which you'll be able to tell us because, you know, because it's the antiderivative, this is saying that like, OK, we have this function F.


uh, you get the


if you go back up to the


the first graph I did, you say we have this, we have this graph F.


OK. We want to know, assuming this is like, let's just say this is F.


right? This is the derivative function


and we want to know if, since this is the, this, this tells us the change in the behavior of F. We want to know, is it possible to


take the, like, take the and derivative, do the opposite derivative, and using the change in the behavior to be able to find out what F is.


But we in the way we do that is we kind of take several approaches and which are just approximations and we try to see


uh by taking the limit or by summing up all the infinitesimal values. Is this an accurate representation


of the function


And we know that for the derivative, any value, any Y value,


in the derivative tells you the slope at any point in, in the origin the parent function F.


But when we do that, we can say that the change, you go to Delta X and X


XI and Xi + 1.


using the trapezoid rule


and we say


if we find the average


of the two, let's say, um, let's see.


Yeah, they say average


F, right, over some range delta X.


right? You see how this makes little.


Yeah, we can say that this is an approximation


to


the uh


change in F.


right?


Because we know that um


since the F is the slope.


Let me try to, let me get some space. Let me move that graph a little lower.


Oh, I can't grab everything


I


very frustrating


Mm


All right


move this


down so I can


remove all of this.


up.


Right


there, close enough


right


And you got room in the bottom


All right. So we know


F


F of X, any point is just the slope M at any point X


All right, just a slope


And we know that the slope is just equal to our change in F, right? The change and the elevation.


divided by the change in our X


And we want to know, is this area under the curve? Can we approximate the area under the curve cause we want to know what Delta F is.


because that's, that's our change in the function.


right?


So we do, so using the trapezoid rule, you're right, trapezoid.


Trapezoidal rule


So we can say um using our average F, right? Over the, over the step size of Delta F.


We say that Delta F is approximately equal to our


average step size, which this one would be F of X_I, X of XI.


plus F of XI + 1.


over 2


times our step


right


is approximately equal to Delta F.


which is the area under the curve.


like the entire, the full area under the curve from X1 to Xn.


is what our integral would be.


But you notice that this is missing


Let me use a different color


Um, like green, you know, that this is missing.


this gap right here


So this is our


our error value


right? And that's why there's Delta F remains an approximation.


But again, if we um


if we take the limit


of delta X as it approaches 0, right?


of this delta F function.


right?


Then we can reasonably get


hopefully


this, the accurate del delta F.


right? Now you notice in the equation it's multiplied, so it'd probably be equal to 0.


but we would, uh


figure out, try to figure out, I guess it'd be


the minute it goes


as the error function goes to the, we'd be there, evaluate the error function.


the secondary error function.


We would want, so we, as a limit of De tax budget there are stepsize gets infinitesimally small. We want our error function.


to be zero


which is what this gap is, the area under the curve, and our trapezoidal rule.


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.


right, depending on, depends on how the function changes.


But, you know, using the right hand or left Riemann thumbs.


you might, uh, one of them might be an, one of them is gonna be an overestimation


and the other is going to be an underestimation


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?


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


Delta X


right


then you'd see that the error would also be smaller.


if you can see that. Sorry, I know it's really small.


but I was just trying to prove a point


Anyway, that's sort of the idea


of, you know, why we are doing


these different methodologies, these different approaches. Like we're just trying to find


um


the most accurate, the exact value of what the area under the curve is.


And these are very, and these are various attempts at approximating what it is. The left rhyming sums, the right-hand rhyming sums.


and then the trapezoidal rule


Now, the right, I think I didn't, I didn't show it, but


the right rhyming sums, um,


is to take


the, let me do another graph.


I don't make it too small so you can actually see me.


Right. What is the air function? Well rather than the air function, I'd say the error value. I'm sorry.


So the error value would be what the actual delta F is with the actual change in function is.


is equal to the uh what we found, the trap using the trapezoidal rule, we just say that, we just say F bar.


F bar. So whenever you use bar, that means you're talking about the average


times delta X.


plus the error function.


The error value


I'm sorry, in the error value could be um I guess positive or negative.


right? It depends on, it depends on if this value is an over overestimation or underestimation?


right? So we evaluated for the error value, it'd be air values equal to delta


I'm sorry


Is that right, doctor Delta F.


I think really what you'd want is just some kind of delta F


minus RF.


delta X


right


OK, this makes more sense


Does it? So if Delta X goes to 0, or you'd be left with is Delta F.


If you don't want your air, you want your air to cancel. You want these two to equal each other.


What I'm missing


Hm


Oh, I'm sorry, that's not NG, that's average. That's supposed to be AVG.


which is supposed to be just kind of F bar. I use, I remember what it was, the notation for it.


later on.


So yeah, sorry, not, um, I misspoke, not air function, error value.


Um, and this value is in terms of the units of the output.


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


um


the units of the output are


cause you wouldn't have, you wouldn't have errors in our exes.


or our T's cause those are our inputs and we define our inputs.


It's usually their output that we want to know.


and that's where, um, that's where the air comes in because we're approximating our output because we don't know


what our uh


function is


and our function by definition is just an order of operations.


of


it's an order of operations that once you put an input in, it does some magic and you get an output.


right?


So we want to know what is the order of operations. If we put in inputs and we get these outputs


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


of this function F.


that you don't know


Is there any way that you can use the behavior to find out what the function is.


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


So we say, OK, we we know what the change in F is at any point in X, right? Cause that's the derivative.


Then is there any way we can um


for any interval of Xi to Xi + 1 or just some delta X. Can we find out what the actual change in F is.


And with the methods we're using, we don't


we can't be sure that they're acts that they're um


or we know that they're just approximations.


depending on how big our Delta X is, right? Just like a derivative. The bigger step size you take,


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


what's going to happen to F, right? It's too much gray area.


Um.


So that's the kind of


the trapezoidal rule, the the mentality behind it


You're just taking an approach to


evaluating the area under the curve.


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.


but that's only because the function is increasing.


If the function were decreasing, let me show you here. Say


like this, the function like that, and you were to do the trapezoidal rule.


then you would find that your trapezoidal rule is an over approximation.


right, because you'd have this


error value here, that's too much. And this is where I said that the error value could be plus or minus.


because it depends on if it's an overestimation or underestimation.


But the idea is that we want


we know that if we can um


or we have an idea that if we can get this step size,


to be infinitesimally small.


then we can get our error value.


to be infinitesimally small too.


Infinitesimally close to zero.


and that would mean that um


our


either a trapezoid rule or right rim and sums, whatever approximation method we're using would become very close


I was like infinitesimy close to what the actual change in function is.


Infinitesimally. There you go, cause say it properly.


Let me give you a couple of minutes


just to kind of mull over


um everything we've talked about


and, uh, give you time to kind of come up with any questions or anything you don't understand.


And then we can go on to the practical application and just kind of working on the problem.


and what it and what it means, what we're actually doing.


Do you want me to keep the um


calculations on the board or can I get rid of some stuff because we're gonna need some space later.


for a i is 7000.5


Um


I can't confirm it and I haven't actually evaluated yet. Let me, um,


and b i is 7000


take a look at it. Uh.


and B you got 6295.


All right. Give me one moment.


So


Yeah


yeah its good?


I got 6625.5.5 for a


ummm


are you sure..


can you show your work


on the board


kk


Uh


By the way, one thing you can do for your um


calculations


is if you notice


that this summation


um for I


that delta T is independent of I. The step size remains the same for every iteration.


So what this means is


is that you can factor out.


this delta T term


so that you just have this delta T times the sum.


of every average value.


over the interval


All right? So it'll just be kind of like this delta V over 2.


All right let me just do this. Var.


for I


That's a little shorthand for you


ok can you see my work


its in green at the top of the board


i can't see it


Let me try refreshing the page to see if that works.


ok lmk


So I refreshed the page but I don't see your work. Also, I got 6295 for B.


I can't see your work. Let me show you, let me show you mine, I guess.


I don't know why I can't see your work


Um, so I said that


I said S is equal to.


plus


times


the average of


each, so I got


125 plus


155 plus.


140 plus.


105 + 100.


And that's how I got my answer


for a


hello?


Hello


Yeah, I'm not sure why you're, I can't see your work.


can you see my work in word form


Oh


never mind. I think you're right. I put in the wrong value


I got 6625.5 instead 7000.


because our answer is we'are 7000 off, so I think you're right. Let me see where you wrote it down.


You got 6625.5.5. You got 6295.


Yeah. So your A is correct.


And what did I get for B


oh ok ok


6995. So it's 2662.


Let me double check


Can I see your work in word form


OK.


I see, I see the text. I don't see all of it. I just see T equals 2


and 1.5


Let me double check. What did I do? So this should be


6295 for b


I don't know why I got 6625.5 from. I'm just uh


I'm kind of dyslexia


So let me redo my maths.


its ok


Yeah, I got 6625.5 for


um it looks like A.


and then let me look at


Sorry for confusing you


I


Yeah, 6295 for me.


I think you just, yeah


So you're right


I apologize


I'm, I'm, I bow to your expertise.


Sorry, you're returning your work in.


its ok haha


Um, but did, did that make sense kind of why we're doing this kind of why they use the word approximation when they talk about this


because there is um


yeah i get it


there is an error value.


that is entirely dependent on what your step size is.


And the bigger your step size, you're kind of delta X or DeltaT, whatever your delta input is.


the less accurate


your method becomes.


And all this stems from physics.


like kind of observing the world and just saying that like, hey, if I know


if I know an object either position


Is there any way I can relate that to its philosophy or the opposite. If I knew an object's velocity,


what is the relationship with its position.


assuming you don't, assuming you only know one and not the other.


And the derivatives and antiderivatives


um were the solutions to that.


Because knowing, like in this example, you know its velocity at any point in time.


and now you are approximating


its position at any point in time.


And the smaller the step size,


um, the more accurate you can get to what its actual position is for any point next. Right now we have a step size of like 1.5. So for every 90 seconds, you know where this object is, a hot air balloon.


the elevation


But in between those 90 seconds, you don't know.


right? That's where the approximation comes in. That's the kind of gray area where we don't know what's happening


Um, we have an idea of what's happening


right? Based on what we can see the what the changes to the velocity.


but we don't know what the actual values are.


Hello


hello


I want to learn geometry


I am not sure what you mean?


nvm I meant that I want to deepen my understanding of tenth grade geometry since I'm in tenth grade and geometry is currently a struggle for me


do you have a problem that you are working on?


no


Is there a particular topic that you wanted to go over?


yeahh


what topic is that?


idk how to explain the topic but I put a question on the board and that question is like the topic I sturggle


I think the topic is just called lines


Parallel lines


You need to know what kinds of angles of equal and what kind are supplementary. Do you know what supplementary means?


