I'm 75 and had advanced math courses. I have been doing his popup quizzes for 2 months as an attempt to remain nimble of mind. He has math courses for all MATH. Why is MATH important? 1. CALCULATORS become a crutch but not always the answer. 2. Many schools offer "more interesting" courses in place of MATH. 3. We need to exercise are body and our MIND! 4. You gain self-esteem by being proficient in MATH. 5. Success in MATH yields success in LIFE. Keep up the good work.
I am a 79 year-old Ivy League Spanish major - (Cornell '67) - never took calculus, and found this video perfectly understandable. The derivative and the calculation of derivative finally make perfect sense to me. I now feel ready to learn more math from TabletClass!
I was so scared of math when I was young I didn't dare take higher math. So at 70, I realized that those fears were because I had a weak concept of Algebra 2. It still makes my head spin, but I appreciate computers and videos that I can pause and proceed at my own speed.
@@markellinghaus8901 LOL .. I started to study it "JB" and now is helping me at work being better at troubleshooting problems and organizing better the aproach to fix them. Good to hear about this community.
I am 68 y.o. with degree in mechanical engineering which means 4 semesters of Calculus in college I took almost 50 years ago. I stumbled upon this video and what I did when the problem was posited, I froze the video to see if I could solve this elementary calculus problem. I did and got the right answer...first derivative and setting the first derivative = 0 and solving for x. Calculus throughout my life has been something I thought about but rarely used. The gift of learning is the greatest gift of all and I am forever in debt to my brilliant college professors and colleagues I worked with throughout my career. My major in college used the highest level of math which was partial differential equations and even when very young it was most challenging. Thanks for channel. You are a great teacher.
In high school and college I was always intimidated by higher math, not because I didn't understand it but rather because I have severe test anxiety. It sucks. But I'm now 51 years old and have been a software engineer since 24. Math has been integral in my career. These videos make revisiting math fun and enjoyable. 🙂👍 I'm so glad my kids do not share my anxieties. Thank God.
I can appreciate this instructor's enthusiasm. That's one sign of a good teacher. But I believe that he misspoke toward the end of this video when says that the top of the parabola is (4,-7), when it is actually the other way around, (-4,7).
@@JakeRichardsong On watching the video again beginning at roughly 16:46 you’ll see that he quickly erases the -7 and rewrites a dashed line that represents the line y = 7, and not -7. And he also has his coordinates written as (-4, 7), which is correct, -4 = x, and 7 = y.
This is the way calculus should be presented--pique interest and understanding first, before getting into the application. I dropped my first calculus class in college because the prof started into a problem with a "well, let's see" approach when I didn't even know what he was doing.
I enjoyed this video. I was a math minor in college many years ago and have forgotten much about calculus. After being a pilot for my career, yes, there wouldn't be aviation without calculus. I think I'll go find my calculus book and play around with it. It's better than solitaire.
I accidentally came across your video .. and speaking as a former Calculus Professor with 40 years experience... this is one of the best videos on youtube introducing Calculus to 1st year college kids! Excellent Job!
I took calculus in university 54 years ago. I never got to use it after graduation. I can't believe how I have completely lost it after watching this video. Amazing how we deteriorate.
Thanks for the lesson. You mentioned the big picture in the beginning and I think that is so important for the student to comprehend what they are doing. We had a professor in engineering school who always advised when taking derivatives what exactly we are looking at and that is the rate of change if one variable with respect to another. He mentioned this over and over again to beat it through our heads what exactly we are looking at
I took calc all the way through differential equations when I was in college. According to K-12 standardized test scores I had great aptitude, but I except for 8th grade algebra when I had a gifted, fabulous teacher I always avoided math because I never really could make it fit together. I thought I was just stupid. Then in my first college math course I got very lucky again and found a teacher that changed my learning life. I had him for college math and algebra but then had to take my first calculus class without him. I dropped it. It was then that I realized a lot of my issues really did have to do with the teachers and the way they taught, and I wasn’t just stupid. But I had finally discovered that I really did love math, so I tried again. After I transferred to a different college I got another fabulous teacher and stuck with her from Calc 1 through “diffy q’s”. I had to work really hard... I took detailed, voluminous notes, and did all my homework three times, Lol...but I got straight A’s. I also became a successful tutor for a while. Many of my students told me I broke it down better than their teachers and their test scores improved dramatically. I considered going into teaching but decided against it, largely because of the curriculum for educating teachers. Comprehending math changed my life. It structured my thinking and even helped my writing ability. I was able to write things more clearly because math and specifically calculus had taught me how to pick a problem apart and put it back together to arrive at an orderly, effective solution. Calculus is bar none one of the great brain growing tools of all time. It’s been 20 years since I finished my DE class. This popped up on my TH-cam feed so I decided to give it a watch for a quick, fun refresher. And I gotta say, while I admire the spirit in which this guy is trying to help, if I was a neophyte, struggling student with no previous knowledge of calculus, this wouldn’t have made much sense to me at all. I would have zeroed right out of his Calc class in short order. He’s all over the map. It’s obvious that math and Calc is something that just always kind of made sense to him, and he doesn’t break it down in a way that’s conducive to the kind of structured, almost rote approach that many of us knuckle draggers need to really comprehend and internalize math. The majority of students are knuckle draggers when it comes to math. I can say that because I was one. One of the things I realized during my math journey in college is that I did not need to grasp the concept being taught immediately in order to be successful. Concepts are important, but it’s not always imperative that they be completely understood immediately in order to achieve success in the classroom. My Calc teacher would sometimes tell us that she was going to teach us the mechanics first and circle back around to the concepts, and also why she was doing it that way for this particular section. It worked! I learned that if I was given a formula and I could memorize it well enough to break a problem down well enough get the right answer for a test, the concept ALWAYS came to me eventually. Sometimes it was sooner, and sometimes it was later. A couple of times the aha moment actually happened during the test while I was working a problem. Lol. The point is that if I could memorize and learn the mechanics well enough to achieve outstanding success on the test, I ALWAYS internalized the concept behind the mechanics at some point. This guy doesn’t spend enough time on the mechanics, IMO. Rote learning has fallen out of favor, but there’s a huge gap in our education process and our education outcomes because of it. The insistence on complete conceptualization at the expense of the hard work of rote learning is where education in general but math education in particular has been failing our young people for over 40 years. Rote learning is a necessary prerequisite to success for all but a fortunate few. And most of us are never going to go on to be engineers or physicists, so we will never use algebra, trig, or calculus for anything practical in our lives. So if the arcane concepts of slope and area under a curve fade for us over time...that is to say, if the larger concepts fade...its not really any harm. But the restructuring of our brains incurred by learning how to pick a problem apart at the mechanical level and arrive at a correct solution, will never leave us. That kind of learning is crucial in life. It’s not popular in modern teaching theory and it’s not popular with students cuz the mechanics are hard work, but teachers especially math teachers who don’t ultimately grasp this concept are doing their students a grave disservice. Just sayin’ in case that helps. This vid gets an A for effort and heart, because it’s in the right place. I think this guy loves his students and he loves math. But it gets just a C for overall effectiveness, which is what really counts.
