Datasets:
NationalTutoringObservatory
/
MathEd-PII

Dataset card Data Studio Files
xet
 Community
1
role
stringclasses
2 values
content
stringlengths
0
2.1k
session_id
int64
10
21.7k
sequence_id
int64
0
2.38k
annotations
listlengths
0
8
volunteer
which is what this gap is, the area under the curve, and our trapezoidal rule.
16,993
448
[]
volunteer
Now, there might be other instances, um, using the travel's order world, you may not have as much overlap. It might be an underestimation.
16,993
449
[]
volunteer
right, depending on, depends on how the function changes.
16,993
450
[]
volunteer
But, you know, using the right hand or left Riemann thumbs.
16,993
451
[]
volunteer
you might, uh, one of them might be an, one of them is gonna be an overestimation
16,993
452
[]
volunteer
and the other is going to be an underestimation
16,993
453
[]
volunteer
of what the area under the curve is. It's not gonna, it's not going to be exact, um, because it depends on your step size. And if you notice, the bigger step size you use, right?
16,993
454
[]
volunteer
Like I use, look at how big my, I only use the two steps in this function, but say I use like a, a really, really small
16,993
455
[]
volunteer
Delta X
16,993
456
[]
volunteer
right
16,993
457
[]
volunteer
then you'd see that the error would also be smaller.
16,993
458
[]
volunteer
if you can see that. Sorry, I know it's really small.
16,993
459
[]
volunteer
but I was just trying to prove a point
16,993
460
[]
volunteer
Anyway, that's sort of the idea
16,993
461
[]
volunteer
of, you know, why we are doing
16,993
462
[]
volunteer
these different methodologies, these different approaches. Like we're just trying to find
16,993
463
[]
volunteer
um
16,993
464
[]
volunteer
the most accurate, the exact value of what the area under the curve is.
16,993
465
[]
volunteer
And these are very, and these are various attempts at approximating what it is. The left rhyming sums, the right-hand rhyming sums.
16,993
466
[]
volunteer
and then the trapezoidal rule
16,993
467
[]
volunteer
Now, the right, I think I didn't, I didn't show it, but
16,993
468
[]
volunteer
the right rhyming sums, um,
16,993
469
[]
volunteer
is to take
16,993
470
[]
volunteer
the, let me do another graph.
16,993
471
[]
volunteer
I don't make it too small so you can actually see me.
16,993
472
[]
volunteer
Right. What is the air function? Well rather than the air function, I'd say the error value. I'm sorry.
16,993
473
[]
volunteer
So the error value would be what the actual delta F is with the actual change in function is.
16,993
474
[]
volunteer
um
16,993
475
[]
volunteer
is equal to the uh what we found, the trap using the trapezoidal rule, we just say that, we just say F bar.
16,993
476
[]
volunteer
F bar. So whenever you use bar, that means you're talking about the average
16,993
477
[]
volunteer
times delta X.
16,993
478
[]
volunteer
plus the error function.
16,993
479
[]
volunteer
The error value
16,993
480
[]
volunteer
I'm sorry, in the error value could be um I guess positive or negative.
16,993
481
[]
volunteer
right? It depends on, it depends on if this value is an over overestimation or underestimation?
16,993
482
[]
volunteer
right? So we evaluated for the error value, it'd be air values equal to delta
16,993
483
[]
volunteer
I'm sorry
16,993
484
[]
volunteer
Is that right, doctor Delta F.
16,993
485
[]
volunteer
I think really what you'd want is just some kind of delta F
16,993
486
[]
volunteer
minus RF.
16,993
487
[]
volunteer
delta X
16,993
488
[]
volunteer
right
16,993
489
[]
volunteer
OK, this makes more sense
16,993
490
[]
volunteer
Does it? So if Delta X goes to 0, or you'd be left with is Delta F.
16,993
491
[]
volunteer
If you don't want your air, you want your air to cancel. You want these two to equal each other.
16,993
492
[]
volunteer
What I'm missing
16,993
493
[]
volunteer
Hm
16,993
494
[]
volunteer
Oh, I'm sorry, that's not NG, that's average. That's supposed to be AVG.
16,993
495
[]
volunteer
which is supposed to be just kind of F bar. I use, I remember what it was, the notation for it.