I think supplementary is when two angles equal 180 degrees


yes.  Also, they can form a straight angle.


okayy


In the diagram, angles and b together form a straight angle and they add 180°


gotcha


angles that have the same relationship to the parallel lines and the transversal are called corresponding angles


So, for example, the two red angles are both formed in the same way by the transversal and one of the parallel lines


the same goes for the black angles and the green angles.


do you see that?


yes


when you have parallel lines and a transversal, corresponding angles are equal


Angles C and F are both inside the parallel lines but on opposite sides of the transversal, right?


C and F are called alternate interior angles.


I understand


the interior means they are inside the parallel lines and the "alternate" means they are on opposite sides of the transversal.


ok


with parallel lines, alternate interior angles are equal


angles A and H are also on "alternate" sides of the transversal, but they are outside (exterior) to the parallel lines.


so A and H are alternate exterior angles


when you have parallel lines. alternate exterior angles are allso equal.


How about angles E and F. What do you think we can say about them?


u want me to say the relationship between angles e and f?


aren't they like the angles a and b in the diagram that I drew above?


yeah


both angles are on a straight line


so what do you think angles E and F add up to ?


degrees


In the diagram for your problem, I drew in a new angle "e". Do you see it?


yes


What can you say about angles "a" and "e", do you think?


not sure?


im confused


but I think they are on the same line


they are actually just like angles A and E in the big diagram below


they are corresponding angles


OHH I'm so dumb how didn't I notice that\


therefore, the measure of a equals the measure of e


bruh I should have kneww


Well, you need to spend some time looking at a good diagram of parallel lines with alt interior and exterior and corresponding angles until you get used to visualizing


them


ok


now, we said that e and b add up 180 degrees, right?


oh yeah


because they are supplementary angles


they tell us that b = 2x +13, right


so we can replace b with the expression 2x +13


does that make sense?


yes


they also tell us that a =3x +12


but now we know that e is the same measure as a, right?


yup


so we can replace e by 3x +12


ohhh and then u just combine a like terms


yes, now you solve that equation for x


ok easy 3x + 12 + 2x + 13, 5x + 25 = 180, 180 - 25 = 155, 155 = 5x, 155/5 = 31 = x


yes, very good!


yay


Sorry, but something has come up, and I am going to have to go. Sorry.


it ok thanks


You are very welcome.


Have a good week!


you to


bye


Hello!


Hello


Uh, hello, Alex.


Can you hear me


Oh yeah, no, I can hear you.


Uh, um, OK. How are you doing?


Good. How about you?


Pretty good


Um, can you tell me more about this assignment?


Yeah, um


so


it's like with real numbers and I'm finding the difference between these two numbers on a number line.


If it helps, I can write down the questions. I'll probably make more sense.


OK.


Uh, yeah, sure. So it's find the difference between two numbers on a number line.


Yeah, but it's like with like these real numbers, and like I understand how to do the real numbers, but just know how to put them on the number one.


Oh, I see, I see. OK, yeah.


Um


so what numbers do you have?


Um, I'm writing them down right now.


All righty


7/3, OK.


And then


yeah


uh


OK


Yeah


Oh what


you wait, this is ninth grade, Alex, where you're all wow squay already. OK.


Wait, do you know what square root is?


Yes


Oh, OK


Ah.


Oh


I honestly forgot what I first learned about square root, but that's impressive that you're already learning it.


So do you want me to solve this and then like


Hm.


you can help me put it on the number one.


I solve it. What, wait, wait, what?


So it says


So it's asking you to solve the square root first and then put it on that number.


find the distance between 7/3 and 25/4 on a number line.


It


oh


Mm, OK


Well, first, yeah, let's first draw the number line.


OK.


OK


Yeah


And then


root of 25/4.


Oh gosh, I need a refresher on this. Um, well,


try placing 7/3 on the number line first.


Um, OK


I say just putting down a point that you call 0.


Huh?


I just placing 0 somewhere on the number line. So you have a sense of where to push, uh, put the other numbers.


Uh, jeez, um.


There you go, yeah.


yes, or I get lost. Um.


yeah


You know, that's fine. So now, now you have 0. What, what should 7/3 go?


um


past the 0, so like that way.


Uh, that


OK


I'll give you a left of 0 is negative, and right of 0 is positive.


Oh, I didn't know that. So right.


Yeah, the general rule for making a number line is the right is always the positive side or the more positive.


and the left is the more negative side.


OK, I remember that.


So, somewhere over here.


Yeah.


Can you start like, um, let me think.


So let's ignore the root of 25/4 part and just focus on the 7/3. It is just, this was just about 7/3.


How would you go about making the number line and putting it on the number line.


Um


I would


You stuck


yeah, I'm terrible with number ones.


No, don't worry. They, they take a lot for a little. Do you have any ideas?


I think it would be like


here because like the 3 is like the denominators, so like it would go off of the denominator.


Yeah, that's true


We're making a number on so just point down reference numbers Algebra 101, whatever like phone numbers.


um


oh


OK. Um


you still there


Yeah


OK. Did, uh, did what I said make sense or do you, do you want more explanation?


um, on like to put like


numbers, like


Yeah, where you put, where you put 7/3.


yeah, like I put it here


So this is one


yeah.


and then


if that's one, then somewhere over here, what would that be?


Nice, yeah


the 27 3 would be over there.


I want the sauce.


Oops


I didn't mean to draw that


Wait, how I


It's OK


Wait, how do I move from side to side on this.


whiteboard


Um


anarchy.


I'm trying to figure that out too.


I don't think you can


OK


but this is, that's not very well designed. That's interesting.


I can just do this real quick.


1 2nd


All right, yeah


Is that gonna be the news er?


Yeah


OK, yes, that's good, yeah.


And then.


um


you know we can keep


I think we can keep those marks there


but we just have to renumber them


uh, let's hope my.


so


I, I


excuse me, uh, why isn't that?


Hm.


Dude, I'm not even that old yet. I feel so trying to use this technology. It might be over.


OK


OK, sorry, yeah, what, what would 7/3 be on the thing we have right now, the number we have right now.


Um


if it is with the denominator, then I would say it would be like


like over here.


Over here


Yeah.


Mm, let me think


OK, so the thing is


the


denominator, uh, let me, let me try to take a way to explain this. So


just because it's 3 on the denominator doesn't mean it's going to be bigger than 3. The for example, if you have um 3/3, right? That would be


and then it will be just 3 of 1/3, right?


Yeah


So


uh, what's 3 divided by 3?


30 divided by 3 is 1.


Right. So one is technically also pre-divided by 3, right?


And


excuse me, so at 1/3 is 3.


Wait, sorry, 3/3 is 3 times


1/3, right? Because there's 3 of them.


that makes sense. Does that make sense to you?


Yeah, that makes sense. I understand.


So,


I'll drop this little bit for you.


So you see how despite the 3 being on the denominator.


That doesn't automatically mean it's going to be 3 plus something.


because with fractions, it's actually division.


essentially, what, what this is supposed to show is the top number divided.


by that's great


Yeah.


Because the fraction sign is really just a division sign.


Yeah, yeah, yeah, exactly, exactly. You're right. So you just take the top number and divide it by the bottom number.


So I, I know you can't do 7/3 because that's not a whole number. But if you were to take a guest, where would that like be roughly like bigger than 2, smaller than 2?


Um


like


this


right here


Yeah, exactly. Good job. Yeah, that's literally 7/3.


Um, I guess another way you can look at this.


is to see how many times 7 go into 23, without going over.


So how many times does 7 go into 3 without going over.


Um, let me think.


That's fine


Oh, no, no wait, no, sorry, no, I didn't instru, I didn't, um, that wasn't a very good question.


What what's the closest that 3 has to like


So, um


what's the closest visible number.


Yeah, sorry, what's that


I would say 3 times 2 gets it to 6, and that's pretty close to 7.


Yeah


right, so you got 3 x 2


and in terms of if you're running it over like, for example, you put a number over here, over 3.


What, what would, um, what number should go up here to make something divided by 3 goes 2.


Um


Something divided by 3 is equal to 2.


The 5th


Well, what, yeah.


over 3


Oh


OK, cause they're not equals the 2.


Yes, and then now you have a remainder because obviously 7


m 16 and you have 1


and then


2 and 1/3 is equal to 7/3, you got, you, um, you got 6/3, and you know 6/3 is equal to 2.


Yeah, so it would


And then you add, add in the, yeah.


Does, does that help you understand fractions a little more.


yeah, I understand


Yeah, I understand that


OK. Yeah, I'm, I'm glad you understand it. OK, so let's do a practice problem. Let me think.


OK


If I were to tell you


to put, uh, let's work with 4 this time.


And let's say I 15/4. What would that go?


Oh yeah, yeah


OK, so one question about this before. So if I'm finding the distance between two numbers, would I like find where they


both go and then like count how many like spaces.


I guess you wouldn't count the spaces, you would just subtract them from each other.


Yeah


Um, do you have any other questions? Sorry, I should have asked that.


so that's it


OK, OK


So before we move on to this, you know.


so yeah, before we move on to root 25/4, try, try putting 15/4 on the number line.


It.


Just as an example or practice, I guess.


15/4, so.


I first have to define.


If you want, you can try using this method where you try to find how many times it goes into.


And like this big number can be your basis.


or you can use


uh what is it? Actually, no, I think the wait, no, sorry, I'm


by the way, this is like, I think my first time teaching eighth grade math. So that would be for a little bit.


It's OK


Um


so


wouldn't it be 3. like 7 something.


It is somethingmething, so because um


pre, well yeah, you, you know.


12 goes into 43 times.


Yeah


and then


you have to, yeah, correct.


It's


yeah, 3.7, 3.75, which is 3 3/4.


OK


So yeah, OK. So you, congratulations, you now know how to put numbers on the number line. Now, let's tackle


this thing in my jig.


Um, first of all, do you know what the, the little weird looking slope


thing in my jig do


um


This missable


uh, what is it, uh, I don't know what it's called.


Oh wait


this


No, this symbol


oh, I don't know what it's called, but like,


I know


Yeah


I


like square root


yeah


or you can call it root for short. Um, so essentially, square root


is two numbers that are the same that multiply to make the number inside the square root.


OK.


So


So what is, what is two numbers that are the same number that multiplies to make 4.


Good.