I think this is an interesting review. It also points to a lot of issues with math teachers. Like you, I have had the good, bad, and ugly when it comes to having a good teacher, especially math. To this point (for the sake of intelligent conversation), I find myself agreeing with you and simultaneously disagreeing with you (pertaining to me, specifically). In some facets, math was easy. In others, insanely difficult. And not because the math was difficult, but it was too rote; too meaningless; too arbitrary. For me, I needed to understand the concept before the mechanics. I needed to have a reason. There had to be a story. While I agree that a story cannot exist without letters (surely one can not write a story without knowing their letters), it provided a direction for my thinking. Otherwise, there was no use in understanding the letters because the story didn't have meaning anyway; interest was lost. It seems people that teach math are those that tend to be good at math. In other words, it came natural to them. For those types of people, it is difficult to understand how us knuckledraggers (love that word) are not getting the concepts. Consequently, they teach as they "see" things. And it's really like this in all academia but math is entirely a different language. It demands structure (which as you indicated, one of the very beneficial things about learning math). For example, I teach psychology. My first class gave me terrible reviews, lol. I had no clue why. As it turns out, I talked over everyone's head but did not have a clue until later talking to the students. Why? It is my passion; something that I understand and place together very easily. It was difficult for me (even having experienced this same issue from math teachers) to see that I was doing the same thing to those poor souls. In fact, psychology is one of the reasons I came to appreciate math (as odd as that may seem). I could see the patterns but was lost in the mechanics/language. Which is why I agree with you about the mechanics being important, but most people do not have the same success from rote memorization (statistically speaking). This is why understanding from your insane work ethic (which is awesome, btw) improved your ability so much; it clicked and you got it! Sadly, I was never taught either sufficiently. I learned more from youtube videos than I ever have from any classroom, though I have had some good teachers years ago. And, it wasn't required for me to learn deep maths, but I have always been the guy who learns things differently, often in reverse of everyone else. Anyway, I found your comment intriguing. Made me evaluate things....which is the predicate of mathematics, ironically. So thank you.
May I suggest that you make a correction to this video? You wrote dx/dy, instead of the proper dy/dx. I noticed it immediately but uncritically because that kind of mistake can happen to any of us in the flow of discussion. I kept expecting you to notice it yourself and correct it, but you didn't. As the discussion progressed, you erased it, then, to my amazement, you wrote it again and verbalized it. The repetition may really cement it into the minds of true beginners. I suppose the effect on them is likely to be trivial; they'll no doubt get it sorted out quickly enough if they continue to pursue the study of calculus.
Hi!! Thank you so much for this video. I have to take a calculus class for my BA major. I haven’t taken a math class in 5 years. I was scared but watching this video made me realize that it’s not so scary and I can do this! ❤
I’m in the 5th grade (like you probably depending on where you live) I’m also eleven. I want to be a mathematician and I’m trying to get ahead. So I’m watching math videos all day
8o years old and just now understand what the quadratic equation was for. If I had only been taught properly my life could have took a very different turn. This should be a good lesson for teachers "sit up & take notice". Thank you Mark, I intend to follow your teaching of the calculus course and hope I can remember enough algebra to get through,:-) Please don't take any notice of the 'downers' who like to complain about everything, bob
Just a year your junior. Have recently learned martrix and now starting on calculus. But I am yet to know when/where I shall apply them? I'm a simple farmer and have a mere O-Level maths,1967, but so much in love with maths. Greetings from Zanzibar
I never thought I would need any of this while in school, though in the recent years I have had to tackle many task which needs not only calculus though lots of other styles of math also. Almost everything in the 3D designing and game developing requires such knowledge, and same with sound engineering, though the cool thing about sound engineering mostly everything on the energy frequency spectrum is universal, so if your knowledgeable with radio waves then you will do well with microwaves.
Surprisingly game development is actually becoming far more straightforward. Writing engines and then shaders that take three dimensional calculations and then reverting them to two dimensional rendering was really difficult, woo matrices... don't get me wrong, all the power behind the new free to use stuff that does 99% of the backend stuff for you is awesome, but it's the sheer computational ability of modern gpus and cpus to forego alot of the necessity to save every bit(no pun intended) you could and take every shortcut you could that allows things to truly shine now. I laugh at some old engines I've written that can run at 200,000 fps uncapped now when they used to run at 80 nicely lol. Also have a big pile of burnt out components... don't write bad code kids.
"never thought I would need any of this while in school". This is the misinterpretation often heard. It is what you learn, it is the learning process you are learning at school. And off course this comes in handy if you want to be a doctor or a chemist or a financial expert, etc. It is all about learning, not if you need this in future.
I’m still trying to re-learn algebra, it’s been 23 years since I took a class. This lesson blew me out of the water, so I really need to learn algebra better to ‘get’ this. I’m really enjoying this lesson and I’m looking forward to learning more.
I had to take business calculus at the beginning of Covid and let me tell you no one knew how to deal with this. We spent two weeks not doing anything and then finally I think the school said just try and get them to pass a class anyway possible and my professor said since we are doing everything online now you get to use your notes you get to help each other and you get to pair up with people taking the final and I passed the class with a B. That was pretty rough.
It is amazing how much we forget over life. It you had asked me to solve this problem, I wouldn't have had a clue. In college, 45 years ago, I was the only "A" in Calculus II, which is not saying much since after Calculus I, there was only 14 of us left :) I am not really smart and I'm certainly no math wiz, but I had my algebra and trigonometry down tight from practice, practice, practice. Because of this, the math of the Calculus wasn't that hard for me. There were people in my class way more gifted in mathematics than I was. I got the "A," despite not being that smart, because I grasped the concepts presented and I understood the problem being asked. Many in my class didn't. That said, if properly prepared for Calculus, it isn't the math that is hard, it is the concepts (like rate of change of water draining from a cone) that boggles the mind.
I liked that he showed several ways of writing "the derivative of". Having different ways which mean the same thing can be a major problem to a new learner. It was for me a long time ago when the professor in a 2nd year calculus class used just a CAPITAL "D" to mean "the derivative".
From basic math conventions, y is always the dependent variable, and equals f(x), and x is the independent variable. This convention is observed to this day in most areas of math ranging from algebra, calculus, statistics, all the way through machine learning to deep learning. Accordingly, a better notation would be y = f(x), dy/dx rather than dx/dy. Especially these days with widespread use of AI/DL/ML, it would be good to train young minds right from the start to use the correct conventions. Also, while this is great, it seems like it could be much more concise. just my .000002
I doubt the teacher John knows that the entire engineering is based on dy/dx (gives deflection) and second derivative is d^2y/dx^2 (bending moment) etc.
Great introduction to the concept of a derivative. As an engineer, this is fairly basic for me but I wish folks wouldn't be so quick to adopt the "i;m lousy at math" mindset. For young folks, all that's needed is to adopt a stick with it attitude when learning new and different concepts. As to the presenter's example used, one need only imagine a baseball being thrown to home plate from left field. The throw is likely to follow an upward/downward trajectory. The function f(x) used simply to define the path the ball follows from the time it leaves the outfielder's hand to the time it is caught by the catcher. Each and every one of you should be able to visualize a rollercoaster following an up/down path. If it is going upward, it must stop before proceeding downward. In other words, play out the concept of the f(x) expression as if it represented the path a baseball or a rollercoaster follows during its travel. Get help if needed but do accept the fact that basic concepts such as those used in this video is essential to building bridges, designing power distribution systems, etc. They are not weird concepts used by geeks to make life unbearable for those who proclaim to be "lousy" in math. You are not but most likely you have not challenged yourself to learn the practical aspects hidden in the germaine looking equations.
Thank you so much!... the way you explain is beyond good, you have a talent for this!... I am taking advanced macroeconomics now, and I started watching this for fun, and I stayed because you are actually putting a picture on my head!
It would be helpful to demonstrate exactly what problem this exercise actually solves. As I understand things, if a cannonball shoots up in this kind of parabolic arc, then after the halfway point (where the slope is zero), it takes the same amount of time to descend as it took to ascend up to that halfway point. You could inject something like that into the video -- or perhaps predict how far away the cannonball will land. It's still only a vaguely practical aspect, but it would get students thinking about the applied usefulness of these problems.
I agree. Calculus teaching often seems to miss the "but why would I want to do this" aspect. For instance, at 4:55 this video says Integration is the way we can find area under a curve. But WHY would we want to find the area under a curve? What's the practicality of this for a school student? Sadly this isn't explained to the beginner here.
I absolutely agree, as a current highschool student in Ap classes not a single teacher actually explained what these are used for. So I'm having to relearn the subject basically so that I understand what's happening. I can solve numbers not irl things.
If you change the word "slope" for "rate of change", then you start to get into real world applications. For example. the rate of change of distance with respect to time is velocity. And the rate of change of velocity with respect to time is acceleration.
