Calculus Symbols and Notation - Basic Introduction to Calculus

[@johnchurch2632]

73 -year-old retired math teacher here. I am enjoying watching John's hard work teaching math. I have my own crazy way of doing things but the clarity and fun of watching another mathematician is a pure joy and I look forward to soak up as much as I can. Great job explaining a subject I love.

[@johnallen1684]

My Calculus teacher at lau lost me when I missed 2 weeks of class and five classes a week while attending National Guard 2 week Sumner camp . No fun catching up

[@leonelfederico245]

Decades ago I stumbled through calculus with a passing grade. At least now I understand what the heck I was working towards. I love this guys videos and he inspired me to purchase an algebra book to see what I missed the first time. I'm going to work my way towards understanding calculus once and for all. Thank you Tablet Math.

[@johnplong3644]

I am doing the same thing I had semesters of calculus I forgot got a lot of Algebra 1 Algebra 2 and Trig Oh my God Pre Calculus doing Limits ..No way at this point It has been almost 40 years As I do the work It is coming back to me It May take me a year or 2 to be prepared to do calculus again

[@uberfalcon1965]

I failed, grading on the curve got me a D+😮 The Prof just didn't care. People asked questions and blew through anyway.

[@JulianShagworthy]

Learning begins at 5:05

[@razalasreficul6902]

Thank you, this guy talks too much

[@golden.lights.twinkle2329]

It's way too long-winded.

[@blitzblutz]

Thank God!

[@phyllismyre3040]

Yes he wastes time with to much BS thank you!

[@Xi_Jing_ping]

Tqq

[@junebrown8207]

This has been an awesome video. Thank you so much for making this video. It certainly makes me want to know more about Calculus. Please continue making these videos as they really do help people to love maths.

[@r.d.boschung8374]

When integrating, you're adding up the area under the curve that is made up of multiplying the value of x in the function 3x^2, times the change in x, which is represented by "dx" in the integrand portion of the integral, from 2 to 5. Area here is the y value of the function 3x^2, multiplied by the change in x. So 3x^2 is the height of one of the infinitely small rectangles, times the width of that rectangle, which is represented by "dx". You're adding up all the infinitely small areas produced by (3x^2) x (dx), from 2 to 5. That's what "3x^2 dx" means. Think of it as "(3x^2)(dx). Hopefully this make the "dx" in the integrand better understood. Thanks, B

[@davidl3846]

Thanks, your explanation was great. I understand what calculus does, but I have always struggled with the notation.

[@frankroper3274]

I tuned in to learn some calculus tips but 5 minutes into the video this guy is just yammering on and on!

[@TerribleChesss]

I can verify as the judge, he is the best!

[@legna6802]

Very delightful and pleasurable explanations

[@quakers200]

I took two semesters of calc. From someone who obviously had English as a second language. In any case I finished up taking non calc statistics which for a Bio major was much more fitting. The hardest part of calc for me was not exactly the mechanics, you can sort of memorize your way through that without much understanding. What was very hard was applying real world problems to calculus. A lot of students end up with tutoring for calculus and I would recommend it to anyone struggling with it.

[@BBCNIGHT]

I’m in 5th grade and do math team I like hard math and easy so I started to learn calculus I’m only 11

[@BrightonMorupane-m6u]

Start from 4:30 if u want to avoid the intro

[@cadester2004]

Thank you for going so slow and explaining everything, I think this is one of the only channels I actually follow because it’s beginner friendly, I’m horrid at math.

[@janibeg3247]

when i took calculus, i had to buy a book called "Introduction to Calculus with Analytic Geometry" by Andree to explain my massive course textbook. That was decades ago.

[@kweku3946]

Excellent explanation

[@velona509]

Very well explained. Thank you so much.

[@alejandremeollo1569]

TOO MUCH TALKING. I NEED THE RESULT OF THE PROBLEM YOU ARE SOLVING.

[@funmilayoadewuyi3438]

tf

[@RASHMIKANTSINGH-m4w]

I am in class 5 and i understood just simplifying and explaning like that will surely make anybody understand it. I am really thankful to you sir (I thought it will be hard, but it wasn't!!)❤❤❤

[@rashidissa5887]

Me the beginner to the subject care less with regards to time. Thanks dear tutor

[@jerrymiller2367]

What if you need to find the area under a very squiggly line? Is there an equation for every type of line?

