#readwise # The Essence of Calculus  ## Metadata - Author: [[3Blue1Brown]] - Full Title: The Essence of Calculus - URL: https://www.youtube.com/watch?v=WUvTyaaNkzM ## Summary Calculus helps us understand shapes and areas, like the area of a circle. By breaking the circle into thin rings, we can approximate its area using simpler shapes. This process leads us to the formula for the area of a circle, which is $\pi r^2$. Calculus connects these approximations to bigger ideas like integrals and derivatives, which are important for solving many math problems. This is the first part of a bigger video series that includes these other videos as well: - [[Integration and The Fundamental Theorem of Calculus]] - [[What Does Area Have To Do With Slope]] ## Document Notes 1. How does the concept of approximating the area of a circle through infinitesimally small rings lead to a deeper understanding of integrals in calculus? 2. In what ways do the relationships between derivatives and integrals provide insight into solving complex mathematical problems, especially those involving areas under curves? 3. What philosophical or historical perspectives can be drawn from the challenges early mathematicians faced when conceptualizing and formalizing the principles of calculus? ## Highlights This is the first video in a series on the essence of calculus, and I'll be publishing the following videos once per day for the next 10 days. The goal here, as the name suggests, is to really get to the heart of the subject in one binge-watchable set. But with a topic that's as broad as calculus, there are many things that can mean, so here's what I have in mind specifically. Calculus has a lot of rules and formulas which are often presented as things to be memorized. There are lots of derivative formulas, the product rule, the chain rule, implicit differentiation, and the fact that integrals and derivatives are opposites, Taylor series, and just a lot of things like that. My goal is for you to come away feeling like you could have invented calculus yourself. That is, cover all those core ideas, but in a way that makes clear where they actually come from and what they really mean, using an all-around visual approach. ([View Highlight](https://read.readwise.io/read/01jfmqm5hxgcjk3w8r95x5ybkv)) - Note: Quote at the top: "The art of doing mathematics is finding that special case that contains all the germs of generality." –David Hilbert --- I want to show how you might stumble into the core ideas of calculus by thinking very deeply about one specific bit of geometry: the area of a circle. Maybe you know that this is $\pi r^2$, but why? Is there a nice way to think about where this formula comes from? Well, contemplating this problem and leaving yourself open to exploring the interesting thoughts that arise can actually lead you to a glimpse of three big ideas in calculus: integrals, derivatives, and the fact that they're opposites. ([View Highlight](https://read.readwise.io/read/01jfmqnmkc4dtwryfs29a2hx32)) ^2fl1p7 - Tags: [[👻 ai highlighted]] --- For smaller and smaller choices of $dr$, you might at first think that turns the problem into a monstrously large sum. I mean, there are many rectangles to consider, and the decimal precision of each one of their areas is going to be an absolute nightmare. But notice, all of their areas in aggregate just look like the area under a graph. And that portion under the graph is just a triangle, a triangle with a base of 3 and a height that's $2 \pi \times 3$. So its area, $\frac 1 2 bh$, works out to be exactly $\pi \times 3^2$. Or if the radius of our original circle was some other value, $R$, that area comes out to be $\pi \times R^2$. And that's the formula for the area of a circle. ([View Highlight](https://read.readwise.io/read/01jfmqv3t9nmv83sk2cj4jxcjy)) ^t6xmvd - Tags: [[👻 ai highlighted]] --- You had this problem that could be approximated with the sum of many small numbers, each of which looked like $2\pi r \times dr$, for values of $r$ ranging between 0 and 3. Remember, the small number $dr$ here represents our choice for the thickness of each ring, for example, 0.1. And there are two important things to note here. First of all, not only is $dr$ a factor in the quantities we're adding up, $2\pi r \times dr$, it also gives the spacing between the different values of $r$. And secondly, the smaller our choice for $dr$, the better the approximation. Adding all of those numbers could be seen in a different, pretty clever way as adding the areas of many thin rectangles sitting underneath a graph, the graph of the function $2\pi r$ in this case. Then, and this is key, by considering smaller and smaller choices for $dr$, Corresponding to better and better approximations of the original problem, the sum, thought of as the aggregate area of those rectangles, approaches the area under the graph. ([View Highlight](https://read.readwise.io/read/01jfmqq69hdtpjh2gbn5c061bx)) ^fjfdnj --- if you want to think like a mathematician here, you don't just care about finding the answer, you care about developing general problem-solving tools and techniques. ([View Highlight](https://read.readwise.io/read/01jfm2af4wkmyggzcd321z56ej)) - Tags: [[👻 ai highlighted]] --- A lot of other hard problems in math and science can be broken down and approximated as the sum of many small quantities ... This happens whenever the quantities you’re adding up, the ones whose sum approximates the original problem, can be thought of as the areas of many thin rectangles sitting side by side. If finer and finer approximations of the original problem correspond to thinner and thinner rectangles, then the original problem is equivalent to finding the area under some graph. ^5dpw3u --- Spend some time building up familiarity with the interplay between the function defining the graph, in this case $x^2$, and the function giving the area. In that playful spirit, if you're lucky, here's something you might notice. When you slightly increase $x$ by some tiny nudge $dx$, look at the resulting change in area, represented with this sliver I'm going to call $dA$ for a tiny difference in area. That sliver can be pretty well approximated with a rectangle, one whose height is $x^2$ and whose width is $dx$. And the smaller the size of that nudge $dx$, the more that sliver actually looks like a rectangle. This gives us an interesting way to think about how $A(x)$ is related to $x^2$. A change to the output of $A$, this little $dA$, is about equal to $x^2 \times dx$, where $x$ is whatever input you started at, and $dx$ the little nudge to the input that caused $A$ to change. Or rearranged, $\frac {dA} {dx}$, the ratio of a tiny change in $A$ to the tiny change in $x$ that caused it, is approximately whatever $x^2$ is at that point. And that’s an approximation that should get better and better for smaller and smaller choices of $dx$. In other words, we don’t know what $A(x)$ is; that remains a mystery. But we do know a property that this mystery function must have. ^6bzelb When you look at two nearby points, for example, 3 and 3.001, consider the change to the output of $A$ between those two points, the difference between the mystery function evaluated at 3.001 and 3. That change, divided by the difference in the input values, which in this case is 0.001, should be about equal to the value of $x^2$ for the starting input, in this case, 3.001. And this relationship between tiny changes to the mystery function and the values of $x^2$ itself is true at all inputs, not just 3. ([View Highlight](https://read.readwise.io/read/01jfmqzt7wdhkra7wkg9bvx5g8)) --- This back and forth between integrals and derivatives, where the derivative of a function for the area under a graph gives you back the function defining the graph itself, is called the fundamental theorem of calculus. It ties together the two big ideas of integrals and derivatives and shows how each one is an inverse of the other. ([View Highlight](https://read.readwise.io/read/01jfmr0kfyjn9ctajh7j61csk8)) ^lsfn5e ---