You could make math obtainable to anyone interested. If you developed a program that tutored a student while doing problems. The key would be to have the program recognize a mistake and tell the student. But most importantly tell the student what the mistake is and what concept he is missing. I’m a retired lawyer when I went to law school we had very basic computer computers. But we had one program for evidence. In it you would have an issue which you’d rule on if you were wrong it would tell you why. It was basic but powerful. You could really get the hang of it. I think a math teacher and a computer programmer could revolutionize teaching of math. At first I thought it would be good for calculus and it would but it would be great for any level math. Also video type games could be developed to give practical application too. I wish I knew a math teacher and a programmer who wanted to bring calculus to the world. Calculus is the big separator of intelligent people. Many intelligent people can’t get it. It’s a shame but a constant tutor available would make it doable for any above average intelligent person! 17:42
i had no mentors & teacher's when i was young to teach me basic Calculus & just like open the door for learning of Calculus because all of my relatives & ancestors never had this subject before but i wander why i got cousins who pass an engineering course & me i love to become an engineer of any fields like civil, mechanical, technology, marine environment or warfare technology but i was so dumb of learning this interesting subject, the teacher of this course explain very precise & if i cannot get the answer he opens another link to get in to my dumb mind, thank you teacher of this course.
When I was learning calculus in high school, the teacher demonstrated the power rule. It was simple, but it confused me because I am the type of person that likes to understand things. I went home that evening, and found the derivative by definition, AFTER which I started applying the power rule. I do not understand why I would tell people I know some calculus if I do not understand why it works!
Wish they taught math this way when I was in school. I actually enjoyed the puzzle solving. Now I learn to help my daughter. She starts high school this fall. I will be ready!!!
I think you did a good job. With just one minor slip. I was a part of a pre-cal cal team in high school. What I liked most was at this level everyone wants to learn because they need it for college.
Probably will be my last Calculus problem!!!!!! LOL!!! Thanks for the video, took Calculus many years ago and didn't do too bad at the time and was applied to our Electronics designs. It is kind of math if you don't use it, you loose it. Just like Algebra and Geometry - all the rules. Thanks for the video, take care
FWLIW: The frustration I've always had with 'higher' maths as someone who has no real interest or need but would at least like to understand: -Here are the concepts - ok, no problem. Please don't muddy things by restating half a dozen different ways, do it right just once. -Here's a bunch of different ways to write those concepts - ok, but just those required for now would be less confusing. -Now we'll do a bunch of unexplained manipulations, and there's the result - that might as well have been magic! It's been my experience that most maths teachers will happily define the concepts multiple ways on the misapprehension that is where their students are confused and more explanation must surely be better rather than more confusing. They then demonstrate all the ways in which the problem may be stated under the equal misapprehension that their students are as fascinated as themselves - nobody is fascinated by having the expectation that they are about to be made to feel stupid rubbed in their face. The 'teacher' will then inevitably go on to show the process like a conjurer performing a trick as if said process itself is the most obvious thing in the world, culminating in a result that will seem to be impenetrable magic to most of their students. The process is never explained, just presented. It is these ill explained nuts and bolts that link the start and finish where all the confusion is to be found. Very frustrating! Some advice for budding teachers, especially maths teachers who tend to be those who most desperately need such advice: -Define the concept(s) required as clearly and succinctly as possible, ONCE, and in only ONE way. Check CAREFULLY that EVERY student gets it and if not CAREFULLY enquire what it is they are struggling with and address those specific issues ONLY. -Once EVERYBODY is on board introduce any new tools required and ONLY those tools. If it's a maths concept or problem DO NOT introduce multiple notations or methods, stick with ONE and choose the most conceptually straightforward if there is one. -Make absolutely sure EVERYBODY understands the purpose those new tools, especially their meaning, before ANY further steps are taken. -Only now is the time to work through illustrations and examples. The following is where 90% of maths teachers loose 90% of their students. -DO NOT under ANY circumstances run through your examples as if they will be as obvious to your students as they are to you. Many will be instantly turned off as they see the widening gap between their lack of understanding and the impression their poor teacher is giving that it should be easy. Of those few still following along most will have an unnecessarily frustrating devil of a time keeping up. The very few who have no issues will fool you in to thinking all is well - they won't understand what is wrong, destined as they are to regrettably become the next generation of 'teachers'. -INSTEAD go through EVERY step and symbol in excruciating detail, being very sure that every assumption, implication, and procedure, is unpacked for the clear understanding of all. It is at THIS POINT that NEARLY EVERY maths TEACHER FAILS NEARLY EVERY STUDENT - take your time and do it properly! -The bells and whistles may be imparted later to those who care and have the ability to grasp them.
this and this and this again . I have no problem with the notation, explained over and over again. I can remember how to magically manipulate the equation. I have no idea of what I am doing because the fundamental concepts underlying the operation are missing.
Something else I found interesting: how should -x^2 be interpreted, from a PEMDAS perspective? I interpreted it the same as our esteemed host as -1*(x^2). But for grins I put the function in Excel to plot it, and to my surprise it interpreted -A2^2 such that it first negated the value in A2 (which is x) then took the square. So in Excel operation priority, negation comes before exponentials. Note that Negation is not a function in PEMDAS, so there is some interesting debate on where it fits.
In Dutch we call it differentiation and integration. I remember having learned integration before going to university ( but not in order to study mathematics : ) )
Derivative is rate of change. Rate of change is just another way of talking about a slope and to find the secant (average slope) of a function you take y2-y1/x2-x1. So, the slope ends up being some kind of y/x. Taking the derivative of a function finds the rate of change as well (like a slope) so Derivative is dy/dx.
It’s not even dy/dx - in this case it should be df(x)/dx. dx/dy is obviously meaningless in this case, as that would be the rate of change in x due to a change in y - yet there is no y in this equation!
Dude, two issues: 1. It's dy/dx, meaning the rate at which the values of y are changing with respect to x. To be sure, there's also dx/dy but it's a different issue all together. 2. By now, you should've noticed the problem and perhaps make another video.
This is a great opportunity of you show me something that great to understand the drop of a bult as in a 16 in gun the fact of the distance for there landing. Thank for this
Actually without explaining what the constant term is =0 is in my view a huge error and negates all of the video. I can't think of one maths teacher would not explain why it is equal to 0. It will lead to students being confused.
Constants (like your 9) can be neglected in a lot of factors because their overall impact is ....negligible. It's not that it has become zero, but on a grand/microscopic scale like the derivatives, they are simply ignored.
@@smanqele All you said is all so incorrect! A linear function, that is, a straight line function, has a constant slope, its slope doesn’t change throughout the entire length of its graphs, but it’s function values change throughout. But a linear function, that is, a straight line function whose function values are the same throughout the entire length of its graph and therefore without change in its function values is called a straight line, or linear ‘constant function.’ And as such, and treated separately and as a separate function, as done with all the other terms in the expression presented in this particular problem, the derivative of the ‘constant function’ y = 9 is dy/dx = 0. Since all of its function values are exactly the same, the slope of the function is 0. Therefore, the derivative of the 9 term in the expression is 0 simply and only because 9 represents the ‘constant function’ y = 9, which is a constant itself, is a number and therefore a ‘constant function’ having a slope of y2 - y1/x2 - x1 = 0 because all function values of y gives 9 - 9/ x2 - x1 = 0/x2 - x1 = 0! So, the derivative of a ‘constant function’ is 0 because all its function values are the same as its function value at any point x. The derivative of a ‘constant function’ is 0 because the function itself is not only a constant, that is, a number, but also because all of its function values are the same throughout the entire length of its graph and are just that number, just that constant! All horizontal lines have 0 slope because their function values, the value of the constant, the number, are the same throughout its graph. For instance, for the graph, that is, the function y = x, it’s derivative, y,’ or dy/dx is 1. So y = x has a slope of 1, it has a rate of change of 1 throughout its entire graph, length. But for the function, the equation y = 9, which is simply a horizontal line, that line being y = 9, y,’ or, dy/dx = 0 simply because a horizontal line has a slope of 0. That is, the function values of the function y = 9 never changes. It always gives, (y2 - y1)/(x2 - x1) = 0 because (9 - 9)/(x2 - x1) = 0 for all values of x!
not for Leibniz notation, the independent variable goes on the bottom and dependent variable goes on top. so dy / dx for a function of y = - x^2 - 8x - 9 is dy / dx = - 2x - 8. dy / dx is read as " the derivative of y with respect to x"
This video seems to me is very useful. But, according to this equation, I would like to know about -9 . I didn’t see you mentioning? Thank you for teaching me. Michael Mai
It was the rules in getting the derivatives of a function, -9 is a constant number and according to the rules on getting the derivatives all the constant numbers is equals to zero. Hope you understand it.