[@goodpun]

I am unsure of this myself, however I think the answer to your question could be one or two things; the first possible answer could be that, well, yes, there could be an equation for every squiggly line, however that is very unlikely. The most likely thing is that you would do it in segments by dividing the squiggly line up, doing multiple equations for the different segments, and adding it all up. Not sure if either are correct, but hopefully the theory helps.

[@navjotsingh2251]

That seems stochastic, and there is a field called stochastic calculus. But that is way beyond the standard calculus, and involves a lot of probability and measure theory. Stochastic calculus deals with functions that are extremely squiggly and contain a lot of noise.

[@navjotsingh2251]

@@goodpun that is an extremely inefficient way, breaking up an extremely squiggly line is not computationally efficient good luck doing that with complex problems it’d be time consuming breaking something up and repeating the calculations. There’s a field called stochastic calculus with deals with functions that are not smooth and contain a lot of noise. Look into it.

[@julianallagas2251]

Where do you use it in a real situation?

[@robertpagano1203]

I loved learning the Concept of integratiuon, but could you take it a little further and show how you plug in the numbers and work the mechanics of the formula, just to give us a few examples. Thanks.

[@Rem91067]

Many years ago, I thought I was intelligent. I was actually good at math. Then I tried to take calculus and was totally lost. That was 30+ years ago. Now, after not studying and hardly ever using the math I did learn, I actually understood your explanation. If my instructors back then had been as clear in their explanations as you, I might not have given up any ambition to major in the sciences.

[@dennis3449]

5 min before any content!

[@nats50]

Sorry to say, but what I notice about your videos is that by the time you get down to the topic, the video is already half gone! 😆

[@mendozajovy]

=x^3, 5^3-2^3=125-8=117

[@pennstatefan]

The function is x^3 ; Plugging the numbers for the variable x gives us (3)^3 - (2)^3 = 19

[@michaelmcdonagh2725]

Thank you john it great

[@willardneal1810]

Very nice. Thank you.

[@agusspermata4600]

What is the answer n solution....i got confused with your talking

[@goodkawz]

5:00

[@donnafields900]

It is really taking a very very very long time to get to the point. Too wordy.

[@danruth1089]

It's not physics there's no experience other than the graph model.

[@theobgshow]

He does that on every video. It's annoying. It seems to be a male trait on KZbin videos.

[@enricoboldrini5350]

​@@theobgshow of course, because I'm pretty sure only men do that lol

[@theobgshow]

@@enricoboldrini5350 you missed the point

[@enricoboldrini5350]

@@theobgshow "It seems to be a male trait on KZbin videos". No I didn't. You said that it seems to be a male trait on KZbin videos, which implies that woman don't do that on youtube, and I disagree with you.

[@felixramey8939]

How do you find the area

[@leonelfederico245]

lol, I was wondering the same thing. However, the conceptual description is amazing.

[@samconstantinou2335]

Isit necessary to have 2 adverts every 4 minutes?

[@skidog75233]

The area 😊of each rectangle is y times dx, and the area of all is the sum of each as dx approaches 0. This is the function of integration.

[@KNOW529]

But he has not explained this

[@edwardprasad5942]

I find this refreshing

[@HydroponicKale]

This video could have been 5 minutes long, and just as informative.

[@sandynaro1]

Anxiety beyond. I am taking calculus this semester.

[@dareese6778]

When you do the rectangles to get an estimate of area, then you go to smaller rectangles. Why not figure out the left over triangles to get an estimated area?

[@ndailorw5079]