2 x 2


Um, oh yeah, I should, uh, the dot in when you get to like higher level of maths where you start dealing with


because the dot means multiplying, isn't that?


Yeah, OK, OK. Yeah, it's just to clarify when you get to uh higher math, you start dealing with letters, and that's going to be pretty fun.


Oh


And obviously, this is gonna be the letter X and you don't want to miss, like you don't want to misunderstood the letter X and the multiplication symbol. So when you get to higher maps, it just turns into a dot. Also, for me, it's easier to write, but sorry, that I got a little off topic there. Anyways, um


No, it's fine


2 x 2, obviously good.


actually


you find the first part of the problem


So when, when you write like a square, oh, I didn't mean that to go into the two whoopsies. When you write like a square root of a fraction


in this case 25/4.


What it's really asking you


history of 25


over square root of 4


I'll so you have to figure out the square root for both and then divide it.


Um, figure out the screw for both, and which would give you the fraction.


um


so, by the way, when, when you square root, you only write down one of the number that what the ah wait, so yeah, so, security of 4.


is 2 x 2. So you only write down 2.


You you don't have to write anything else, just write it down, yeah.


OK.


And then scare the 5, what are 2 numbers that are the same, that multiplies up to 25.


Uh, 5 times 5.


Good. And how will we read that write that?


in this space


Um.


I'll scroll down a little, sorry. Um


where


gosh, this skirt


Ah.


Don't worry about it


I think I'm bad at technology. Uh


what is going on? OK.


Yeah


Like great


like that


There. OK


And


oh, you got your fraction


Uh


fraction. I don't need to write down for you. You're smart enough for that.


Yeah, so this is your fraction


Ah, and well, what do you think 5 or 2 goes on the number line.


Um, she'd have to do


all right


uh


Yup, correct


So like


right here.


Nice. So this is


um


so this number


and


where do we put what's 07 or 3, yeah.


Actually, have you ever Algebra II? I don't think we have plot 7 or 3. OK, yeah, 1 was 7/3 goal on the number landed.


Um


it was right here.


Uh, it's the


Right here. OK. Oh yeah, yeah, we did, we did, we did plot it, oops.


I forgot. OK


Now.


um, scroll down a little past the


OK


past the square root bit. So now you have 7/3.


and you have uh what 5/2.


Yeah.


Do you know how to find the difference between fractions.


I would think it would be the subtractive.


Different, yeah, difference is subtracting.


So what you have to find like the common denominator thing.


Yup, correct. That is the first step.


So what's a common denominator for 3 and 2.


Um


OK, yeah. All righty. I think you got this stuff. I'll let you work.


Very nice


So the difference be


Uh, it would be


No, well, it's not negative.


but I think, I think the number you got is correct.


But


So just one


wait, wait, one?


Yeah


Like, no, no fraction. Whoa, whoa, whoa, hang on. You do need a fraction.


Oh, that did not


Oh


so it needs to be in a


So I put it into two fractions.


Yeah, there's 2 fractions


So don't I have to subtract them?


Mhm, yeah


The number is correct, but again, you don't need the negative sign. Here's the thing.


You, you can't have negative distance.


between two objects, right? Let's say


there's a, um, let's say there's a lamp 1 ft across from me in my room. Let's just say for example.


OK


Yeah.


Now, 1 ft would mean that group lamp is literally nowhere because in real life


no matter how far away you are from an object, the distance never goes negative, right?


As if you get, if it's like directly on top of you, then maybe it's zero distance, maybe, but.


you can't have negative distance because that's, that's going to be inside the object is inside and um I don't think that's healthy for the human body.


Oh, OK


Uh, yeah, but the main point is, if you're talking about distance, it does not go negative.


So 1/6 is your final answer. Good. Do you want to practice another problem like this?


Um, sure


All right, wait, is there, is there any other problems in your homework though that you have to do first?


Um, all the rest are just the finding distance.


Oh, OK, we can just do those then.


OK, I'll write down the problem.


Do you want me to clear all the stuff we have on the uh thing, or should we just scroll down?


You can clear it if you want, but if it takes a while, then I can just write it.


I can straight up delete everything, but I wonder if you need.


like some of


I guess


you have a basic understanding of how


the square root and how to plot numbers on a number line, yeah?


Yeah


OK, then I'll delete everything


OK


Is that 2.4?


Yes


Oh, and you have fruit of 27. 0.


what


Are you sure this is tenth grade math. What?


Ben, OK. Well, we're cut rooting now, so that's fun.


I can do this part and


What, OK.


Oh


I understand it now. OK, yeah, well, what's the key root of 27.


uh, give me one second.


Uh


Are you putting it into calculators?


No, I'm just just deleting the circle, but we can, we're not allowed to use the calculator, sadly.


Oh, I see


Oh, it's perfect


So it's


well, it's a perfect cube.


Yeah


the numbers 3.


Mhm


I think that's all you do for that.


So your final answer is 3.


Yeah.


That's correct? OK.


Well


well, wait, does it also asking you to plot on this one, or is it just finding the distance or differences.


I've planned the distance between 2.4 and


Cube roots of 27, very close, you know, is this a new topic that you just learned?


Uh


yeah


OK


Right. Well, yeah, cure of 27, you got that native 2.4. Dance native 2.4.


Well, the


do you not you find a distance between the two.


Um


I thought, I think it's the subtracting thing again.


Mm.


but I don't know how I would check those, well,


uh oh.


it's a trap


what, when you're finding the distance.


um, I recommend cutting the bigger number first.


and then subtracting it by the smaller number.


OK, so


So here's here's a funny rule about mathematics. What do you think happens when you subtract an already negative number.


uh, I think I don't remember, but I think 2 negatives equal positive.


Oh wow, correct. Yeah, 22 negatives equal to 1 positive.


So you would add them, I think.


Yup


OK, so 5


weeks


Nice. Yeah. So what, what does the number 5.4 mean?


That's the distance between the two numbers.


Great, yup, 5.4, you have the answer.


OK.


Well, that's, oh


thank you


Yeah. Is there any other problems you have to do?


Nope, that's it


That's literally it. Do you still want to practice more or not?


Uh, I think I understand it.


Yeah, yeah, honestly, I think you seem to understand it very well.


All right, yeah. Um,


do you, wait, do you still have any more questions?


Nope


No? All righty then.


OK


All right, I'll leave now. See you, take care.


Take care


hello!


Hello Emily! How can I help you today?


Hi, i need help with this specific problem on my AP precalc work


I would be happy to help you with that.


thank u!


Would you like to start with a, or is there a specific one you would like clarification on?


honestly the whole problem is not maaking sense bc theres no number, but i can try letter a and you can check?


Sure, that sounds great!


alr is a, [a,b]


Exactly, that is correct!


As we move from a to b, the slope is increasing, so f(x) is concave up over that interval.


Do you want to try b?


wait can intervals be represented in letters?


Yes, in this case, instead of giving you specific x values, they are giving you letters to make the problem more general.


ahhh got uuu


Usually you would use numbers though


okay ill try b


Also, one minor thing, when we are referring to concavity, we typically use parentheses instead of brackets


okkay


Because nothing is happening to the slope when we are AT a or b, only the values in between


alr b is (c,d)


Almost! f(x) is concave down over that interval, but is there anywhere else where it is concave down as well?


the ending of B?


Yes, it is also concave down over (b, c)


Can you explain why that is?


cuz the graph starts the shift


in the direction its moving


like where i circles


Yes, the slope starts decreasing at b


At b we have a slightly positive slope, and as we move towards c it gets closer to zero


correct


And then after c it is still concave down because it keeps decreasing from 0 to a negative number


So what would be the interval for part b?


(b,d)


or infinity?


Great work!


I would go with (b, d) because we don't know what is happening after x = d


true true


alr lemme try c


ok does it even increase at all


Yes, there is an interval where the graph increases. Remember, now we are talking about the graph f(x) increasing, not the slope increasing


ok so i think im lost can you guide me to it so itll help me for d?


Of course no problem!


Is it ok if I erase the drawings from the picture?


ofccc


Ok, lets start by looking at each section of the graph one by one.


kk


Can you circle the point on f(x) where x = a?


Yup, that's the x value x = a; can you circle the point on the f(x) curve? In other words, the point at f(a)


ooopsss


No problem haha


Can you do the same for the point on the f(x) curve where x = b


Great! Looking at the two circled points, which one is greater (higher on the graph/higher y-coordinate)?


b ofc


Right! So did the f(x) have to increase or decrease to get there?


increase!


Yes, so (a, b) is the first increasing interval!


Can you find any other increasing intervals?


ok imma search


(b,c)


Awesome! So what would be the final answer for c?


(a,b) (b,c)


Yes, and we can turn those into one interval in this case since it is increasing over all of (a, c)


Emily


alr let me find outt


the decreasing


Alright!


(c,d)


Nice work! Also one more thing, does the worksheet give anymore information about f(x)?


For example does it say it is a polynomial or anything else?


nope, shocking


Ok, I just wanted to make sure that there are no assumptions about the  end behavior going to infinity or -infinity


Would you like to try part e?


would it be 1


Not quite, the zeroes of a function refers to the points where that function equals 0. In other words, f(x), the y value on this graph, is equal to zero at what x coordinate?


from (b,c)


wait no


Unlike the other questions, zeroes are at invidual points rather than intervals.


Which axis would the zero of a function intersect?


ok umm x axis?


Yes! Where does f(x) intersect the x-axis?


well at 0,0


The x axis is the horizontal axis here. Can you circle the point where f(x) instersects that line (the x-axis)?


is that right?


Nice try! f(x) refers to the curve. where do the curve and the horizontal line (the x-axis) intersect?


oh bumsickkless


that/


There we go, nice job! 🎉


So that would be the zero on this graph, because it is the only place that f(x) = 0.


Can you write the coordinate for that point?


it don't have numbers:(


We can apply the same strategy we used for intervals! Instead of using numbers, we can use the letters they give us. What is the letter they give at the circled x value


d


Yes, so we know our coordinate is (d, ?).


Awesome! The coordinate is (d, 0).


Would you like to try another similar problem to practice?


sure but for the y intercept lemme find it


Ok!


is that the y intercept


Yes! What is the coordinate for that point?


good question lol urmmm...


i dont know actually


Ok lets fill in (?, ?) one at a time. What does the x-coordinate of the y-intercept have to be?


Right! In order for a point to intersect the y-intercept, it can't move left or right, it has to be at x = 0. We can also see this on the graph. Points to the right of the y-axis (the center vertical line) have positive x values, and points to the left have negative x values.