So can you offer more calculus problems which have a practical application to cell phones, airplanes etc. or other every day things and issues we would encounter and explain exactly how calculus would apply? I had a crazy application for the problem you just did though I'm a beginner so I could be wrong. Would a country designing an defensive missile system need to know when the slope of an incoming missile is 0 in order to shoot it down?
Well had a minor in mathematics up through differential calculus that is when it all made sense to me, but it was a struggle up to that point. If I had this understanding earlier would not have spent hours of homework trying to understanding the basics when I first started. Thou rusty now this was nice reminder. :) Well done!
I never took notes even through calculus. Never had a problem. Benefits of a nearly perfect auditory memory just listening in class meant I had an audio recording of it in my head.
Well I’m not going to lie… i‘m So ADD that the entire time you were explaining about the slope and integration I was trying to find the irrational roots of the function. So thank you for forcing me to do a refresher on the quadratic formula 😂
Good explanations. The movie Stand and Deliver is "the story of Jaime Escalante, a high school teacher who successfully inspired his dropout-prone students to learn calculus." I've always been curious if he just taught to a level his students could understand, like your doing here, had tricks or short cuts or just took the time needed to ensure each student understood the material. Any thoughts?
Sorry to go off topic, but my wife and I watched that movie on our Honeymoon in 1989, we were at a cabin in the mountains for a week and I rented us a bunch of movies. I still remember Stand & Deliver, and Gorillas in the Mist after 4 kids and 33 years.
In 1966 I was flunking math in the Knavey’s nuclear power school. When I asked for help the instructor told me, “Read the book!” The text book was terrible, it caused me more confusion than anything else. I flunked out out of nuke school. Which, as I learned later, you really had to be really, really weird to enjoy duty on submarines! I could have used a good instructor like yourself, however, I may have become really, really weird.
13:01. Now I am thoroughly confused. When you’ve got one to the second power, shouldn’t you be multiplying 1×1, which would equal one? You don’t multiply the power by the number? You multiply the number by itself, as many times as the power tells you to? Is this something different?
I completely understand your confusion, John. The rule in finding the derivative is that you multiply the exponent by the coefficient to find the new coefficient. Afterwards, you decrease the original exponent by one to find the new exponent. Here are some examples: 2x^3 becomes 6x^2, 3x^2 becomes 6x, 4x^5 becomes 20x^4, -4x^4 becomes -16x^3, 5x becomes 5, and 3x^-2 becomes -6x^-3. Hope this helps.
@kdjayo1 : He is using a standard formula. If y=x^n then dy/dx= nx^(n-1) This formula can be derived by methods that are known as ‘first principles’ and requires knowledge of the Binomial Theorem and a concept called ‘limits’. So frankly this is not the way to teach the ‘first calculus problem’ as he puts it. This is actually a problem on maxima and minima and comes much later in calculus, there are numerous other things you need to know before you get here. If you really want to understand Calculus, you must learn the first principles. Your fundamentals will be strong then and you can then master this subject.
when you did the tic marks for the 4 calculation where did you come up with that approach, i almost tied that to map reading. or did you tie geometry in there? its been years since I did calculus, interesting though. i could never some how get it in my head, especially algebra!
i’ve always been interested in advanced math like this but haven’t been able to learn in school. this is because of my own mental health issues aiding in depleting my grades, so i’m unable to learn anything new and am forced to redo simple classes until graduation. i likely won’t be able to attend any college after i graduate. the education system doesn’t seem to actually look out for students from a humanistic standpoint but rather as a way to get kids in and out so that they’ll be able to function as slaves to our economic system.
I'm 75 and had advanced math courses.
I have been doing his popup quizzes for 2 months as an attempt to remain nimble of mind.
He has math courses for all MATH. Why is MATH important?
1. CALCULATORS become a crutch but not always the answer.
2. Many schools offer "more interesting" courses in place of MATH.
3. We need to exercise are body and our MIND!
4. You gain self-esteem by being proficient in MATH.
5. Success in MATH yields success in LIFE.
Keep up the good work.
Not sure how I ended up watching this video, but I watched it all the way through. Wish I had a teacher like this for when I struggled with math.
I know why I'm here because I suck at math and I'm trying to listen . Lol 😆
This guy has such a one on one voice I also wish I had a teacher like him
Same.
@Ralph Reilly yeah haha I would humbly suggest this is not a very clear explanation. Much better way to explain.
@@cattnipp So where do I find your videos on the subject?
I am a 79 year-old Ivy League Spanish major - (Cornell '67) - never took calculus, and found this video perfectly understandable. The derivative and the calculation of derivative finally make perfect sense to me. I now feel ready to learn more math from TabletClass!
I was so scared of math when I was young I didn't dare take higher math. So at 70, I realized that those fears were because I had a weak concept of Algebra 2. It still makes my head spin, but I appreciate computers and videos that I can pause and proceed at my own speed.
Are you reviewing math at 70 ? If so.. it's great to know I am not along. LOL
Makes a few of us boomers reviewing this topic.
I'm 72 and decided to learn graph theory, for which I have to learn matrix analysis, for which I have to learn Algebra II and Calc. 😀
@@markellinghaus8901 LOL .. I started to study it "JB" and now is helping me at work being better at troubleshooting problems and organizing better the aproach to fix them. Good to hear about this community.
great to hear - you are not alone, math gets the wheels turning even at 60+ yrs
I am 68 y.o. with degree in mechanical engineering which means 4 semesters of Calculus in college I took almost 50 years ago. I stumbled upon this video and what I did when the problem was posited, I froze the video to see if I could solve this elementary calculus problem. I did and got the right answer...first derivative and setting the first derivative = 0 and solving for x.
Calculus throughout my life has been something I thought about but rarely used. The gift of learning is the greatest gift of all and I am forever in debt to my brilliant college professors and colleagues I worked with throughout my career. My major in college used the highest level of math which was partial differential equations and even when very young it was most challenging.
Thanks for channel. You are a great teacher.
You are my fathers age. He will turn 71 years old this year. He is a retired electrical engineer 👷♀️
right Diffy Eqs! killer!
you are not a lone geezer❤
Joe helped me a lot with algebra 2 and I got fairly good with it.
Warching
This video is just great. I am 70 yrs old and I am just reaching back to my high school math years for pure pleasure. Thank you.
My high school maths teacher didn’t explain calculus this easily and nicely. Excellent lesson. If you know algebra this stuff is a picnic.
In high school and college I was always intimidated by higher math, not because I didn't understand it but rather because I have severe test anxiety. It sucks. But I'm now 51 years old and have been a software engineer since 24. Math has been integral in my career. These videos make revisiting math fun and enjoyable. 🙂👍 I'm so glad my kids do not share my anxieties. Thank God.
(X-9) ( x+1)
What kind of software engineering do you do? I graduated with degrees in computer science and applied mathematics and I rarely used math in my career.
@@bjbell52 I was asking myself the same thing
@@bjbell52 he’s a liar
He is a liar because you are believing a false prophet. All men are liars.
I can appreciate this instructor's enthusiasm. That's one sign of a good teacher. But I believe that he misspoke toward the end of this video when says that the top of the parabola is (4,-7), when it is actually the other way around, (-4,7).
Yes, the 7 is not negative, that error confused me at first.
