to see the entire thing, or at least a good sense of how the function works, before moving on.0163
Next, you choose the point of interest type: what you are specifically looking for.0171
In this case, let's say we are looking for a root, and we want to find out what this root is--that one right there.0174
We go to the menu; we choose "zero."0180
At that point, you identify where the search should occur.0183
This won't be necessarily on all graphing calculators; but many graphing calculators will require you to put an interval:0186
what is the lowest place that we can look from and the highest point that we are looking to?0193
What is the lowest x-value that we have to look from and the highest x-value that we are going to work up to?0198
And then, it will search within that interval.0203
So, we start, and we say, "We know that it is going to be above here and below here for this x-interval,0206
so we are just looking somewhere inside of this interval."0211
Next, you give your graphing calculator a guess.0215
You don't necessarily have to have your guess be perfect; it doesn't have to be right on top of the thing you are going for.0218
But the closer it is to where you are going, the faster your graphing calculator will be able to figure it out.0223
The algorithms that it is using to actually figure this out...it just starts from somewhere that you tell it to start, and then it works out either way, effectively.0228
So, it works out from some starting place; so if your starting place is closer to where you are going, it will make it a faster process.0236
We choose some guess somewhere, and then it cranks through it, and it gets us an answer.0242
We have a result, and it will display it somewhere on your graph.0248
It might display it next to the point; it might display it at the bottom of the window or the top; it depends on your specific calculator.0252
But it will punch out some value; and so, in this case, we managed to get that it happens at 2.81; so we have three decimal places of accuracy.0257
The process works pretty much just like this for if you are looking for the zero, the minimum, or the maximum.0266
But it works a little bit differently if you are looking for the intersection of two functions.0272
For that, you graph both functions; so you will have to graph one of the functions,0276
and then you choose another; you set another function graphing the other one.0280
Then, you will choose "intersect" in your menu (you choose your point of interest type).0283
You identify the two functions: instead of identifying where the search occurs, between what locations,0287
you say, "Here is the first function I care about; here is the second function I care about."0293
And then, you give some guess of about where you think the intersection occurs.0297
And then, once again, it runs through some algorithm, and it figures out where the actual intersection is.0302
There are some advanced techniques that we can talk about, as well.0308
One thing that you might want to do is be able to solve for an arbitrary output value on a function.0310
Occasionally, you need to solve some function and find out what input value will give a certain output.0317
For example, we might have the function f(x) = x^{3} - 27x^{2} + 9.0322
And we want to find all of the x-values such that f(x) = 419, all of the values that we can plug in and get 419 out of it.0326
In other words, what we are trying to do is find all of the solutions to 419 = x^{3} - 27x^{2} + 9.0335
If you are not sure why that is, well, we have f(x) here and f(x) here.0341
What we want f(x) to be is 419; so we swap it out here, and we have the equation 419 = x^{3} - 27x^{2} + 9.0344
So, we are effectively just looking to solve this equation.0353
What are all the x-values that get this equation solved?0356
We can use a graphing calculator to quickly find a close approximation.0360
We can write the above as 0 = x^{3} - 27x^{2} + 9 - 419.0364
We are just moving the 419 to the other side of the equation through subtraction.0370
Now, at this point, if we have 0 = this stuff, well, what we can do is say, "Let's just make this y = this stuff."0374
So, we swap out the 0 for a y; and now we have something that we can graph.0382
We can graph y = x^{3} - 27x^{2} + 9 - 419.0386
So, we plug that into our graphing calculator, and we use the calculator to solve for all zeroes,0393
because, if we find a zero, if we find each of the zeroes to this equation right here,0399
if we find 0 = this equation, well, then we will have satisfied this equation.0405
And if we have satisfied that equation, we know that we have satisfied that equation.0409
And if we have satisfied that equation, we have figured out all of the places where x equals 419.0413
And so, we are done; this is a really great way, if you want to just figure out what input values will cause a function to give certain output values.0419
You can work through this method, and you can solve arbitrarily for anything; it is pretty great.0428
Another advanced technique to know about is for calculus.0433
If you look in the menu that allows you to find the various points of interest,0436
you will normally also find some options for calculus.0439
The derivative, dy/dx, finds a numerical approximation of the derivative at the point you choose.0442
Once again, it is not a precise value; but it is a pretty good numerical approximation.0448
And the integral finds a numerical approximation (once again, not the precise value, but a good approximation) of the definite integral for the interval you choose.0453
That is all of the area underneath some starting location to some ending location underneath your curve.0462
Now, don't worry if these options don't really make sense to you right now.0468
That is perfectly fine; this is for a future course when you get into calculus.0472
It is some interesting stuff now; but don't worry about it right now.0476
You will be learning about these things later on in a calculus course.0478
But when you get to calculus, keep them in mind; keep the fact that your calculator has the ability to figure out derivatives; it has the ability to figure out integrals.0481
They can be really handy for checking your work or for solving difficult problems when you are in a calculus class.0489
They can really help you out in that place.0494
Finally, how do you show your work with this stuff?0497
The problem with all of these methods, and rightly so, is that you don't really do any solving on your own.0500
You are just letting the calculator do all of the work for you.0506
While that is OK in some situations; you still should be able to work out solutions for problems like this on your own.0510
This means that you should be able to do this without a calculator, too.0519
Most teachers won't accept "because my calculator said so" as the work for an answer!0523
Most teachers are going to say, "No, you have to be able to solve this on your own, not just rely on a calculator being able to do it."0529
And that is perfectly reasonable; the point of being in math class is being able to understand how this stuff works.0536
And if we are completely relying on a calculator, it is not that great.0541
You don't want to become dependent on your calculator for all of your solving.0543
You can use it as a way to check your work or solve a problem that you can't figure out algebraically right now.0548
But don't let it become a crutch that replaces thinking.0554
You want your calculator to be a tool that helps you do math, but not the only way that you can do math.0557
It is a great tool, but if it ends up replacing all of your ability to solve stuff on your own, that is kind of not what you are going for.0562
You really want to be able to understand and think, and just appreciate it as a tool.0569
But it is a great way to check your work and to give you a head start on being able to figure out a problem where you are not quite sure which direction to go.0572
It can give you a hint, because you might be able to figure out the answer before you even work towards it.0580
All right, we will see you at Educator.com later--goodbye!0583
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