So now we have (0, ?). Looking at the graph, do you see any extra information you are being given about that point?


the e


Yes! They tell us that when x = 0, f(x) = e. So the final coordinate would be (0, e).


yerpppp


alr


Awesome work on this problem! Would you like to try something similar to e and f just for practice?


you know what sure


i dont mid


mind


Great!


Ok, I put a problem below the assignment.


Would you like to try out a?


ok


the zeroes of the function, (b,0) and (c,0)


Super! What about part b?


okay imma jot thhis down in my journal! and lol where the confetti come from ahha


I was excited for you haha Emily


lol aww thanks aalr lemme figure b


Yes that's it nice job! Is there anything else you wanted to work on?


nope i Emily i've mastered this topic honestly


all thanks to your help


are you like a professor that teaches precalc?


cuz how do you know it ?


Agreed, it was great working with you, I hope to see you again soon!


No, Northwood High School, but I just love tutoring.


wait crazy work


what grade?


if i can ask


I'm a college freshman this year. I'm taking multivariable calculus now so I did this a few years ago.


oh omg u smart smart for real!


Lol I'm just a math person I guess


fr like u js helped a lot, so thank you so much, have a great day!


You too, talk to you soon hopefully!


It was great working with you, lmk if you have any other questions!


Hi, Alex!


hi!


How are you?


great, you?


doing very well! what are you looking for help with?


i need help with some homework!


can you see the screenshot?


alright, perf! yeah I can see it


okay if I move it a bit? my screen's a bit funny


just to here


yea thats fine!


okay, awesome!


what do you want to start with?


number 9!


alright! any preliminary ideas?


not really.. i was sick for the lesson so i might need an extra hand


okay, that's totally fine!


okay, so we're trying to see how a) was transformed, or changed, to get d(x)


so when we enter -1 as x for a), what do we get?


yep! and what about for d(x)?


hmm, so what change might we do to get from -3 to 3 (more than one right answer!)


multiply with a -1


yeah, great thinking!


let's confirm by looking at the other values


like if we put 2 in the functions, for instance, what do we get?


yeah! and what about for d(x)?


does that hold up with the negative one?


no


are you sure?


what happens if you multiply 6 by -1?


so does multiplying by -1 get you from the first function to the next?


mhm!


yeah! and looking at the other values, it seems to check out, right?


yess


okay, so that's the symbolic form of the transformation--multiplying a(x) by -1


what if we think about it graphically?


like this is my terrible representation of a(1) and d(1) lol


so the value of k would be -1?


I'm not quite sure what k means here, it seems to be specific to this worksheet or class


but if it's the thing you transform by, then yeah!


have you used k like this in a class before?


(different classes/teachers just use different letters)


uhhh to be honest im equally lost cause i was absent


i somehow got a 93 on the test but im so lost for thsi worksheet


okay, that's fine!


I would guess it would be -1, like the transforming factor


but yeah I'm not fully sure bc it's not a standard math convention


i realized i completely skipped number 9


would that be a reflection over the x axis?


yes!


exactly lol


as displayed in my terrible drawing


loved that! <3


okay, so now 11


d(x) in terms of a(x)


any preliminary thoughts?


maybe something with the signs..?


yeah, nice thinking!


we've found that multiplying a(x) by negative one gives of d(x), right?


mhm


so could we utilize that fact to write d(x) in terms of a)?


would it be d(x) = -1a(x)


yeah!!!!


because we're just taking the outputs of a) and multiplying them by -1! nice work :)


so basically for that part i just need to know k in order for me to solve it


just add it to the beginning


yes, for these types of problems!


once you get to shifting up and down or left and right, you have to be more careful


but for flipping or scaling up and down, you're just multiplying by one number!


which is what you did here, with -1!


and you can always double-check by plugging values back in


hmmm okok


im still kind of confused on how to solve for k but i think i briefly understand


yeah, let's do the next one and try to solidify things, yeah?


yes!


okay, so let's look at g(x) and f(x) here


any guesses as to what might have been done to f(x) to get g(x)?


a stretch?


yeah! nice intuition


but by how much? let's look at some points


do you see the points I marked in blue?


(you can use any pair of points but these were on lines so they're nice!)


yeah, the distance between them is def 3, but for the scale factor, we can take a different approach


nice thinking though!


what are the coordinates of these points?


1,1 and 1,4


yeah! nice job


so if we think of these as functions, what does the x coordinate represent?


a vertical stretch?


oh yeah, I mistyped, the y coordinate! and yeah, it is a vertical stretch!


how much are you stretching by?


alright, good thinking! but, let's try plugging it in


here, what does f(1)) equal?


(it's the y coordinate of the point with x coordinate 1)


yeah!


and g(1)?


(same process)


isnt it also 1?


when the x coordinate of g(x) = 1, what's the y coordinate?


ohhh 4


yeah! it's a bit confusing on graphs


so scaling vertically is a multiplication transformation


when you multiply 1 by 3, do you get 4?


no


yeah! 3 would 100% work if it was an addition transformation


but since it's a multiplication transformation, we need to figure out what to multiply 1 by to get 4


what do we need to multiply 1 by to get 4?


so k is?


okay so k = 4


yep!


g(x) = 4f(x)


exactly!


and we can plug in 1 to see that this works-- g(1) = 4f(1) so 4 = 4*1 which is true!


and then for 12, how would you describe the transformation?


vertical stretch


yep! in my experience, teachers often like to know by how much though


so it never hurts to add


by what factor are you stretching?


yeah!


so a vertical stretch by a factor of 4


omggg tyyy


amazing work, nice job solving through it!


thank sm lol!


function transformations are cool but can be so annoying!


no frrr


i was absent on the worst days lol!


anyways those were my last two questions


thanks sm!


you're welcome! have a good night :)


oh problem solved


yes!


can we use mics instead if that's okay with you.


yes, of course


Hello


OK


Um, I started, you want to know about the extent extend the uh polynomial, right?


Um, yes


OK. Uh, do you have any like material or something uh problem I can start with or you want me to find this something.


Um, give me a sec.


I wanna start off as a pomo, um, what is it called? Give me a sec.


Sorry.


Could we start by um the Panamo arithmetic.


Oh yeah


All right, thank you. That would be a great start.


Give me one second


OK, I start, hey.


So the basically


um


well, I searched some steps like how to teach child, um


uh, there's some operations you need to know.


addition, substruction, and multiplication and exponents.


Correct. I'll teach the the um algebra one


OK


So you understand and I, um,


well, so you know the definition, I think, um, I can give you some example, um, maybe you can give me the answer like like.


I try which level you guys.


OK, that would be nice.


Uh, I mean, do you know some like the basic laws.


like the exponent laws of


Yeah, experiment and then um where


the inverse and stuff, right?


I've been teaching that the algebra 2, so it, it'll be very easy.


OK. So let's try this one.


I


So how do you feel this one


Um, can you repeat that? I can't hear you.


Uh, how do you feel this one like?


this problem I typed in the whiteboard.


Don't you have to um distribute, distribute.


Yeah


Oh, OK, so you can distribute like terms, right?


yeah


Uh, all right.


Oh, that's a, that's a, um, 000 wait, I messed up.


Sorry


No problem


Is it gonna be uh 1 X X plus X will be 1 X uh PowerPoint 2, right?


Mm


uh


well, that's uh


let's say exponential like when your times X to the power of one.


X to the power of one.


then it means


uh cause the base uh diff


the base are the same. So the base are just X and then you add the power this is to on the last one is just one.


That it's not sign.


It's 2 + 1.


OK


OK. And at least it's uh


I


product


And uh when you do


uh I mean divided, uh, let me think here a second. Oh, actual power divided by X to the power of one. OK.


then this one just a change of sign to minus.


to -1


So you can use it to remember when this one is uh the, the product you just think like I where is divided you just thinking as a subject, substruct.


So what you're saying is first you distribute and then Ala turns and then divide, right?


Uh, no, like for this one, you need to separate in two terms first, uh,


so this is the first term, this is 2nd term, and then in the first term there are two things. The two things that the problem for this, you just think about like every single one you need to tap the for this one.


after I


Yeah.


Oh, so you need to, are you gonna add, or you, you're gonna multiply, right?


Yeah, you're going to multiply, but remember you need to every single time, you know, like everyone, you can't lose anyone and uh


and then you just uh give for X already and then you're not thinking about it too. 2 is also the same thing like 2 times the 1st 12 times the second one, the 3rd 1.


OK


And then you add them together and you see which one is similar like you add one more time to simplify it.


Mhm


So, what I'm used to is um for um algebra 1, I've been taught um polymals and factoring, which is similar, right?


Uh, yeah, similar. It's, uh, this is just the factor format, you know that.


this is


The part I don't really understand is uh um the exponent between the X and the 2.


mhm


When you multiply, does that change, or you have to once you multiply, does that turn into one no X PowerPoint 21, right?


Uh, that's no change.


Uh.


That, that doesn't change, that is the same.


So


Yeah, but I'm a little confusing what's that meaning did you say? Like, let's true power.


2 to the power


02 to the power, OK.


uh, so you, you're confused it as well there is no to before it. It's a your meaning?


Mm, um, so you see, um, um, in the XO what is that? I can't even see, but it's X, is it X minus what?


Uh


yeah


X ms 2 and then X PowerPoint, exponent 2 + X + 1, right?


Yeah


So what I'm confused is I get the part where you have to multiply and on each of them.


But once I multiply the x to the x exponent too.


Mhm


does that change the extra one.


And, well, no.


cause it's less rule you need to follow it.


Oh, OK, I get it.


So, uh, the, the just a single X that means x truly power of 1.


OK


And then when times adds to the power of 2, you just add them together.


So


because it's a product role.


X is


Uh, this is a powerful one.


just an ax


X times


so let me change it quickly.


Uh, you can use uh like the uppercase A and lowercase a to just a type 8.


OK, OK, OK


Oops, give me a second


A times A to the part 2, right?


Uh, well, no, I mean, you can use uh the uh the keyboard like the.


All the text box, OK.


the textbooks


I, I'm very confused because I'm new to, and I'm like.


Yeah, no problem


I really need, I, I like math so much, but I just that once it becomes hard, you can't really focus on any of the units.


Yeah, I understand


Um


give me a second. I'm using.


uh


next


in terms


So X to X is y to the


1 to the exponent of 2, right? I'm very confused because I don't like the X women. It confuses me very much, so.


Oh, OK. So,


hm


OK, this is a single X.


oops.


And then this is the X to the power of 2.


Mhm.


