New version of chapter 1 for Essence of calculus

This video is great, and it much better illustrates that the derivative of the integral of f(x) is f(x), (that integration and differentiation are reverse functions).

Royal with Cheese

Hi Peter, always happy to receive feedback, no matter how nit-picky :)

3blue1brown

Great video! 2 tiny pieces of nitpicky feedback: 1. At 6:40, it's a bit strange that you are generalizing to any R, yet you still have specific numbers on the axes. 2. At 15:40, the position of the double arrow seems really off to the right, I think it would be better closer to the center. (I did warn you I was going to be nitpicky :D )

Péter Mernyei

Oh, and Thomas R Jackson: great point about gentle repetition. Magnasium: I agree. The philosophy is brilliant.

The Delectable Electro The CyberPuppy

Wow, that's a pretty intense introduction (and a great one). There are a lot of concepts covered here. - It might be nice to reassure people that it's okay if they don't follow everything completely, because everything is going to be reviewed in more detail in future episodes. Without that, all these concepts might be overwhelming and intimidating. - I think it's good that you chose an example (a circle's area) where students will already know the formula. When you derive the formula using integration, they will recognise that your 'slices' approach produced the right formula, and that will give them more confidence using 'slices' as a valid technique. - Personally I prefer the method of breaking the circle down into sectors, which can be approximated as triangles. But i can see all sorts of reasons why that's less effective in this context. - I love what you did with the fundamental theorem of calculus. You started with the "area" definition of integration, and defined differentiation in terms of finding the height of a slice of that area, only then moving onto the idea of differentiation as the slope of a graph. I've never seen it done that way, but I think it's great. This way, the connection between the two operations is baked into our understanding from the beginning.

The Delectable Electro The CyberPuppy

At 2:55, "trapezoid" is misspelled. Otherwise, great video! I'm so glad you decided to introduce the connection between differentiation and integration early in the series.

Sohan

Thanks very much for your videos, 3Blue1Brown. For some videos, we need to have some prerequsite math knowledge to fully understand it. Do you have a full list of math textbooks to recommend? Thanks.

Yes! Nice emphasis on the discovery-based approach right off the bat.

Kathryn Schmiedicke

This whole series restores a beauty to calculus that is lost in numerical analysis using a programed loop function. But on the other hand, digital calculations using simple addition/subtraction in a loop return to the basic concepts of small quantity manipulations applied many times. Newton/Leibniz meets Turning.

I love the new philosophical focus on being able to derive the maths on your own. That's exactly how I go about learning maths. I think demonstrating the potential of calculus via the area of a circle is nice for a scholar, and using a real-world example (like the distance covered by a car with a linear velocity function) would be better for someone not so academically involved. I don't know which type of audience you're catering for; I'm happy either way. @4.39. I don't think it was necessary to make the rectangles so tall that they overflowed the screen, and then you had to explain that you needed to rescale them to fit. I think it's fine and much simpler if you directly put them so that they would all fit.

Magnasium

Hi! I really love this video, however I don't see it as an introduction but as a video on the fundamental theorem of calculus that might belong a bit later as an intro to integrals. With that objective, it is better than the previous version, however in the previous version you did both derivatives and integrals and it seemed it got a wider view of what was to come. As a suggestion, maybe you could make a 2min video that gives a glimpse of what the series is about and then just dive into it.

Javier Almeida

There is so much to like about this introduction. One thing off the bat, I think you hit the pacing just right. You also included a lot of gentle repetition, which I think helps people grasp concepts while watching, rather than pausing to mull things over so much. You always have great graphics, but these seem to go a step higher in terms of visual clarity. I like how you're handling the language in this video as well. You introduce a bit of the jargon, but don't really dwell on it or get bogged down in formal definitions. Part of me wanted to shout out "it is dr because it is the distance of change", but there wasn't really a need at this point. I think your choice is better. Most of all, the video achieved what a good introduction should. You put forward the goals of the series, the things you are going to cover, and provided a good example of the tone and style of the series. I think it will be great for the targe audience. Good job!

This is a very exciting introduction! Very well done. To comment on other comments: I think "rectangleish" is sufficient considering the goal of the video, and that using the formula for the area of a trapezoid would be distracting. As for whether the formula for the area of a circle is sufficiently interesting: I think that it depends on the student, but it probably does the job. When I hadn't learned any calculus, I remember wondering about how area formulas came about and I'm pretty sure I would have been drawn into your series by this video.

Ebrahim Ebrahim

At the three minute mark, maybe instead of calling the figure rectangleish, maybe let the figure be a trapazoid with one base equal to 2*pi*r and another base equal to 2*pi*(r+dr) and then use the area formula for a trapazoid to show that for smaller and smaller choices of dr it becomes the formula for the area of a rectangle.

The formula for the area of a circle is something most people learn early in their mathematical education, so using such a roundabout way to compute it feels a bit underwhelming and pointless. Of course, later in the video we're given the justification that the technique that was developed is useful for other things, but until the viewer has faced a problem that requires it, it doesn't feel that powerful. I think focusing more on solving a full realistic example instead of introducing names like "derivative" and "integral" might make for a more enjoyable viewing experience for uninitiated minds. On the other hand, most of the audience won't be seeking these videos for fun, but rather because they need to learn the theory - in which case being given an intuition for those concepts right away is most helpful. EDIT: to put it another way, what I like most about the casual videos is that they generally propose a question or a problem, and then develop "new" mathematics as a means to model and solve it. This video seems to develop new math for the sake of having a more complete theory of math, which is alright for a scholar, but might bore a student.

André Mello

Beautiful.

Don Sanderson