@@JakeRichardsong
On watching the video again beginning at roughly 16:46 you’ll see that he quickly erases the -7 and rewrites a dashed line that represents the line y = 7, and not -7. And he also has his coordinates written as (-4, 7), which is correct, -4 = x, and 7 = y.
I had a terrific Functions/Trig teacher like this when I went back to refresh... to continue with Cal 1 and Cal 2. Great memory, terrific teacher.
This is the way calculus should be presented--pique interest and understanding first, before getting into the application. I dropped my first calculus class in college because the prof started into a problem with a "well, let's see" approach when I didn't even know what he was doing.
I enjoyed this video. I was a math minor in college many years ago and have forgotten much about calculus. After being a pilot for my career, yes, there wouldn't be aviation without calculus. I think I'll go find my calculus book and play around with it. It's better than solitaire.
I wish all instructors were as clear and provided simple instructions. Great work!
I accidentally came across your video .. and speaking as a former Calculus Professor with 40 years experience... this is one of the best videos on youtube introducing Calculus to 1st year college kids! Excellent Job!
Oh? See my comment on the video. The calculation of the derivative is incomplete! An excellent job?
I took calculus in university 54 years ago. I never got to use it after graduation. I can't believe how I have completely lost it after watching this video. Amazing how we deteriorate.
Thanks for the lesson. You mentioned the big picture in the beginning and I think that is so important for the student to comprehend what they are doing. We had a professor in engineering school who always advised when taking derivatives what exactly we are looking at and that is the rate of change if one variable with respect to another. He mentioned this over and over again to beat it through our heads what exactly we are looking at
I agree, I’m learning all these calculations, solving problems, but don’t know how to apply the information into making real-world designs
You😅
I took calc all the way through differential equations when I was in college. According to K-12 standardized test scores I had great aptitude, but I except for 8th grade algebra when I had a gifted, fabulous teacher I always avoided math because I never really could make it fit together. I thought I was just stupid. Then in my first college math course I got very lucky again and found a teacher that changed my learning life. I had him for college math and algebra but then had to take my first calculus class without him. I dropped it. It was then that I realized a lot of my issues really did have to do with the teachers and the way they taught, and I wasn’t just stupid. But I had finally discovered that I really did love math, so I tried again.
After I transferred to a different college I got another fabulous teacher and stuck with her from Calc 1 through “diffy q’s”. I had to work really hard... I took detailed, voluminous notes, and did all my homework three times, Lol...but I got straight A’s. I also became a successful tutor for a while. Many of my students told me I broke it down better than their teachers and their test scores improved dramatically. I considered going into teaching but decided against it, largely because of the curriculum for educating teachers.
Comprehending math changed my life. It structured my thinking and even helped my writing ability. I was able to write things more clearly because math and specifically calculus had taught me how to pick a problem apart and put it back together to arrive at an orderly, effective solution. Calculus is bar none one of the great brain growing tools of all time.
It’s been 20 years since I finished my DE class. This popped up on my TH-cam feed so I decided to give it a watch for a quick, fun refresher. And I gotta say, while I admire the spirit in which this guy is trying to help, if I was a neophyte, struggling student with no previous knowledge of calculus, this wouldn’t have made much sense to me at all. I would have zeroed right out of his Calc class in short order. He’s all over the map. It’s obvious that math and Calc is something that just always kind of made sense to him, and he doesn’t break it down in a way that’s conducive to the kind of structured, almost rote approach that many of us knuckle draggers need to really comprehend and internalize math. The majority of students are knuckle draggers when it comes to math. I can say that because I was one.
One of the things I realized during my math journey in college is that I did not need to grasp the concept being taught immediately in order to be successful. Concepts are important, but it’s not always imperative that they be completely understood immediately in order to achieve success in the classroom. My Calc teacher would sometimes tell us that she was going to teach us the mechanics first and circle back around to the concepts, and also why she was doing it that way for this particular section. It worked! I learned that if I was given a formula and I could memorize it well enough to break a problem down well enough get the right answer for a test, the concept ALWAYS came to me eventually. Sometimes it was sooner, and sometimes it was later. A couple of times the aha moment actually happened during the test while I was working a problem. Lol. The point is that if I could memorize and learn the mechanics well enough to achieve outstanding success on the test, I ALWAYS internalized the concept behind the mechanics at some point.
This guy doesn’t spend enough time on the mechanics, IMO. Rote learning has fallen out of favor, but there’s a huge gap in our education process and our education outcomes because of it. The insistence on complete conceptualization at the expense of the hard work of rote learning is where education in general but math education in particular has been failing our young people for over 40 years. Rote learning is a necessary prerequisite to success for all but a fortunate few. And most of us are never going to go on to be engineers or physicists, so we will never use algebra, trig, or calculus for anything practical in our lives. So if the arcane concepts of slope and area under a curve fade for us over time...that is to say, if the larger concepts fade...its not really any harm. But the restructuring of our brains incurred by learning how to pick a problem apart at the mechanical level and arrive at a correct solution, will never leave us.
That kind of learning is crucial in life. It’s not popular in modern teaching theory and it’s not popular with students cuz the mechanics are hard work, but teachers especially math teachers who don’t ultimately grasp this concept are doing their students a grave disservice.
Just sayin’ in case that helps. This vid gets an A for effort and heart, because it’s in the right place. I think this guy loves his students and he loves math. But it gets just a C for overall effectiveness, which is what really counts.
I think this is an interesting review. It also points to a lot of issues with math teachers. Like you, I have had the good, bad, and ugly when it comes to having a good teacher, especially math.
To this point (for the sake of intelligent conversation), I find myself agreeing with you and simultaneously disagreeing with you (pertaining to me, specifically). In some facets, math was easy. In others, insanely difficult. And not because the math was difficult, but it was too rote; too meaningless; too arbitrary. For me, I needed to understand the concept before the mechanics. I needed to have a reason. There had to be a story. While I agree that a story cannot exist without letters (surely one can not write a story without knowing their letters), it provided a direction for my thinking. Otherwise, there was no use in understanding the letters because the story didn't have meaning anyway; interest was lost.
It seems people that teach math are those that tend to be good at math. In other words, it came natural to them. For those types of people, it is difficult to understand how us knuckledraggers (love that word) are not getting the concepts. Consequently, they teach as they "see" things. And it's really like this in all academia but math is entirely a different language. It demands structure (which as you indicated, one of the very beneficial things about learning math). For example, I teach psychology. My first class gave me terrible reviews, lol. I had no clue why. As it turns out, I talked over everyone's head but did not have a clue until later talking to the students. Why? It is my passion; something that I understand and place together very easily. It was difficult for me (even having experienced this same issue from math teachers) to see that I was doing the same thing to those poor souls. In fact, psychology is one of the reasons I came to appreciate math (as odd as that may seem). I could see the patterns but was lost in the mechanics/language.
Which is why I agree with you about the mechanics being important, but most people do not have the same success from rote memorization (statistically speaking). This is why understanding from your insane work ethic (which is awesome, btw) improved your ability so much; it clicked and you got it! Sadly, I was never taught either sufficiently. I learned more from youtube videos than I ever have from any classroom, though I have had some good teachers years ago. And, it wasn't required for me to learn deep maths, but I have always been the guy who learns things differently, often in reverse of everyone else.
Anyway, I found your comment intriguing. Made me evaluate things....which is the predicate of mathematics, ironically. So thank you.
I think of how many knuckle draggers could have used u as a teacher. I think u would have been one of the great ones.
May I suggest that you make a correction to this video? You wrote dx/dy, instead of the proper dy/dx. I noticed it immediately but uncritically because that kind of mistake can happen to any of us in the flow of discussion. I kept expecting you to notice it yourself and correct it, but you didn't. As the discussion progressed, you erased it, then, to my amazement, you wrote it again and verbalized it. The repetition may really cement it into the minds of true beginners. I suppose the effect on them is likely to be trivial; they'll no doubt get it sorted out quickly enough if they continue to pursue the study of calculus.