So when your tempt them up.


and the thing is, the rule is likely is


OK


Is that clear


So what you're saying, I'm sorry because I'm very confused. I don't really understand anything about this is that I get the distributing like terms, but once you multiply the other term, once you multiply the first sum to the 2nd term, do you, um, does that change anything? Does that change the exponent?


No, no, no. Cause it just said the same thing when you time them up. You just think about this is our 1st, 1st term, and then second term and another term, you just uh like uh you only have the X and the X to the power of two. Let is their first term to write and after light, the second term is x times X, and the third term is X times 1. And after that you think about 2.


the two, the 2 times the first term of light.


The 2 times X, the 2 times 1.


and there is no change of anything is the problem given. So you won't change it. The thing you change is what you got, what you have got is the result you get.


So the answer is, um, one to the PowerPoint of 2, right? 1 X.


Uh, well, no, cause you remember, uh, the, the first act, X is the one actually one power because when you test them up, you need to add them.


Yeah, let's put that


X. There's a hidden exponent that you don't see when you multiply that it comes out, right?


Yeah, cause so when when you see the axit it's X to the power of Y actually, it's not zero cause when actually zero power is equal to 1.


Yeah


So it's gonna be X or 2 of 3 of 22 of 3.


No, 3


go


Yeah, that's 2 + 3.


Oh, you can add like exponents.


OK, that's


Yeah, cause, cause one product X is's eye because when divided is minus.


OK.


OK, give me another problem. I'll see if I can solve that one.


OK


Uh do you need to do a sample way of this one. I can do a sample.


Yeah, yeah.


Uh, give me one second.


I


so


So this is very confusing. Are the um stars multiplying, right?


Mhm


OK, so X times or, oh yes, yes, I know this one's very easy.


OK. I understand it.


Yes


OK, let's talk next step.


It's gonna be 3 times X to the power of 2, right?


The first one


Mm, yeah, yeah, yeah


No, well, no, cause, uh, higher power will be good first.


The highest power would be, um, yes, but X times X to the power plus 2


times X to the power of 2 when you add together it's gonna be um.


Yeah, yeah, yeah, that's correct. Mhm.


is


me


Yeah, let's uh first.


3 X to the power of 2, right?


Yeah


And then there is another one also.


Yeah.


OK. And the once you add all of them, you're gonna bring it down and you're going to like, you're gonna like a combine like turns, right?


cause one they are similar, you can combine them together.


Yeah, OK


X times X is 6, it's gonna be 33 times x 3 times power to the 2, right? Power to 2.


Mhm.


And then it's gonna be


um 3 + 3 times.


3 times 2.


No


the


This one


well, there is no 3 times to cause uh another um, well,


this is uh


single one. There is no similar to this one because this is the third power, and this one is a constraint. You don't think about X, but there is a to the two power let's say similar and you add lukeer. Another one is just a single X tru the power of one.


And then you add them up


cuts there the same.


Let's uh


distribute


OK


Yeah. So did you get whatless question will be the final result.


Um


can you repeat that


Uh, what did you get the final result for this problem.


I got 33 to 2, and then I got


3 to the uh 3 X.


plus 3 X, and then I got.


Yeah, I'm very wrong. I was on a different side of the math. OK, I got it wrong.


cause I I


No, I'm fine. You, you're correct for this too.


can you can't add, you cannot add X plus a number, right? You can add a um a um.


Yeah, you just keep it. It's OK if you can't eat. Let's a problem. You don't worry about it, not very simplified, you know. If you can't simplify, just leave it.


Yeah, yeah they, correct. That's correct. You just uh uh cause acts to the power of 3, you just leave it right on it. It's OK.


Mhm


Thank you


Um, so is that all for today?


Uh


mm, if you can, you, you want to try another problem, I can, I can write.


Right


cause uh the result set of this one, it should be X to the power of 3 + 3 times x to the power of 2 + 3 times x + 2.


uh, which means there should be


4 terms of the final answer.


OK


Did you understand this


Oh


Um, um, I just can't get along with the exponents cause that really confuses me, especially when you type it on a computer or iPad. It's good for me to do on a worksheet because I understand the symbols more, but once I do on a computer, it's the symbols are kind of different, which confuses me very much.


Oh, OK.


I understand you. Uh, so, so this time will be to the power, um.


well cause I didn't use the iPad, but.


Yeah


OK, that makes, OK, that makes easier. OK, thank you. Um, I'm, I'm gonna go ahead and go to another classroom. Is that OK? Cause I have um used history class to attend.


OK, OK


Thank you very much you're very helpful.


It was nice having you as a teacher. Thank you.


Bye


Hello! How may I help?


sooo i have an assignment and i kinda know how to do it but at the same time dont


Okay, mind explaining it to me and then telling me which part you get and which is confusing?


okay so the first equation is write 0.6 repeating as a fraction


i wasnt at the school when we learned this i do not know how to do this


at all


Okay well let’s start by thinking about 0.6 when it isn’t repeating, what would that be as a fraction?


i thinnkkkk 3/5??


Yes exactly


Let’s leave it as 6/10 though


ok


So we know 0.6 is 6/10 but then 0.66 is bigger than 0.6, so to make a bigger fraction we can make the denominator smaller right? We can try 6/9 or 6/8


okay


wait would that be the answer


Well we know the answer is a fraction a little greater than 6/10 so what do you think it is, try some out


hold on i am thinkingg


You can type some into a calculator and try it out to see if you get .6666 remember though the denominator is less than 10 :)


And sure take your time


zero point six six six repeating is two-thirds, not ‘almost,’ not ‘kinda.’  say x equals zero point six repeating, multiply it by ten and you get ten x equals six point six repeating, then subtract the first one from the second and boom ten x minus x equals six point six repeating minus zero point six repeating, the repeating junk cancels out so it’s just nine x equals six, divide that out and you get x equals six over nine which is two over three, so yeah zero point six six six repeating so it isss two-thirds??


this took me awhile


Yesss! Sorry I didn’t give you the actual formula I just haven’t done it that way in so long (im in college). But super well done


thank you so mucchhhh


how is it in college


Suree! And no worries with time you’ll memorize these now I literally just know the basic decimals into fractions right off the bat. college has been fine :) A lot of homework though!


oh i bet in my school we have homework every night


Lol homework does help you retain knowledge though so I guess it’s acceptable :). Is that all you needed help on for today? If so good luck in the future and I’m glad I could help today


thank you and yes that is all i need byee


Bye!


Tommorow we have a daily quiz


and its on the things we learned such as limits and such things


graphically and numercially


Hi Michael! How can I help you today>


oh i have a test tmr


in calc hons


its about the things we learned


what topic shall we go over?


limits graphically and numercially


okay!


do you have any problems that your teacher assigned based on those topics?


wait i think i can get the work we did in class


thats great! i can help you and explain the work that you did in class!


okay so how do i get it on here


or do i js copy n paste


if you look on the taskbar at the bottom


the 5th button would be an image with an arrow


you would use that to upload the photo


can u see ????


yes i can see thank you!


okay ill point igs


okay this problem


like how am i supposed to know which limits


wait i mean domain &range


for this problem you do not need a limit


domain and range okay.


so for the domain first we would have to factor and simplify the function


as you can see the problem factored and got 3x+1/x+3


ohhhh okay so how would i get the 5/3


to find the range you would need to solve for x


so you see how you have the function


okay


now HERE is where i really struggle at tbh


okay which one do you need help with specifically?


how would i know what to right when it says approaching from the left or right if its looks like the same thing


write*


for example 1


so when you are approaching the limit from the right side it literally means from the right side


for example for part D


it says find the limit as x approaches 2 from the right


you would look at the x value 2


and see what y value it approaches from the right side


you can see its a value of 1


does that make sense?


ohhhh okay so what about part L


like how did we get that answer


L would be 1


because 1 is your limit


oh wait


you are talking about part L


sorry i misunderstood


ur good!!!


okay so for part L


in order for the limit to exist we have to make sure that both sides (aka limit as x approaches 5 form the left and limit as x approaches 5 from the right) are the same


that way the limit as x approaches 5 can exist


so if you look on the left and right side of 5 do they both approach the same value?


oh Okay i see so i would look at the arrow going down right?


yes you would!


okay i think im getting the hang of it


yup!


OKAY NOW THISSSSS


is crazy


ah finding the limit numerically


i would just use the calculator for this though right ???


yes you would


but do you understand what the values mean?


tbh no..


okay lets go over it


okay


so the question is asking to find the limit as x approaches 3 of that function


and if you notice when you plug in 3 you get undefined


so plugging in 3 would not work


yes


we could estimate the limit using values super close to 3


ohhhh okay


i see


so you use numbers super close to 3 and plug it in


thats the whole point correct


to find which one is closet


from the right and left side if they approach the same value the limtit would be that value


you have to make sure that the f(x) is approaching the same number


wait huh?


at x = 2.999 and x = 3.001


f(x) is approashing what


f(x) is the value you get after plugging in those super close numbers


f(x) is the limit


ohhh ookkk


at 2.999 and at 3.001 those values are approaching the same limit


so thats why it works


Okay i see now


wait can i go check 1.1 really quick to see if i need help understanding to


yes!


okay the only question I have about this problem is how would i put this into the calcukator


ok so for this one its asking for average velpcioty from 2 seconds to 8 seconds


do you recall the average velocity formula?


distance/time ?


sum like det


right


its the change in displacement/time


okk


so that means we to find d(8) - (d(2))/ (8-2)


sorry (d(8) - d(2))/(8-2)


do get d(8) you would plug 8 into the distance equation


and same for 2


okayyy i remember that part


so thats all im findning


right or


yes and you put it all together into the average velocity formula


and remember your units


okkkkk thats smart!


okay wait so how would i even start by making a sketch for a problem based on limits n stuff


because I rmb we did it in class but i dnt know about a problem pretaining to it


interesting is this still on limits? or is this for derivatives


limits i think


tbh im not sure


oh wait sorry i misread that v means vehicles not velocity in this context


ok


just give me a second to figure out this problem


no problem, take ur time


okay!


so you need help with part b right?


yes


okay so for this we are just graphing the time vs the number of vehicles


so at each time there would be some number of vehicles


Okay


how do you want to scale it?


every 5 seconds?


sure


okay so for that its basically like making a table!


we will be using times from 0 to 30


plug those into the v(t) equation and plot those points


do you need me to go over how to do that?


okayyy so like the first ones just graphing


umm sure u can


okay so for exmaple lets plug in 5


t = 5


you put that into v(t) so in your calculator you would input 92-15sin(5/3)


and then whatever answer you get you would plot that on your y


does that kind of make sense?


you would repeat for every time value


okay yes


thank uuuu


of course!


do you need me to explain c and d?


i think i should do good on the test now


uh nooo


okay good luck on your test and thank you for letting me tutor!