@ Dareese Banish the thought and idea of the process of “estimating.” The process of calculating the amount of area beneath a curve is approximating, and therefore called approximating. So, and without going into any detail (in any calculus text and located at the beginning chapter on integral calculus you’ll find mathematical proofs for approximating rectangles with regards to area problems as well as to other problems in science. The proofs satisfy our intuition and gut feeling about the matter by showing and giving us the precise definition of the process), you’re not “estimating” here, you’re approximating! The two ideas and their processes with regards to this particular problem, though somewhat intuitively similar, are nevertheless different things and different processes. Since there’s no mathematical formula for finding the area of polygons (many-sided shapes) of the type beneath the curve of this parabola such as the formulas we do have for finding the areas of polygons that are in the shapes of triangles, rectangles, squares, circles, and whatnots, then the exact amount of area under the curve is found instead by the process of approximating the area beneath it, and not “estimating” the area beneath it, using rectangles and the sum of their areas, since a rectangle has a formula for finding its exact area. In other words, since the area under the curve has no formula for finding its area, the sum of the areas of the rectangles inscribed beneath the curve of the parabola within that area and up to all its boundaries on all its sides is used instead to find its exact area. The more rectangles inscribed beneath the curve, the smaller they become as a result, necessarily so, and the better and nearer that process of approximation comes to be to the exact amount of area beneath the curve. And as a further result, and simultaneously, your “left over triangles” also disappear by that process. So that ultimately the entire area beneath the curve is completely filled in by that process of approximating it with rectangles. That is, in doing so, the exact amount of area under the curve serves as the limit of the amount of rectangles that can be inscribed within it. In other words, in reaching that limit, which is the entire area itself, we arrive at the exact answer, and that answer equals the sum of the areas of all the inscribed rectangles. Which is simply to say, though somewhat longer in saying as much, no doubt, that if we know the exact sum of the areas of the rectangles that completely fill the area beneath the curve (since we can’t know the exact amount of the area beneath the curve, since there’s no formula for finding it for shapes like this, without approximating it by using the known area of the sum of the areas of the inscribed rectangles) then we’ve found the exact amount of the area they’re inscribed within when we completely fill in that area with them. The entire area beneath the curve of the parabola, bounded by that section of its curve from x = 2 to x = 5, the lines x = 2 and x = 5 themselves, and the line along the x-axis from x = 2 to x = 5, serves as the limit of the amount of rectangles that can be inscribed within it to fill it entirely. Therefore, the question of the exact amount of area beneath the curve and within those boundaries pointed out above is in effect and indeed reality the answer to its own question when that limit is reached by approximating it with rectangles until that area is ultimately and entirely filled in by those rectangles and the sum of the areas of those rectangles then added to give that area’s exact amount and the answer to the question. Long story short, the exact amount of the area beneath the curve of the parabola equals the sum of the areas of the rectangles inscribed within it that fills it entirely and completely.

[@dennis3449]

Because calculus is EXACT. Estimation is just used to see if we're headed in the right direction

[@worlore1651]

Reminds me of programming. You give a program rules to follow.this would be a loop, you give the loop a set of rules and it follows.

[@davecrooks2609]

Intro too long.

[@rainerkuper9683]

Much too long.

[@kimthiennguyen3710]

Could I skip algebra 2 and go straight to precal?

[@kingkoba5618]

You need a2pc in order to take precalc just like you take precalc for calc. Requirements to strive.

[@maskedmarvyl4774]

My problem with learning "all the little rules" of Calculus in order to "find the answer", is that they are taught by rote, and learned by rote, without explanation of the concepts Behind the rules. Also, rarely, if ever, are practical examples given of why we would want to find things like a certain bounded area under a parabola, and what real life situations require graphing a parabola of a,certain magnitude to begin with. It never occurs to most math teachers that their students would want to understand the underlying concepts and practical applications of these problems.

[@davidl3846]

So true, I'm reasonably intelligent and have a BSc, but without practical applications my mind glazes over at the thought of doing an academic exercise just for fun.

[@davidboyle4798]

Is the answer 16sinx^6

[@davidboyle4798]

Actually

[@yolandaschuenke4384]

what was the answer?

[@shey_clone]

117

[@robertwisne7993]

Way too much talking off the subject! First explain that algebra can find the area of a shapes with straight lines. Integration is needed to find the area of a shape below a curved line.

[@KennethSorling]

"the area under a curve"... Isn't that the operation denoted by big sigma? As I understood it, sigma equals integration, summa equals derivation (which isn't treated here). Am I misinformed, or did you explain the wrong thing?

[@carloskruut423]

I learned recently that sigma notation is for finite summation while integration which is the long S shape is for infinite summation. So because integration has to do with finding the area under a curve which can be thought of as an infinite amount of small lines going down, the area under a curve is found using the integration symbol S

[@dbthree3892]

Why is the function 3 X2 and not 2 X2 or 4 X2 etc.

[@ndailorw5079]

@ DBthree No particular reason. He simply chose that function. He could’ve chosen a host of other quadratic expressions. That one probably just floated his boat at the moment, and it’s a pretty clean and easy one to work with. But you can choose the functions you listed, too. It means nothing beyond that at this point in his course. He simply chose any ole’ function, is all. So, nothing to sweat, he simply chose a function, plain and simple. At this point, he simply wants to show us what things are and how to go about them, and not show us which things to choose.