Hi!! Thank you so much for this video. I have to take a calculus class for my BA major. I haven’t taken a math class in 5 years. I was scared but watching this video made me realize that it’s not so scary and I can do this! ❤
I have been struggling with this since middle school. I'm 65 now and thanx to you I think I get it!
A highly motivational lesson to learn mathematics. Thank a lot
I'm an eleven year old seeking for more advanced blocks of mathematics that exceed my age level, and this video was a great introduction of calculus.
I’m in the 5th grade (like you probably depending on where you live) I’m also eleven. I want to be a mathematician and I’m trying to get ahead. So I’m watching math videos all day
@@alexgfan2011this is excellent guys! Keep up the amazing work 💪🏿
8o years old and just now understand what the quadratic equation was for. If I had only been taught properly my life could have took a very different turn. This should be a good lesson for teachers "sit up & take notice". Thank you Mark, I intend to follow your teaching of the calculus course and hope I can remember enough algebra to get through,:-) Please don't take any notice of the 'downers' who like to complain about everything, bob
Just a year your junior. Have recently learned martrix and now starting on calculus. But I am yet to know when/where I shall apply them? I'm a simple farmer and have a mere O-Level maths,1967, but so much in love with maths. Greetings from Zanzibar
I never thought I would need any of this while in school, though in the recent years I have had to tackle many task which needs not only calculus though lots of other styles of math also. Almost everything in the 3D designing and game developing requires such knowledge, and same with sound engineering, though the cool thing about sound engineering mostly everything on the energy frequency spectrum is universal, so if your knowledgeable with radio waves then you will do well with microwaves.
Surprisingly game development is actually becoming far more straightforward. Writing engines and then shaders that take three dimensional calculations and then reverting them to two dimensional rendering was really difficult, woo matrices... don't get me wrong, all the power behind the new free to use stuff that does 99% of the backend stuff for you is awesome, but it's the sheer computational ability of modern gpus and cpus to forego alot of the necessity to save every bit(no pun intended) you could and take every shortcut you could that allows things to truly shine now. I laugh at some old engines I've written that can run at 200,000 fps uncapped now when they used to run at 80 nicely lol. Also have a big pile of burnt out components... don't write bad code kids.
"never thought I would need any of this while in school". This is the misinterpretation often heard.
It is what you learn, it is the learning process you are learning at school. And off course this comes in handy if you want to be a doctor or a chemist or a financial expert, etc.
It is all about learning, not if you need this in future.
I’m still trying to re-learn algebra, it’s been 23 years since I took a class. This lesson blew me out of the water, so I really need to learn algebra better to ‘get’ this. I’m really enjoying this lesson and I’m looking forward to learning more.
I had to take business calculus at the beginning of Covid and let me tell you no one knew how to deal with this. We spent two weeks not doing anything and then finally I think the school said just try and get them to pass a class anyway possible and my professor said since we are doing everything online now you get to use your notes you get to help each other and you get to pair up with people taking the final and I passed the class with a B. That was pretty rough.
It is amazing how much we forget over life. It you had asked me to solve this problem, I wouldn't have had a clue. In college, 45 years ago, I was the only "A" in Calculus II, which is not saying much since after Calculus I, there was only 14 of us left :)
I am not really smart and I'm certainly no math wiz, but I had my algebra and trigonometry down tight from practice, practice, practice. Because of this, the math of the Calculus wasn't that hard for me. There were people in my class way more gifted in mathematics than I was. I got the "A," despite not being that smart, because I grasped the concepts presented and I understood the problem being asked. Many in my class didn't. That said, if properly prepared for Calculus, it isn't the math that is hard, it is the concepts (like rate of change of water draining from a cone) that boggles the mind.
I liked that he showed several ways of writing "the derivative of". Having different ways which mean the same thing can be a major problem to a new learner. It was for me a long time ago when the professor in a 2nd year calculus class used just a CAPITAL "D" to mean "the derivative".
From basic math conventions, y is always the dependent variable, and equals f(x), and x is the independent variable. This convention is observed to this day in most areas of math ranging from algebra, calculus, statistics, all the way through machine learning to deep learning. Accordingly, a better notation would be y = f(x), dy/dx rather than dx/dy. Especially these days with widespread use of AI/DL/ML, it would be good to train young minds right from the start to use the correct conventions. Also, while this is great, it seems like it could be much more concise. just my .000002
Good catch! I agree with you. the convention is like you stated above. y = f(x), dy/dx rather than dx/dy.
I think you meant to say that y is always the dependent variable, not the independent variable as you wrote in your comment.
I doubt the teacher John knows that the entire engineering is based on dy/dx (gives deflection) and second derivative is d^2y/dx^2 (bending moment) etc.
Great introduction to the concept of a derivative. As an engineer, this is fairly basic for me but I wish folks wouldn't be so quick to adopt the "i;m lousy at math" mindset. For young folks, all that's needed is to adopt a stick with it attitude when learning new and different concepts. As to the presenter's example used, one need only imagine a baseball being thrown to home plate from left field. The throw is likely to follow an upward/downward trajectory. The function f(x) used simply to define the path the ball follows from the time it leaves the outfielder's hand to the time it is caught by the catcher. Each and every one of you should be able to visualize a rollercoaster following an up/down path. If it is going upward, it must stop before proceeding downward. In other words, play out the concept of the f(x) expression as if it represented the path a baseball or a rollercoaster follows during its travel. Get help if needed but do accept the fact that basic concepts such as those used in this video is essential to building bridges, designing power distribution systems, etc. They are not weird concepts used by geeks to make life unbearable for those who proclaim to be "lousy" in math. You are not but most likely you have not challenged yourself to learn the practical aspects hidden in the germaine looking equations.
I'm better at English, and I humbly say you spelled "germaine" wrong. It's "germane." 🙂
While other calculus videos aim to discourage us to take calculus, you encourage us.
This is better than my math teacher‘s explanation thank you so much
I use this math everyday :) Sometimes I forget fundamentals. Appreciate you!
Thank you so much!... the way you explain is beyond good, you have a talent for this!... I am taking advanced macroeconomics now, and I started watching this for fun, and I stayed because you are actually putting a picture on my head!
Sir, you simply have induced the love for calculus in me..thanks a lot.
Liked it
It would be helpful to demonstrate exactly what problem this exercise actually solves. As I understand things, if a cannonball shoots up in this kind of parabolic arc, then after the halfway point (where the slope is zero), it takes the same amount of time to descend as it took to ascend up to that halfway point. You could inject something like that into the video -- or perhaps predict how far away the cannonball will land. It's still only a vaguely practical aspect, but it would get students thinking about the applied usefulness of these problems.
I agree. Calculus teaching often seems to miss the "but why would I want to do this" aspect. For instance, at 4:55 this video says Integration is the way we can find area under a curve.
But WHY would we want to find the area under a curve? What's the practicality of this for a school student? Sadly this isn't explained to the beginner here.
I absolutely agree, as a current highschool student in Ap classes not a single teacher actually explained what these are used for. So I'm having to relearn the subject basically so that I understand what's happening. I can solve numbers not irl things.
If you change the word "slope" for "rate of change", then you start to get into real world applications.
For example. the rate of change of distance with respect to time is velocity. And the rate of change of velocity with respect to time is acceleration.
You are a great teacher. I can listen, watch you all day
You could make math obtainable to anyone interested. If you developed a program that tutored a student while doing problems. The key would be to have the program recognize a mistake and tell the student. But most importantly tell the student what the mistake is and what concept he is missing. I’m a retired lawyer when I went to law school we had very basic computer computers. But we had one program for evidence. In it you would have an issue which you’d rule on if you were wrong it would tell you why. It was basic but powerful. You could really get the hang of it. I think a math teacher and a computer programmer could revolutionize teaching of math. At first I thought it would be good for calculus and it would but it would be great for any level math. Also video type games could be developed to give practical application too.
