From: 3blue1brown
This article introduces a visual approach to understanding fundamental calculus concepts, aiming to make complex ideas intuitive and accessible.
The Essence of Calculus: A Visual Journey
The goal of this series is to uncover the core ideas of calculus in a way that clarifies their origins and meanings through an all-around visual approach [01:01:00]. The intent is for the viewer to feel they “could have invented calculus themselves” [00:55:00], by pondering ideas and drawing diagrams, making the truths seem reasonable to stumble upon [01:22:00]. This emphasizes the creative approaches and the role of visual reasoning in mathematical discovery.
Unveiling Core Ideas Through Geometry: Area of a Circle
The journey into calculus begins by deeply exploring a specific geometric problem: the area of a circle [01:31:00]. Contemplating this problem visually can lead to a glimpse of three major calculus concepts: integrals, derivatives, and their inverse relationship [01:53:00]. This demonstrates how visualizing mathematical concepts can aid in understanding difficult math problems.
Slicing the Circle into Concentric Rings
To find the area of a circle, one might consider slicing it into many concentric rings [02:16:00]. This approach respects the circle’s symmetry, a principle often rewarded in mathematics [02:25:00].
Approximating Ring Area as a Rectangle
Each ring, with an inner radius r and a tiny thickness dr, can be imagined straightened out and approximated as a thin rectangle [02:46:00].
- The width of this rectangle is the circumference of the ring:
2πr[03:03:00]. - The thickness is
dr, representing a tiny difference in radius [03:20:00]. - The area of this approximate rectangle is
2πr * dr[03:32:00].
This approximation becomes increasingly accurate as dr gets smaller [03:42:00].
Summing Ring Areas Visually
The areas of all these approximate rings can be visualized as thin, upright rectangles placed side by side along an axis representing r [04:46:00].
- Each rectangle has a thickness
dr[04:50:00]. - The height of each rectangle at a given
ris2πr, representing the circumference of the corresponding ring [05:01:00].
This arrangement forms a graphical intuition under the graph of the function y = 2πr, which is a straight line with slope 2π [05:26:00].
From Approximation to Precision: The Integral Concept
As dr becomes smaller and smaller, the sum of the areas of these many rectangles approaches the exact area under the graph [05:51:00]. In the case of y = 2πr, the area under the graph (from r=0 to the circle’s radius R) forms a triangle [06:15:00].
The area of this triangle (½ * base * height) is:
½ * R * (2πR) = πR² [06:24:00].
This visual method precisely derives the formula for the area of a circle. This transition from an approximate sum to a precise area under a graph is a core concept in calculus, leading to the idea of an integral [07:07:00]. Many problems in math and science, such as calculating distance from velocity, can be approximated by summing small quantities and then reframed as finding the area under a graph [08:30:00]. An integral function, denoted a(x), gives the area under a curve between a fixed point and a variable point x [10:26:00].
Introduction to Derivatives: Measuring Sensitivity
While finding an integral directly can be difficult, examining how the area under a graph changes with a tiny nudge to the input reveals another key concept: derivatives [11:17:00].
If we consider the area under a graph y = x², and slightly increase x by dx, the resulting change in area (da) can be approximated by a thin rectangle [11:48:00].
- This rectangle’s height is
x²[12:05:00]. - Its width is
dx[12:05:00]. - So,
da ≈ x² * dx[12:22:00].
Rearranging this, the ratio da / dx (a tiny change in area divided by a tiny change in x) is approximately x² [12:34:00]. This approximation improves as dx gets smaller [12:48:00].
This ratio, da/dx, is called the derivative of a(x) [14:17:00]. Loosely, a derivative measures how sensitive a function’s output is to small changes in its input [14:32:00]. There are many ways to visualize a derivative depending on the function and how tiny nudges are considered [14:37:00].
The Fundamental Theorem of Calculus: Integrals and Derivatives as Inverses
The relationship observed between the area function a(x) and the original graph x² (where da/dx = x²) is fundamental. Once methods for computing derivatives are understood, one can reverse-engineer a function if its derivative is known [15:07:00].
This interconnectedness, where the derivative of a function representing the area under a graph gives back the function defining the graph itself, is known as the Fundamental Theorem of Calculus [15:20:00]. It reveals that integrals and derivatives are inverse operations [15:38:00]. This highlights the importance of spatial intuition in math and visual proofs in building a robust understanding of these concepts.