16,993
496
[]
volunteer
later on.
16,993
497
[]
volunteer
So yeah, sorry, not, um, I misspoke, not air function, error value.
16,993
498
[]
volunteer
Um, and this value is in terms of the units of the output.
16,993
499
[]
volunteer
So say you're talking about speed, then it could be, you know, the error value could be in MPH, meters per second, or whatever the
16,993
500
[]
volunteer
um
16,993
501
[]
volunteer
the units of the output are
16,993
502
[]
volunteer
cause you wouldn't have, you wouldn't have errors in our exes.
16,993
503
[]
volunteer
or our T's cause those are our inputs and we define our inputs.
16,993
504
[]
volunteer
It's usually their output that we want to know.
16,993
505
[]
volunteer
and that's where, um, that's where the air comes in because we're approximating our output because we don't know
16,993
506
[]
volunteer
what our uh
16,993
507
[]
volunteer
function is
16,993
508
[]
volunteer
and our function by definition is just an order of operations.
16,993
509
[]
volunteer
of
16,993
510
[]
volunteer
it's an order of operations that once you put an input in, it does some magic and you get an output.
16,993
511
[]
volunteer
right?
16,993
512
[]
volunteer
So we want to know what is the order of operations. If we put in inputs and we get these outputs
16,993
513
[]
volunteer
we want to know what the order of operation is so we can properly define, you know, what's the behavior or what's happening. But right now, when they give you, we're looking for area under the curve. They're basically telling you, hey, here's the behavior
16,993
514
[]
volunteer
of this function F.
16,993
515
[]
volunteer
that you don't know
16,993
516
[]
volunteer
Is there any way that you can use the behavior to find out what the function is.
16,993
517
[]
volunteer
And the behavior in this instance is we know what the change in F is or the rate of change of F at any point in X, any point X
16,993
518
[]
volunteer
So we say, OK, we we know what the change in F is at any point in X, right? Cause that's the derivative.
16,993
519
[]
volunteer
Then is there any way we can um
16,993
520
[]
volunteer
for any interval of Xi to Xi + 1 or just some delta X. Can we find out what the actual change in F is.
16,993
521
[]
volunteer
And with the methods we're using, we don't
16,993
522
[]
volunteer
we can't be sure that they're acts that they're um
16,993
523
[]
volunteer
or we know that they're just approximations.
16,993
524
[]
volunteer
depending on how big our Delta X is, right? Just like a derivative. The bigger step size you take,
16,993
525
[]
volunteer
the bigger the error value is going to be because you're just, you're too far away from the tangent line in terms of derivative. You're too far away from the tangent line to know
16,993
526
[]
volunteer
what's going to happen to F, right? It's too much gray area.
16,993
527
[]
volunteer
Um.
16,993
528
[]
volunteer
So that's the kind of
16,993
529
[]
volunteer
the trapezoidal rule, the the mentality behind it
16,993
530
[]
volunteer
You're just taking an approach to
16,993
531
[]
volunteer
evaluating the area under the curve.
16,993
532
[]
volunteer
Um, oh, and you can have with the trapezoidal rule, you can't, it can't, I know I'm looking at the graph now, it looks like an under approximation.
16,993
533
[]
volunteer
but that's only because the function is increasing.
16,993
534
[]
volunteer
If the function were decreasing, let me show you here. Say
16,993
535
[]
volunteer
like this, the function like that, and you were to do the trapezoidal rule.
16,993
536
[]
volunteer
then you would find that your trapezoidal rule is an over approximation.
16,993
537
[]
volunteer
right, because you'd have this
16,993
538
[]
volunteer
error value here, that's too much. And this is where I said that the error value could be plus or minus.
16,993
539
[]
volunteer
because it depends on if it's an overestimation or underestimation.
16,993
540
[]
volunteer
But the idea is that we want
16,993
541
[]
volunteer
we know that if we can um
16,993
542
[]
volunteer
or we have an idea that if we can get this step size,
16,993
543
[]
volunteer
to be infinitesimally small.
16,993
544
[]
volunteer
then we can get our error value.
16,993
545
[]
volunteer
to be infinitesimally small too.
16,993
546
[]
volunteer
Infinitesimally close to zero.
16,993
547
[]