I wish I knew a math teacher and a programmer who wanted to bring calculus to the world. Calculus is the big separator of intelligent people. Many intelligent people can’t get it. It’s a shame but a constant tutor available would make it doable for any above average intelligent person! 17:42
I Failed Calculus in 2010, your video came out 1 year ago, this would have helped so much
Using dx/dy is incorrect, it should be dy/dx
Yes huge mistake and very sloppy, it is hard to think that any maths teacher would make this error.
Yeah Change in Y / Change in X
@@zomisintu Correct. He has his change in Y confused with his change in X. He has to swap (invert) both to get "dY / dX."
However, dx / dy = 1 / ( dy / dx ) legitimately.
i had no mentors & teacher's when i was young to teach me basic Calculus & just like open the door for learning of Calculus because all of my relatives & ancestors never had this subject before but i wander why i got cousins who pass an engineering course & me i love to become an engineer of any fields like civil, mechanical, technology, marine environment or warfare technology but i was so dumb of learning this interesting subject, the teacher of this course explain very precise & if i cannot get the answer he opens another link to get in to my dumb mind, thank you teacher of this course.
You’re amazing, taking precalculus 1 right now. Kind of a struggle, your videos really help out. I wish I had you as my professor. Thank you!!
Three minutes in, you FINALLY get to the problem. WAAAAAY too much time spent promoting yourself, your site, and whatever else I skipped over.
This is so easy. I was taught the second part of that question in Singapore when I was only 10 years old. I still live and study in Singapore.
When I was learning calculus in high school, the teacher demonstrated the power rule. It was simple, but it confused me because I am the type of person that likes to understand things. I went home that evening, and found the derivative by definition, AFTER which I started applying the power rule. I do not understand why I would tell people I know some calculus if I do not understand why it works!
Young man, in mathematics you don't understand things. You just get used to them. - John von Neumann
Wish they taught math this way when I was in school. I actually enjoyed the puzzle solving. Now I learn to help my daughter. She starts high school this fall. I will be ready!!!
I think you did a good job. With just one minor slip. I was a part of a pre-cal cal team in high school. What I liked most was at this level everyone wants to learn because they need it for college.
Where is your calculus 1 and 2 courses on your website? Thanks!
Great explanation. Thank you so much
Probably will be my last Calculus problem!!!!!! LOL!!!
Thanks for the video, took Calculus many years ago and didn't do too bad at the time and was applied to our Electronics designs. It is kind of math if you don't use it, you loose it. Just like Algebra and Geometry - all the rules.
Thanks for the video, take care
a very instructive video in calculus. good job!!!
I'm very interested in learning more, you made it easy to understand wish you could teach my class. Thank you!
FWLIW: The frustration I've always had with 'higher' maths as someone who has no real interest or need but would at least like to understand:
-Here are the concepts - ok, no problem. Please don't muddy things by restating half a dozen different ways, do it right just once.
-Here's a bunch of different ways to write those concepts - ok, but just those required for now would be less confusing.
-Now we'll do a bunch of unexplained manipulations, and there's the result - that might as well have been magic!
It's been my experience that most maths teachers will happily define the concepts multiple ways on the misapprehension that is where their students are confused and more explanation must surely be better rather than more confusing.
They then demonstrate all the ways in which the problem may be stated under the equal misapprehension that their students are as fascinated as themselves - nobody is fascinated by having the expectation that they are about to be made to feel stupid rubbed in their face.
The 'teacher' will then inevitably go on to show the process like a conjurer performing a trick as if said process itself is the most obvious thing in the world, culminating in a result that will seem to be impenetrable magic to most of their students.
The process is never explained, just presented. It is these ill explained nuts and bolts that link the start and finish where all the confusion is to be found.
Very frustrating!
Some advice for budding teachers, especially maths teachers who tend to be those who most desperately need such advice:
-Define the concept(s) required as clearly and succinctly as possible, ONCE, and in only ONE way. Check CAREFULLY that EVERY student gets it and if not CAREFULLY enquire what it is they are struggling with and address those specific issues ONLY.
-Once EVERYBODY is on board introduce any new tools required and ONLY those tools. If it's a maths concept or problem DO NOT introduce multiple notations or methods, stick with ONE and choose the most conceptually straightforward if there is one.
-Make absolutely sure EVERYBODY understands the purpose those new tools, especially their meaning, before ANY further steps are taken.
-Only now is the time to work through illustrations and examples. The following is where 90% of maths teachers loose 90% of their students.
-DO NOT under ANY circumstances run through your examples as if they will be as obvious to your students as they are to you. Many will be instantly turned off as they see the widening gap between their lack of understanding and the impression their poor teacher is giving that it should be easy. Of those few still following along most will have an unnecessarily frustrating devil of a time keeping up. The very few who have no issues will fool you in to thinking all is well - they won't understand what is wrong, destined as they are to regrettably become the next generation of 'teachers'.
-INSTEAD go through EVERY step and symbol in excruciating detail, being very sure that every assumption, implication, and procedure, is unpacked for the clear understanding of all. It is at THIS POINT that NEARLY EVERY maths TEACHER FAILS NEARLY EVERY STUDENT - take your time and do it properly!
-The bells and whistles may be imparted later to those who care and have the ability to grasp them.
this and this and this again . I have no problem with the notation, explained over and over again. I can remember how to magically manipulate the equation. I have no idea of what I am doing because the fundamental concepts underlying the operation are missing.
Would be great to do a series going from introducing calculus like this to progressing to haf. Hard🏴🙋♂️
You must be a Catholic?
Something else I found interesting: how should -x^2 be interpreted, from a PEMDAS perspective? I interpreted it the same as our esteemed host as -1*(x^2). But for grins I put the function in Excel to plot it, and to my surprise it interpreted -A2^2 such that it first negated the value in A2 (which is x) then took the square. So in Excel operation priority, negation comes before exponentials. Note that Negation is not a function in PEMDAS, so there is some interesting debate on where it fits.
Never thought i'd follow through the whole video. He has a way of making you wanna stay....2 thumbs 👍🏽👍🏽⬆️⬆️
Graph the line -2x-8 and show that the vertex of -x^2-8x-9 occurs at the x-intercept of the the line.
In Dutch we call it differentiation and integration. I remember having learned integration before going to university ( but not in order to study mathematics : ) )
I am professional Engineer. I love it the way you explain. I started watching your show. I am going to forward to some of young students. Thanks
Derivative is rate of change. Rate of change is just another way of talking about a slope and to find the secant (average slope) of a function you take y2-y1/x2-x1. So, the slope ends up being some kind of y/x. Taking the derivative of a function finds the rate of change as well (like a slope) so Derivative is dy/dx.
In this video, you were saying “dx/dy.” Actually, it is “dy/dx.”
Exactly
@@acbulgin2 , yes I have noticed that. At first I thought it was just a human mistake...then I realized it is really lack of knowledge....so, I left
It’s not even dy/dx - in this case it should be df(x)/dx. dx/dy is obviously meaningless in this case, as that would be the rate of change in x due to a change in y - yet there is no y in this equation!
@@stevenproud6167 , that is even more correct
Dude, two issues:
1. It's dy/dx, meaning the rate at which the values of y are changing with respect to x. To be sure, there's also dx/dy but it's a different issue all together.
2. By now, you should've noticed the problem and perhaps make another video.
Awesome man. I wish we had all these TH-cam videos around when I went to college. Nevertheless you are a great teacher…👍👏👏👏🙏
Wow that helped taking classes now and they never said drop the last number for the derivative to make it easier.
Great video. Great refresher. Thank you.
Thank you Sir. I got it. Never really understood calculus at school. Awsome 😊
The algorithm suggested this video. It was really helpful. Thank you so much 😊
Very well done!
This is a great opportunity of you show me something that great to understand the drop of a bult as in a 16 in gun the fact of the distance for there landing.
Thank for this
Thank you thank you thank you, please continue to publish calculus videos.
Leibniz' commonly used notation for the derivative would be dy/dx, not dx/dy as described here....
AWESOME. !!! what a huge help. !! thank. YOU. so much. !!