[@tomherd4179]

Was waiting fort the integration, but it never came.

[@eod9910]

Wow 5 minutes into the video before you got to the point.

[@oldgraybeard3659]

I "DISLIKE-ed" this video for two reasons: 1.) It's a bit verbose and circumlocutous. I felt like you were talking to elementary school kids. 2.) With the 20 minutes of this video, you could have explained both elongated "s" and sigma symbols, and differentiated when each is used. I'm in college, and never had this explained to me before. Please, remake this video.

[@jerrymiller2367]

He IS talking to elementary school students! Or at least it's where they can start, at that age, to learn these concepts. If you're an adult watching these then you probably went off the rails with math sometime around age 12, and he's trying to pick up where you left off by teaching you the basic calculus concepts. These videos are great if you know nothing about calculus. You can try some other KZbin videos on calculus, actual videos of college calculus classes, where they have five times the material at ten times the speed and see how well you understand those instead.

[@blyat2477]

🤓🤓🤓🤓🤓

[@theobgshow]

This 19 minute video, could have been 4 minutes, if he didn't love the sound of his own voice and his repetitive long intros

[@jeffharrison1090]

So are there REAL answers like the AREA of a rectangle 3x4=12...or is the answer just a "formula" which is what you started with? Total confusion!

[@user-ub5su9vj6d]

5:05

[@4XLibelle]

Ugh!! Love your videos but I clicked on this one specifically to learn why the dx is there!! Darn; can someone please explain?!

[@ndailorw5079]

@ 4XLibelle First, integration (the integral) and differentiation (the derivative) are inverses of each other. One undoes what the other did. In Oder to get back to an original function we started with we either take the derivative of an integrated function ( that process is called taking the Antiderivative), or we take the integral of a derivative (that process is called integration). Let’s say we have a function f(x) = x^2. Then it’s derivative is, df(x)/dx = 2x. Now, and to answer your question, when we multiply both sides of that equation by dx, in order to get dx on the right side of the equation to indicate that the function’s being differentiated (taking the derivative) with respect to x, and to also cancel it out on the left side of the equation by that process, we then end up with a differential equation df(x) = 2xdx. Which equals f’(x) = 2xdx. Next, if we take the integral of that equation on both sides of the equation (but right here understand that I’ll be using capital S as our elongated integral sign since we don’t have an integral symbol or sign on our keyboards) we then get S f’(x) = S 2xdx! And that’s one reason why we see dx next to the integrand of the integral ( the integrand is 2x for this particular function) for this particular function and how it gets there. And taking things a step farther to its final step we get S 2xdx = x^2 + C. And I’ll stop there, hoping that I’ve answered your question.

[@4XLibelle]

Perfect! Thank you so much!

[@terrywix6844]

Let's learn something that you'll never need or use in your job...

[@TokyoExpress867]

He should have had a million views , but he takks too much, maybe i am dumb

[@rasheednesbitt8667]

You talk wayyyy too much! Try to be more concise.

[@ajmiller7102]

Sometimes one can over simplify, use too many words to make a point.

[@Philosophia-yise]

but that means you will never know the exact area of a space under a curve...no matter how precise it is never exact

[@FrenchmansFlats51]

correct it is never exact. think about the ratio of a circle circumference to its diameter, pi. Or as derived by integrating over a circle formula. 3.14159626…..

[@nikolastsikalakis4235]

its 117

[@geniegenie2453]

Very annoying pointless first 5 min, and still no clue what dx is at the end. Disappointed.

[@felixramey8939]

Of that particular math problem

[@hugoh9914]

Only calculus video I can understand

[@kingbee48185]

I hated algebra, it was so boring solving for X. I am finding calculus a nice challenge for my love of problem solving skills. The only issue I have had so far is solving differential equations. These guys break apart the equations but don't really explain the special rules they are using to isolate the numbers and assume beginners like me know what they are and how they apply them.

[@akhileshwarthakur9397]

Its answer is 117

[@leonardoamadorfatimagarcia35]

So ₁ ∫ 30 dx = 30 ⁰ I learned it without the video😂

[@Ananya_anoop]

125 - 8 = 117

[@kokayiufanifu8309]

Will music help your brain learn math!?

[@TKRM2007]

Too much talking

[@jerrymiller2367]

Not enough listening.