Alex, and ofcccc:)


Hi hi


Hii! How are you doing?


I'm good. Day was good. How was yours?


It was fine. sorry, My network was disconnected


gimme a min


ok


im back!


Hooray!


yay!


pizza party!!


lol i wish


I had some!


Yay!


how should we start?


We can go over the standard derivatives and then do problems


Best way to get understanding


Were you able to look over polynomial and trig derivatives?


We also need to address the chain rule


yes, I look over the cheat sheet. I memorized the trig derivatives


awesome


You can test.


do i open the textbook?


IYKYK


can you solve these


yes!


done! it was easy


good


oh sorry, I didn't notice you were writting.


its ok


i'm confusing myself


how?


where is it confusing?


I can do it but I forgot the notation for it


i think you should have take dg


no its dx


that way dg cancels


does that make sense


yes, It does.


you have an imbedded function


this dg under df is not correct.


it should be dx


ohh, I got it what you did. It is right.


thanks for checking me


shouldn't it be like this, like you explained earier


well lets confirm it


using cos2x, we can use product rule


okay


there should be a "+" in between


no


oh ok! right, i confused it with product rule


can you use product rule to solve the second problem


example*


yes


ok


is that right?


I didn't know i had to substitute y later on.


its ok


sometimes it isn't an easy thing to substitute y in and its easier just to keep y


yeah, depends upon question.


you can see how it would be easier to do x^y and y=tan(x)^2 as just x^y first


yes, that's right


so, we first differentiate f with respect to dg. then differentiate g respect to x?


ok, gimme a min


in xy, who is f and g?


I changed it


Though I seem to have done derivative wrong as I get the wrong answer


where did you get wrong answer?


I got 2x^2 up top


didn't make sense to me


ohh, gimme a min


how did you get 2x and x?


dy/dx=2x, x is a constant in df/dy


how do you know x is a constant?


derivative with respect to y


you can apply product rule to that?


where are you look at?


looking*


where you took eg - f=xy and y =x^2


above cos2x


yes here


right, so when you take derivative with respect to y, x is a constant


eg.
f=ax ==> df/dx=a


yes, right


but below in blue writing, you wrote df/dx


yes here, dy/dx


right


shouldn't y be constant here?


It because we have to consider both variables here


I got it, what you were doing wrong


when you got 2x^2


what was it?


where did i err


Im writing on board


good job


thanks! you got it where you mistake?


keep going


good


arigato!


how do we know if an alphabet is a constant?


so there is something called implicit differentiation


that is where you assume every letter is a variable


most of the time we only use one variable, and every other letter is a constant


You won't deal with implicit differentiation until multivariable calculus


can you tell if i have implicit differentiation in my syllabus?


It has second order derivative


derivative of a function with respect to another function is chain rule


You don't have implicit differentiation


We still have to go over second order derivatives and maxima and minima


actually scroll down to section 6


differential equations is where we start implicit differentiation


well hold on


I remember it from multivariable calculus, thats where we deal with 3d space xyz


diff eq was after multivariable


ohh, three variables?


yes


stuff like that


this represents a plane


oh no, that's bad!


I don't see anything about 3d space here


spheres, planes, prisms


there is. another section for it


those are 3d shapes


where?


what section was it?


section 4


ok


I highlighted it with green


i see it


which textbook is it?


class 11 chapter 12


so chapter before derivatives


oh, you asking for chapter


yeah


can you send me link please?


for the 3d space?


same textbook we are using


Calculus 201


Dr. Smith – In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e.g. y = f(x) and yet we will still need to know what f'(x) is. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates (the next section).


sorry, network issue


gimme a min


im back!


btw what program, or type of school are you in to be taking these classes


I didn'XYZ University (multivariable, diff eq and linear algebra)


the exam I mentioned is basically AP Calculus, so I’ve been going over some higher-level math this summer to get ready for it


and for other topics, we have 2 substitute teachers that covers those topics


AB or BC?


bc


so we have gone over everything up to chain rule


next would be derivative of a composite function and second order derivatives


critical points


okay, do i stop casting?


sure


ok


Im covering vector. and i read about xyz planes


when?


you are doing vector space now?


dot product and cross product?


yes, I started like 10 days ago. I studied dot product, cross product


magnitude of vector, direction of vector


unit, null, collinear vector


right


good


yeah, Im studing from this textbook class 12. it has lot of things.


sounds like it, this is setup for physics


Newtonian mechanics


collisions, forces


ohh, I see. will i be studing them later on this year?


depends if you are taking physics


I think its A and C you cover it, projectile motion


I think B has wave and optics


I may be entirely wrong


Wait, 3 topics or courses are (1) electricity and magnetism, (2) newtonian mechanis and (3) wave and optics


not in that order


ohh, Im not sure If i want to take physics. i don't like it.


haha, no need to be shy


unless you don't need to take it


yes, I will have to research about it. how much it is necessary


State University major


im thinking about computer science


You may need at least one physics course


but it depends on the school and their syllabus


oh, that's bad! why do we need physics in cs


I don't think you will need a physics course


you definitely need linear algebra


but probably don't need diff eq


again depends on your school


yeah, I'll have to look it up.


are you a cs major?


no me


meche


oh, That's cool. do you make robots?


no, I manufacture parts


oh awesome!


what do you want to do with cs?


i wanna go in cybersecurity.


i think it's cool.


That is cool, Red or Blue team?


im still thinking about it


im gonna start with blue team tho.


good ole ethical hacking


yeah


gimme 5 min pls


yea


im back!


welcome back


domo arigato!


I want to highlight topics we have covered


ok


im sorry, my netowrk doesn't seem to wanna work at all.


NOOOOOOOOOOOOOOOO


we'll do that tomorrow


ok. is this goodbye


good night?


yup! i guess


Alex?


John~Emily


Have a great night and morrow


Chris


Sarah


ready


Hey Maya!


How can I help?


Hi! 👋


I’m going into tenth grade, but my math is around late-ninth grade level. I want to catch up and get ahead. Can you give me tips or a plan to study?


Yeah, sure!


Schools around the country all have different standards for math, so what does late-ninth grade math mean for you?


My MCAS results show that my math skills are at a late-ninth grade level. I want to use that information to figure out what I should focus on to catch up as soon as possible.


Gotcha, I see


Honestly, I think some of the best resources you'll have are probably your teachersf


*teachers


They'll know your strengths and weaknesses with respect to mathematics much better than I do, since they know the context of your school and the context of your performance


My first piece of advice would be to ask them for possible resources to work ahead/enrich yourself so that you can work your way back to grade level


Yeah, you’re right


Thanks for the feedback


Mhm!


Of course, there's also online resources


I’ll get working now on math. Again, thanks for the advice


Hey Ben!


How can I help?


can you help me solve this problem step by step


determine the length of the missing leg and hypotenuse of each the special right triangles


ok


hello?


Hi!


Is that all?


yes


Alright, gotcha


Let's start by identifying what kind of special right triangle this is


What do you think it is?


I'm not sure?


That's okay!


How familiar are you with special right triangles?


not that well


Would it help if we reviewed them together?


sure!


Awesome!


Alright, done!


Let's go top to bottom with the 3 common special right triangles


To start, we have the 30-60-90 triangle (named after its angles)


Its sides will always have the ratio 1 : sqrt3 : 2


Next, we have the 45-45-90 triangle


ok


Its sides will always have the ratio 1 : 1 : sqrt2


Finally, we have the 3-4-5 triangle


This one's a bit different because it doesn't have clean angles, so we just name it using its sides


It is a very famous pythagorean triple, so it is pretty well known as a special right triangle


yes


Alright, that's it for now!


okidki


Now, do you think you can identify which type of right triangle this is?


the last one?


Close!


Take a peak at the angles


the first one?


Yeah!


Do you understand why?


explain why please


Alright, sure


helo?


My internet isn't great, sorry


its okay mines isn't either don't worry


Gotcha, thank you


The triangle we have is 30-60-90 because its angles match that of a 30-60-90 triangle


right


You can find that the other angle in our triangle is 60 degrees by doing 180 - 90 - 30 = 60


It isn't 45-45-90 because the angles aren't the same


It also isn't 3-4-5 because the angles aren't the same


ok


The angles in a 3-4-5 triangle happen to be 36.87..., 53.13..., and 90


It's really messy, which is why no one actually makes you memorize those angles


Does that make sense?


yes


Great!


Now that you know what type of triangle this is, let's look at the ratios


okk


Since this is a 30-60-90 triangle, we notice that the ratios of the sides are 1 : sqrt3 : 2


That means that the ratio between the side with length 6 and the sqrt3 is the same as the ratio between x and 2


Do you see why?


Do you think you can express that in the form of an equation that you can solve?


I don't think so


Alright, I can walk you through it


Ratios are fundamentally just fractions


thxs


The ratio 1:2 just means 1/2


2:3 = 2/3


And so on


kk


This means, if we let our hypotenuse be a variable x, we can use the ratios we have to say that since
x : 6 = 2 : sqrt3,
x / 6 = 2 / sqrt3


Now, you can solve that equation to find the hypotenuse


What do you think it would be?


nooooo


I definitely need help


That's okay!


Do you get my reasoning from earlier?


yes I do


Great!


So, we have this equation where x is the hypotenuse


okay


We want to find x, since that's what the hypotenuse is


so multiple 6 both sides?


Yep!


x= 2 root3 times 6


2 / sqrt3 * 6, yeah


Do you know how to simplify that?


nope


That's alright, it looks harder than it is


First of all, you don't want a square root in your denominator, so you multiply both the numerator and denominator by sqrt3


I'm writing this on the whiteboard btw


okk


I see thank you


Do you see how to continue?


does the root 3 disappear


The two sqrt3s turn into a 3


That's because sqrt3 * sqrt3 = 3


ok


then?


Then, you just do the calculations


2 * 6 / 3 = 4


how you get 4


2 * 6 = 12, and 12 / 3 = 4


does the  root3 disappear


No, the root3 in the numerator is still there


I just calculated the green boxed portion


ohhhhh okayyy I see I see


thank you


Of course, is this making more sense


yes


does the 4root 3 go where?


That's what x is, so that is your hypotenuse!


so instead of 2x its 4 root 3


Sorry, the numbers are a bit misleading


The red numbers are meant to be the ratios, while the black numbers are the actual side lengths


okayy


so now we just need one length right


Yep!