Great review. Looking forward to doing more.
Clear explanation and example.of how it use after solved equation.thanks
I am trying again on TABE test and found you videos that hope I will learn from and pass next time
If you master algebra, not just be mediocre at it, you will do very well in pre calc and calc.
There’s a synergy between these skills. I became much better at algebra (and trig) while studying calculus.
Loved algebra but struggled a little with Matrices. Calculus seems more interesting
The explanation is great. It would have been exquisite if you could explain why the constant 9 gets eliminated!
Actually without explaining what the constant term is =0 is in my view a huge error and negates all of the video. I can't think of one maths teacher would not explain why it is equal to 0. It will lead to students being confused.
As he says in 18:42, it is not a full course - but it would have been nice ;)
Constants (like your 9) can be neglected in a lot of factors because their overall impact is ....negligible. It's not that it has become zero, but on a grand/microscopic scale like the derivatives, they are simply ignored.
Because all the constant does is raise or lower the parabola. It provides no information about the slope.
@@smanqele
All you said is all so incorrect! A linear function, that is, a straight line function, has a constant slope, its slope doesn’t change throughout the entire length of its graphs, but it’s function values change throughout. But a linear function, that is, a straight line function whose function values are the same throughout the entire length of its graph and therefore without change in its function values is called a straight line, or linear ‘constant function.’ And as such, and treated separately and as a separate function, as done with all the other terms in the expression presented in this particular problem, the derivative of the ‘constant function’ y = 9 is dy/dx = 0. Since all of its function values are exactly the same, the slope of the function is 0. Therefore, the derivative of the 9 term in the expression is 0 simply and only because 9 represents the ‘constant function’ y = 9, which is a constant itself, is a number and therefore a ‘constant function’ having a slope of y2 - y1/x2 - x1 = 0 because all function values of y gives 9 - 9/ x2 - x1 = 0/x2 - x1 = 0! So, the derivative of a ‘constant function’ is 0 because all its function values are the same as its function value at any point x. The derivative of a ‘constant function’ is 0 because the function itself is not only a constant, that is, a number, but also because all of its function values are the same throughout the entire length of its graph and are just that number, just that constant! All horizontal lines have 0 slope because their function values, the value of the constant, the number, are the same throughout its graph. For instance, for the graph, that is, the function y = x, it’s derivative, y,’ or dy/dx is 1. So y = x has a slope of 1, it has a rate of change of 1 throughout its entire graph, length. But for the function, the equation y = 9, which is simply a horizontal line, that line being y = 9, y,’ or, dy/dx = 0 simply because a horizontal line has a slope of 0. That is, the function values of the function y = 9 never changes. It always gives, (y2 - y1)/(x2 - x1) = 0 because (9 - 9)/(x2 - x1) = 0 for all values of x!
not for Leibniz notation, the independent variable goes on the bottom and dependent variable goes on top.
so dy / dx for a function of y = - x^2 - 8x - 9 is dy / dx = - 2x - 8.
dy / dx is read as " the derivative of y with respect to x"
you can remember this easily by remembering the formula for slope from beginning high school : m = delta y / delta x
Thank you for such a clear explanation
This video seems to me is very useful. But, according to this equation, I would like to know about -9 . I didn’t see you mentioning? Thank you for teaching me. Michael Mai
It was the rules in getting the derivatives of a function, -9 is a constant number and according to the rules on getting the derivatives all the constant numbers is equals to zero. Hope you understand it.
@@johnpaulveniegas5443 Technically, isn't it 9x^0 (which = 9), so you take the zero times the 9x and the whoe term becomes zero.
So can you offer more calculus problems which have a practical application to cell phones, airplanes etc. or other every day things and issues we would encounter and explain exactly how calculus would apply? I had a crazy application for the problem you just did though I'm a beginner so I could be wrong. Would a country designing an defensive missile system need to know when the slope of an incoming missile is 0 in order to shoot it down?
Nicely done. Advert punctuated my concentration right the point I needed to follow. Thanks for posting.
Will this be on the test?
Well had a minor in mathematics up through differential calculus that is when it all made sense to me, but it was a struggle up to that point. If I had this understanding earlier would not have spent hours of homework trying to understanding the basics when I first started. Thou rusty now this was nice reminder. :) Well done!
How do you know, hence, why is the slope formula the one that you picked? How did the formula get to be -2x and so on?
I never took notes even through calculus. Never had a problem. Benefits of a nearly perfect auditory memory just listening in class meant I had an audio recording of it in my head.
For those that are interested, SKIP to 12:30 of video. Nothing said before that!
all i can say is i enjoy your videos, Thank you, you are helping.
feels great to learn calculus .....great teacher.
Begins at 12:20
He starts every video the same. 10 plus minutes of telling you to take good notes and show your workings.
Well I’m not going to lie… i‘m So ADD that the entire time you were explaining about the slope and integration I was trying to find the irrational roots of the function. So thank you for forcing me to do a refresher on the quadratic formula 😂
The quality of this video is impressive.
Good explanations. The movie Stand and Deliver is "the story of Jaime Escalante, a high school teacher who successfully inspired his dropout-prone students to learn calculus." I've always been curious if he just taught to a level his students could understand, like your doing here, had tricks or short cuts or just took the time needed to ensure each student understood the material. Any thoughts?
Sorry to go off topic, but my wife and I watched that movie on our Honeymoon in 1989, we were at a cabin in the mountains for a week and I rented us a bunch of movies. I still remember Stand & Deliver, and Gorillas in the Mist after 4 kids and 33 years.
@@Thebigboramyou watched movies on your honeymoon? Maybe I’m crazy but that is nuts.
Could calculus cover irregular lines without any formulas defining it ??
In 1966 I was flunking math in the Knavey’s nuclear power school. When I asked for help the instructor told me, “Read the book!” The text book was terrible, it caused me more confusion than anything else. I flunked out out of nuke school. Which, as I learned later, you really had to be really, really weird to enjoy duty on submarines! I could have used a good instructor like yourself, however, I may have become really, really weird.
Most teachers suck. If a student could "just read the book", then there would be no need for teachers.
Excellent teacher!
I don’t understand where you got the derivative from, you made it up for a example?
13:01. Now I am thoroughly confused. When you’ve got one to the second power, shouldn’t you be multiplying 1×1, which would equal one? You don’t multiply the power by the number? You multiply the number by itself, as many times as the power tells you to? Is this something different?
I completely understand your confusion, John. The rule in finding the derivative is that you multiply the exponent by the coefficient to find the new coefficient. Afterwards, you decrease the original exponent by one to find the new exponent. Here are some examples: 2x^3 becomes 6x^2, 3x^2 becomes 6x, 4x^5 becomes 20x^4, -4x^4 becomes -16x^3, 5x becomes 5, and 3x^-2 becomes -6x^-3. Hope this helps.
@@Generatorman59
Thank you!
@kdjayo1 : He is using a standard formula. If y=x^n then dy/dx= nx^(n-1)
This formula can be derived by methods that are known as ‘first principles’ and requires knowledge of the Binomial Theorem and a concept called ‘limits’.
So frankly this is not the way to teach the ‘first calculus problem’ as he puts it. This is actually a problem on maxima and minima and comes much later in calculus, there are numerous other things you need to know before you get here.
If you really want to understand Calculus, you must learn the first principles. Your fundamentals will be strong then and you can then master this subject.
Better. How does anyone know the original equation for the curve?
when you did the tic marks for the 4 calculation where did you come up with that approach, i almost tied that to map reading. or did you tie geometry in there? its been years since I did calculus, interesting though. i could never some how get it in my head, especially algebra!
i’ve always been interested in advanced math like this but haven’t been able to learn in school. this is because of my own mental health issues aiding in depleting my grades, so i’m unable to learn anything new and am forced to redo simple classes until graduation. i likely won’t be able to attend any college after i graduate. the education system doesn’t seem to actually look out for students from a humanistic standpoint but rather as a way to get kids in and out so that they’ll be able to function as slaves to our economic system.