[@josephlaura7387]

117

[@janicehemi8983]

rambler

[@selahattinkara-o5h]

👏👏👏👏

[@scottbarber9374]

Twenty minutes to say: "This symbol means add up the area under the curve of this particular function, bound on the X axis by these two numbers."

[@ndailorw5079]

@ Scott Barber But to say that is to go to the other extreme and oversimplify things and the whole entire matter at hand here. What, for example, would you add up here?

[@scottbarber9374]

@@ndailorw5079 I said what it is that would be added up: The area under the curve.

[@ndailorw5079]

@@scottbarber9374 The guy is simply going from point a to point b, and necessarily so, to show how integral calculus was discovered and developed over many centuries from the ancient but endless process of adding rectangles under the curves of functions like the one we have here. He’s showing how the integral inherently uses that very same process but in an incomparably quicker and more powerful way to arrive at the same answer. In other words, much of the “rambling and babel,” though not all of it, I’ll agree, is more than necessary and obligatory if we’re to understand where we’re going and how to get there, and even why we get there. Besides, your last reply only begs again the question I asked in my first reply to you.

[@jimjohnston6848]

I enjoy the material, but there is far too much extraneous commentary. Perhaps simply getting to the point would be a better option...

[@robertstuart6645]

If I'm correct, the answer is 117.

[@johnkim9936]

disappointed. no conclusion for the question

[@davidcorbett9851]

Math good grammar spelling economists polititions religion cops judges lawyers colleges rich people have never fixed the world for most nor even half

[@jimsperlakis5634]

That took 10 minutes to say what should have taken 3. Just say what each symbol represents. Don't generalize.

[@brumman2623]

stop rambling on - get on with the math!

[@Opt-OutTheMovement]

You talk too much hurry up teach the lesson. Video starts at 5:20

[@jacobkeemink5268]

Lots of talk and very little explination

[@kenesu1281]

✨💖

[@tvsettv]

Too much talking about nothing

[@audreydaleski1067]

1st grade calc.

[@antonyhaase918]

5 mins of upselling before content is too much. Get to the learning and upsell in middle or end.

[@nancydelu4061]

Math is a language. I found that kids who spoke more than one language were better at math. Weird, huh?

[@ndailorw5079]

Perhaps because they understand and know that the alphabet and mechanics and the rules (the grammar of a language) of a foreign language must first be learned in order to learn and speak (do) that foreign language. And you’re absolutely right, the very same thing applies to mathematics, it’s a foreign language simply because it’s a symbolic language. Human beings are naturally used to speaking grammatically, and not symbolically. Hence the difficulty in learning math… we let the symbols scare us instead of learning what mean in order to use them properly, so we run from math… But It’s just a simple matter of learning the symbolic grammar, learning the language of mathematics and how to speak and use it in order to excel in it. And it’s all down hill from there and as easy as pie from that point on.

[@nancydelu4061]

@@ndailorw5079 yes, if one can conceive it one can speak it. And when you can say it and you have mastered writing and reading in language, you can start to even "read" symbols. What the hey, it works for parrots and chimps! You are spot on !

[@edwardprasad5942]

He's trying to explain the science of calculus..hence wordy

[@luiscecenas6681]

Dude first of all thankyou, second I dont have time loose, 6 minutes of blabber talk, do it like the khan acadamy sell ur course in 3 seconds after u given value

[@noelcc6606]

Bla bla bla, you talk a lot!

[@johndonaghy9226]

Way too much talking. This so called 'teacher' really likes to listen to himself. Shut up and get to the point... Then do your self promoting...

[@yoshiyukishibuya]

idiotically lengthy 16:56

[@alexdcruz3682]

Too long an introduction. Grt to the point please! Click!

[@ronaldjorgensen6839]

NEW EACH /ALL FORGOT NEED RELEARN EACH/ALL

[@markryan850]

Too long winded....get to your topic before my interest wanes

[@julioacosta5592]

THANKS

[@CarlWinter-oy8uf]

What does d x stand for ? You never answer your students questions ---and you call yourself a math teacher ? Shame on you !

[@FrenchmansFlats51]

not a good calculus class

[@avistmasa]

Talk too much nonsenses to make confusion!

[@arnoldvillastique7828]

Corny

[@TomUlugia]

Talk to much

[@cannotsay5505]

WARNING: instructor talks too much about himself, skip ahead 5 mins

[@phyllismyre3040]

You talk too much off the subject! You should just get straight to work without to much BS!