It's the same exact process


Just with different ratios


wait can we practice it again with those ratios


Elaborate?


can you help me find the last length


Yeah, sure!


The ratios are a bit different though


okay


Looking at the red numbers, what would the ratio be?


Exactly!


Now, do you remember how to set up the equation?


Let's say x is the unknown side


ok


Can you set up the equation?


no help me


is it 1/root3


That's one side, yes!


is the other 1/2


Not quite, remember that you now want to use the numbers and variables that you know


Here, you know one of the sides is 6 and the other side is x


Does that ring a bell?


so 1/2=1/root3


so x/6=1/root3?


The second equation is correct, yep!


How did you get the first equation?


so in this case it would be like=ke this I'ma write on the board


Gotcha, sure!


You're totally correct so far!


I need help on what to do after


What can you multiply both the numerator and the denominator by to get rid of the sqrt3 in the denominator?


is that right?


Yes!


can you help me after


Sure, I'll try to nudge you in the right direction


You can simplify the denominators, what's sqrt3 * sqrt3?


wait so would that leave 3 by its self


That would leave 3 in the denominator


and after


Now, you don't have to do anything with the sqrt3 except let it hang there


However, you can simplify 6/3


as 2 right


right


Since that's what x is, that'll be what your last side is


x is equal to 2sqrt3, so it'll just be 2sqrt3


Do you mean multiplication when you write x?


yes


Ah, gotcha


In that case, you're totally right!


I wrote it at the top where the rest of the lengths are is that correct


Yep!


Do you understand the process a bit more now?


yess I do I do


Great!


Is that all?


is that all I do?


yes that's all for me on my part


determine the length of the missing leg and hypotenuse of each the special right triangles


If this is the problem, I don't see what else there is to do


You found the missing leg and hypotenuse


yess that's all thank you so much


Of course, I'm glad I could help!


for your help and patience have a good rest of your day


You too, it was a joy working with you


Cya!


bye


Hi Alex!


Hello my name is Alex


Oh sorry, hey Alex


Sarah is my mom


Okay, how can I help you today?


Before do the work i have a pet to watch us


Okay cool


Is there any question you need help with? Or a general topic?


The pet is coming


Okay


I’m going to send the work pic after


Okay, whenever you're ready


Okay should we start with question 1?


Yes


Oh is 1 8


When they tell us what x is, all we need to do is swap out x for the number


5 is negative so we need to keep that in mind!


So what would 3-5 be?


Then it is -2


Yes!


Okay let's move on to 2


Can you finish the equation?


11 1/2


Great!


How would you write out the a= equation for question 3


Wait


Not quite, let's take it step by step


First let's do (-2-3)/5


What would -2-3 be?


Yup!


And then we have -5/5 which is


Yes -1


Then it should be-15


When we divide the same number the answer is always 1


And when the top number is negative, while the bottom number is positive the number is negative too


Thats why here we have -1


All good with that?


Yes


Is the answer -15


yeah the final answer is -15


You got it in the end :)


Yes!


Okay let's start with -1^2


Are we on 6


Yes or no


Not yet, we're on 5


Do you know how to calculate -1 squared?


Not quite, let's take it step by step


What is -1 squared


Do we follow pemdas


Yes, so we first want to see what -1 squared is


And it is 1


So we have 3 x 1


Now we have -(-1) which would be


Yes


And now we just have -7


So what is 4 - 7?


Awesome


Do you have an idea how you would write an algebraic equation for 6?


Just the equation for now, without solving it yet


No


Okay


Wait


Let's make p stand for pineapples and m stand for mango


You can continue


So we want to just focus on this part for now


we have p and m and we know that there are 8 less p than m


How could we write that out?


No m=p-8


Good thinking but not quite


We do use the subtraction that's true


What would we do to m to make it the same number as p


which one would you pick a or b


What is a


And whaat is b


a) p = m + 8 pr b) p = m -8


B


Yes


Okay let's go through the difference between your answer and b before we move forward


Let's say p is 10


We just need the eq


But yes


Yes but what we need more importantly, is for you to understand how to get there without my help so you can be ready for class and tests


So with your equation we have that m is 2 and p is 10


But does this satisfy our original rule that p has 8 less than m?


But is m is 2, then it's m that's 8 less than m


Nod


But what we need, is p to be eight digits smaller than m


So what do we change to your equation, to make sure that p is eight smaller than m instead


Yes that works!


Awesome


Now i think i have an eq for 7


Okay, great type it out for me


J=(b-12)divide 2


Yeah that's right


Do you want to go over how you calculated that


That becomes the - and the divide


Yup


Great work!


The answer key says it is b=2j+12


I feel like you're doing really well with these. My advice would be to double check as you're going along because one small blunder can impact the whole equation, and have more confidence


Yes b = 2j + 12 is your equation rearranged


Which is the opposite


Do you want me to show you how to rearrange it into the answer guide


You just flip the signs into + and x


But x is 2j


2j is 2xj


Yes!


I think we are done


Okay, good luck with everything and always feel free to reach out. I feel like you know this stuff well and can do it!


Bye bye


Bye! :)


Hello!


Hey Jamie, what do you need help with today?


I need help with one question. I'll upload it on the board!


Alright, what about this question confuses you?


I'm not sure how to start the problem


Alright, how about we try rearranging the equation?


Do you see any way we can make the equation easier for us to utilize?


Is that right?


Well instead of replacing the terms, how about before we do that step, we just rearrange where the terms are?


do you know what a constant term is?


not really


I missed the class on this topic so I'm not sure


Ok let me give you  an example


now in this equation, what do you think the constant term is


Now, that means that when x = 0, J(x) = 3, correct?


yes


Now, understandinrtg that, lets rearrange the equation a bit


Do you understand how I rearranged the equation, and why?


Yes


Is it to find what the value of g(x) is?


Well not g(x)


that wouold be a function


we would be finding g(0) with this


now how would you go about doing that?


with this new equation, knowing what J(0) and g(0)) and x) equal


okay


There you go!


Is there anything else you have any issues with or do not understand?


nope!


Alright then, have a good day!


Thank you, you too!


Hi! how are you doing?


Hello, I am fine, how are you?


great!


so, how can i help you today?


are you able to see the request i said when it asked me my goals for tody


yes, i can, so you want algebra questions to match the Florida ones?


yeah from the same cirriculum as Florida algrba one classes


could you be more precise with what lesson you need help with?


or i can research the curriculum but that can take some time


oh i dont need help just to get me a headstart because of the summer


im going into 9th grade so i just need a little reboot of my math skills again


let me take a minute to research a bit.


k


or you could just give me regular algebra one pornmbles


Northwood High School, particularly to Algebra I. Northwood High School, Northwood High School credit–bearing course and requires passing the statewide Algebra I End‑of‑Course (EOC) assessment, which counts for 30% Northwood High School credit, though it doesn’t affect promotion beyond that grade level
A student who completes Algebra I in 7th may progress to Geometry in 8th grade, paving the way to Calculas


pretty sure there simalar anyway


here is what i found. if you also need help with what to revise tell me


yeah that sounds about right


okay, one min


1. Foundational Pre-Algebra Skills
These are must-know basics:

Operations with integers: positive and negative numbers, order of operations (PEMDAS)

Fractions and decimals: operations, conversions, simplifying

Ratios, proportions, and percentages

Exponents and square roots (basic)

Solving one-step and two-step equations

Properties of operations: distributive, associative, commutative

Understanding variables and expressions

Basic coordinate plane skills: plotting points, reading graphs

2. Preview of Algebra I Concepts
These are good to introduce during summer to reduce difficulty later:

Simplifying algebraic expressions: combining like terms, using distributive property

Solving multi-step linear equations

Understanding functions: input/output, function notation, interpreting tables

Graphing linear equations: slope-intercept form
y=mx+bSolving inequalities and representing them on a number line

Basics of quadratic expressions (just the idea, not factoring yet)

Real-life application problems using equations and graphs


here is what you can revise over the summer to be ready. you can find plenty of materials and excercises online to help you


huh


what


What's wrong?


i need the actual problems


so i cna solve them


oh okay. i can create some right now. one min


i asked to like find some algebra one problems


ok great thanks


could you write it on the board


or paste the picture of it


Ok, so i created some equations and functions. Do you want me to write them on the board?


yes please


i will do them one by one so you can solve them


ook


you can do this


r you sure this is algebra one


this seems too easy


we can start easy


ok


still simpifying


Yes


now, solve for x


k


are these separate problems?


yes


sorry my power went out for like 5 minutes


i think


for the first?


yeah


yes, it is


and 6


for 2


yes, correct.


now i will write in the chat two systems bcs it is taking me forever to clean this board lol.


k srry


1
y=2x+3
3x+y=9


2
2x-3y=12
4x+3y=6


you can work on these


x,y


Yes


and 3,,-2


yes, correct


first one was a bit trickier


now I will give you a function


also theres a featur with the three dots


at the tools at the bottom


if you click the three dots you can delete the whole board or undoo everything


Yeah, I saw it. You are my first math student sorry


its fine


this is algebra?


looks funky for algebra one


these are functions.


ok


you can try them


whats p and f


Where do you see p?


f is function


you basically have to convert the x with 0 and 2


so for the first one find


Thats an f


oh


why are there two functions


sorry my writing is bad


you have to find the function when x is 0 and when x is 2


x is 0 and 2 at the same time


So it is for the first one: 3*0^2 - 2*0 +1


you have two points there


one where x is 0


and one when x is 2


ok


do the math and the first should =1


find the second


ahh ok ok


i understand


it now


thats good


ill be right back


Yes


okay sure


i am back


ok, so now i prepared a problem


hmm did you connect?


The population of a bacteria culture doubles every hour. If there 100 bacterias in the start, how many will there be after 5 hours


hmmm simple


Is the point (3,-4) a solution to the equation y=-2x+2


this is the last one i prepared


yes


ok, perfect


so, do you have any questions now?


nope


do you have blooket


account


no, never heard of it


oh ok


if you ever need help, just in the feedback favorite me, and you can request another session with me


you can favorite


and request?


i can give additional material here and you will be emailed, but its better if you request it


ok


bye thanks for the help


bye, anytime!


Hi! How are you?


Good and you?


Greaet


What can I do for you today :)


Is this for AP Precalc


I want to understand how to solve these questions


O


Yes


For the first question, I think we can simplify it first


First we can move all the x to one side, ie |y+2| = 2x + |y-2|


Then the y terms


We can then get |y+2| - |y-2| = 2x, which can be further simplified to 0.5 (|y+2| - |y-2|) = x


You divided by two why aren’t they under the y values


I multiplied by 0.5


Oh


I understand


I don't have a stylus on me so I'll be typing a bit


Okay


With the equation we get we can see that if we plug in y = -4 and values around it, we can visualize what it can look like


Lemme get a desmos snipit real quick


Ok


Back to the equation, we can see that with the two absolute values of |y+2| - |y-2|, if y is negative, x will also be negative


for example if y = -1. we will end up with |1| - |-3| = 1 - 3 = -4


So for one, we know that near y = -4, the implicit equation will be negative


Oh okay makes sense


next, we can see that at y = -2 and 2, the equation suddenly jumps to an infinitely steep slope


but if we analyze the equation


we can see that |y+2| - |y-2| will lead to x being 2 or -2 past y = 2 and -2


Okay makes sense


Basically with this problem it's best to simplify and then plug in values near the point specified


for problem 2 we can compare rate of change.


Using the y_2-y_1/x_2-x_1 for slope


We can plug in and see that we get slope = N - 0 / -1.1-(-1)


or N / -0.1


N can’t be negative and can’t be zero so B CD automatically are eliminated


absolutely!


Okay that one was easy


So with -0.1 in the denominator we get this one


It's been a while since I've done parametrics so I might need a moment


It’s okay


Okay so first step would be to get t, so we can sub in x = t


we then get y * e^t = 3, and simplifying, we can get y = 3/e^t


so, with x = t and y = 3/e^t


x has a slope of 1 (with x = t) and y is a negative exponential


Thus y will have a negative slope


I posted an image of the parametric


For most of the parametric problems, it'll ask either to analyze the x and y components


Oh okay


Thank you


btw are you comfortable with isolating equations for the x and y components for parametrics


Can you give me an example


Let's say I want you to give me the equation y = x^2 in the form of (x(t), y(t)))


I will make x equal to t


And since y equals x squared then it’s turned to y equals t squared


nice


For ap precalc I think for parametrics the most difficult part is knowing how to find the x and y components


past that when you have them isolated you can often just look at the components individually for what the problem's asking for


Okay


for example a circle, x^2 + y ^2 = r, it adds in trigonometry when getting the x and y components


An ellipse too


it'll look something like x = rcos(t), y = rsin(t)


yeah


ap precalc is mostly an algebra review course of calculus


So a lot of practice is good


I am going to take calculus in a week for my sophomore year … is it hard


I would say if you got the fundamentals for algebra down it'll be pretty light for the first half


Derivatives and limits are very similar to the rest of algebra


Oh okay


There's some really good sources online


I would recommend youtubers like blackpenredpen


he posts videos like doing 100 derivatives, integrals, precalc equations, etc


and he explains the concepts really well


Oh wow


Thank you so much


Will that be everything today?


Yes


You really helped me a lot


No problem, have a good day!


You too


Hello


Hello, how is it going?


What can I help you with today?


I need help with the problems on the board


Okay.


Which part are you confused on?


How to find the correct answer..?


First, tell me what a domain is.


Al the x values


*All


..hello?


Hi, sorry.


Yes, that is correct.


Now, for the first question, what are all the x-values possible?


I’m not sure


Where does the graph start and end (on the x-axis)?


Which number?


The greatest is 6 and the lowest is -5


In x value


Correct.


*s


So, technically the range is from -5 to 6. But, we need to account for any open circles within the range.


Are there any?


We can also eliminate choices a and b, since those are the not the correct numbers.


Ok


There’s two open circles


We won't consider open circles that start the function. Only those that are within the function.


In this case, there is only one.


Which number is it at?


Or just -4, we only need the x-value.


Now, we can determine what the domain is.


An open circle means paranthesis, while a closed circle means brackets.


Can you determine the domain now?


What doe the -4 do


*es


It's a hole in the graph, meaning it doesn't exist as part of the domain.


So does it affect the notation..?


It still does affect the notation, yes.


Look at answer choices c and d


They end with -4 (for the first part) and start with four (for the second part).


This indicates that -4 is restricted from the domain, but is still included in the notation.


What do you mean that it starts and ends with four?


(-5, -4) U (-4, 6]


(-5, -4) - this ends with -4.


(-4, 6] - this starts with -4.


Oh


Okay


Yes.


So, can you now determine which is the answer choice?


Is it c


Yes, it is.


okay


Next question?


Yes


Okay.


Do you want to try this one on your own first?


is it a?


Nice job.


yay


How do you solve the last one?


First, for the domain, identify the smallest and largest x-value.


Uhm


Negative Infinity and positive infinity..??


Yes?


Yes.


But, we have a hole in our graph.


Or a break.


Which number does this occur at?


Correct.


So, what would our domain be?


still not sure


(negative inf, 3) U (3, inf).


This is what I think it is.


oh


ohh.


Wait does that represent just the domain


Yes, not the range. Just the domain.


Hi, are you there?


Hey Mr. Lee


Hello


Before we start


can you talk with your mic?


is it fine?


I prefer typing.


ok is it fine if i use my mic


ok


All right. So, I wanted to give you a um


Question.


And ask you, can you like give me a question on a topic, so like I can understand it better. The topic is like


Um


Converting


This 2 decimals.


It's converting fractions to decimals.


I want to ask you, yeah, if you can give me um a um


Question on it so I can understand my topic.


OK.


I put one on the board


OK, so, uh, this will be 75% because 3 out of 44 would be a quarter. A quarter is 25.


So 3 years, 75 out of 44 will will equal. 4 out of 4 be.


So, yeah.


What would it be as a decimal?


As a decimal, it would be um


OK, Mr. Lee, uh, these type of questions are good.


yes, that's right


Like these are


Yeah. So these questions are good. Now, can you give me like harder?


uh, questions like that?


If you can.


So. OK, um,


So, what we want to do first is because I


What you want to do first is to


multiplied by like 7 like that. You wanna.


to put


About 40s.


And then what we want to see is


How many times 7 goals and 26, 6, it cannot go. So it can go to 60, so that would be


About 5 times, 7 times 5 is, it's 35.


I.


So that I can


Ah, so 35. Well, we can go more.


7 times 9 maybe. I think Yeah, that won't work, so maybe 8.


Um yeah, that would be a 56, so I think that's our best bet. So 8:56.


Minus that.


OK.


Borrow, I mean, no, that's going to be a 5, my bad. That's a 10, that's a 4 now, 50 or 40, uh, to get this, you would, uh, do 5 this time.


So this would be a 5 right here. Can't forget the 0 over there 7 times 5 is 35.


You minus that.


Take this 3. This will be 1, that's a 5. That's a 0. Now we just bring down that 50 now uh 50 would be mm not a 5.


Here's the thing that's probably here for engineers.


Maybe 4867 times 7.


What happens is if there is a difference between them Mr. Lee, you hear?


I'm


so far so good


Alright, uh, that would be


whoops


Thanks, man. Oh, that'll be 49. My bad. Yeah, I realized that after.


because essentially, so.


Right.


We're introducing the 49, uh, what you, my bad, if you hear some voices behind. That's my brother, OK, so.


Minus that and then you get 4, take that, that would be a 4, give that one. That's a one, it's 0 again. Take down another 10.


Right. Another 10.


Um


Now we're going to see sometime, how many times 7 goes into that, that would be sometimes 10, no, 7 to 70, 7 times 8, no.


We can go, no, 7 times 1177.


only bring down one zero


Sometimes 2.


I think 7 times.


Um.


100, well, we could do sometimes 1498.


Only bring down 10.


Uh, what do you mean by that?


so it would be how times does 7 go into 10?


So it'd be uh


To the 7 going 10.


Oh,


Oh yeah.


My mind turned off. My brain just shut down. OK, let me do that again.


OK. So, yes, you're right. I forgot about that.


These little mistakes.


OK, so, some someone would go into that one time, 7. That's 17, take that.


That's um


Bye-bye.


3. So 30.


This is a long process, but we're getting there. Uh 7 goes into 30 about not 5 times, 4 times, 28, 4, we would put this right here.


Or


Sometimes 428.


What we have is leftover is.


No, no 0.


Take it, uh, 7 can go into 20.


Not 2 times.


A she? Yeah, 2 times because sometimes 2 is 14, sometimes 3 would be a little bit over.


So what we would do would be


7 will go into 20.


2 times.


So 27.


That's a 14.


Oh man, it's a lot of work.


OK, so one.


Well that's


That's 6.


You're getting there.


That's going to be a 60 now.


So 7 goes into 60.


We did this before, how much did it go in?


I think he was.


Nafo.


That's sometimes.


Uh


Sometimes 8.


Right? So


would go into


That


It's the closest number, 7 times 8.


Now would be a 50.


OK, so I understand.


OK.


Mr. Lee, you wanna say anything?


so you are back to 60, which is where you started,


so now it's going to repeat


Yeah, you're right. Actually, I didn't see that nice eagle eye. um.


Yeah, you're right, so we're, we would stop would just be um


The two


So we can just


Take this one out.


The 8, and I think that would be it. So, Mr. Lee, what do you think? Am I right?


Or close.


Yes that's right


Let's go.


I feel good now.


Alright, Mr. Lee, um.


The number will keep on repeating


If you can stay with me for a moment. Uh, I have another topic.


Yeah, yeah, you're right, I can see that one now.


OK, so what I, my next topic is, um, this is all for the uh SHSAT. This is why I need to know it.


Um, next one will be converting decimals to fractions.


Which we kind of did actually, if I'm not going to lie, we actually did that.


So


Uh, let's see the next one.


Fraction 2


that.


Um, give me one sec, Mr. Lee.


Um


My next one was converting.


to fractions and decimals. I want to ask you, is that the same as we did?


Mr. Lee


papers.


What is your question?


Well, my question was, um,


Uh, is


Converting uh.


Converting percents to fractions and decimal the same as what we did my first topic, like the same as my first topic, which was uh converting.


Converting fractions to the small is this the similar or the same topic?


Did we cover it in the first topic?


is the topic converting decimals to fractions?


I don't, um,


No, Mr. Lee, uh, it's fine. Let's just do a different topic. Um, my next topic was


Give me one sec. Let me, let me take it out real quick.


All right, Mr. Lee, next one is uh ratios and proportions, as HSCT math, um, what we need to do is learning the concepts. It will take about 15 to 25 minutes if that's fine with you.


I want to know if that's actually fine with you.


It's not so good, I understand.


Do you have a problem to go over?


Well, um, yeah, so, let me actually give you the, the question.


Give me 1 2nd, Mr. Lee.


Hi, Mr. Lee, I'm going to write it over here, here.


Yes sir