Analogy: The Calculus Camera

假设你是一名摄影师。你会遇到一个美丽的反光物体:一个金属球体，一个平静的湖泊，一个镜子房。看起来是个不错的拍照机会，对吧?

Sure, except your photos come out like this:

(An old-timey selfie.Source.)

The dilemma: You need a camera for the photo, but don't want the camerainthe photo. The instrument shouldn't appear inside the subject. (Hubert, you're leaving scalpels in the patient again.)

So, we need an isolated photo of a shiny object. What can we do?

• Shrink it down: Make the camera as small as possible. Microscopic, a fleck of dust. But even that speck will show up on the sphere. Take the photo, and fix the blemish with our best guess given the surrounding pixels.

• Go invisible: Make the camera unobservable to the subject: make it from perfectly transparent glass, or actively camouflage with your surroundings (like anoctopus). The camera is there, looking at the subject, but the subject cannot notice it.

In Calculus, a function like $f(x) = x^2$ is our subject. Limits (shrinking) and infinitesimals (invisibility) are how we take photos without our reflection getting in the way.

Taking a Photo of a Function

Consider a function like $f(x) = 2x + 3$. If I take a photo with my camera, I get:

• Before: $f(x) = 2x + 3$
• After: $f(x + \text{camera}) = 2(x + \text{camera}) + 3 = 2x + 3 + 2\text{camera}$

We have the original function, and put the camera into the scene. The result is the original function ($2x + 3$) and the camera observing "2". That is, the camera thinks "2" is how much the function has changed. (And yes, $\frac{d}{dx} 2x + 3 = 2$.)

Ok. Now take a function like $f(x) = x^2$. Again, let's put the camera into the scene to observe changes:

• Before: $f(x) = x^2$
• After: $f(x + \text{camera}) = (x + \text{camera})^2 = x^2 + 2x \cdot \text{camera} + \text{camera}^2$

人力资源管理。相机直接观察到一些变化($2x \cdot \text{camera}$)但是还有另一个$\text{camera}^2$ term:相机正在观察自己的反射!$\text{camera}^2$这个术语之所以存在是因为我们首先有一个相机。这是一个错觉。

What's the fix?

1. List all the changes the camera sees:
   
   

2. 弄清楚摄像头直接观察到了什么。We divide to see what was "attached" to the camera:
   
   

3. Remove "reflections" where the camera saw itself:
   
   

From a technical perspective, the last step happens by shrinking the camera to zero (limits) or letting the camera be invisible (infinitesimals).

整洁的,是吗?我们重新定义了寻找衍生品的过程:做出改变，看看直接的影响是什么，去除人为因素。“相机直接看到的东西”和相机的“反射”的概念帮助我下定决心扔掉那些出现在那里的术语。

In math, we have fancy terms likelinearandnon-linearfunctions. We can think in terms of "shiny" or "dull" functions.

Linear functions are dull because they only have terms like $x$ or constant values -- the camera can attach directly, and there's no reflection. Non-linear functions have self-interactions (like $x^2$) which means the camera has a chance to see itself. Reflections need to be removed.

With multiple subjects [$f(x)$, $g(x)$] or multiple cameras (for the x, y and z axis) we get cross terms like $df \cdot dg$ or $dx \cdot dy$. The goal is the same: remove unnecessary self- and cross-reflections from the final result. Show what the camera directly sees.

A few related blog posts on limits:

• An Intuitive Introduction To Limits

• Why Do We Need Limits and Infinitesimals?

• Building Intuition for the Derivative

The role of dx

Regular calculus books use $dx$ as the camera to detect change. The goal is to introduce a change ($dx$), then get the difference ($f(x + dx) - f(x)$).

This difference (for example, $2x \cdot dx + dx^2$) isolates the changes that $dx$ is directly responsible for ("sees"). We can then divide by $dx$ to get the change as a rate (how much we got out for how much we put in).

The concern is the same: the change $dx$ may have reflections ($dx^2$) that need to be removed.

Real-World Application: The Hawthorne Effect

TheHawthorne Effectis where people behave differently when being studied. The study itself is appearing in the results.

If you ask people to enter a study about their eating, exercise, reading, or sleeping habits, those behaviors will change. (Gotta look good for the camera! Where are those Greek philosophers I'd always meant to read?)

Math gives us a few suggestions:

• Shrink the effect:让研究尽可能不受干扰(就像iPhone被动监控你的脚步)。即便如此，也要弄清楚结果有多大程度的偏差，并为此进行调整。(你又把手机落在洗衣机上了，你这个狡猾的家伙。)

• Make the observations invisible:Imagine you don't know when the study is going to start. "Sometime in the next 20 years we'll silently observe your grocery shopping habits. Sign here." Hrm. You won't change your behavior for 20 years "just in case", so you'll just be you.

认为数学只适用于方程?哈。Only if we don't internalize the underlying concept.

Happy math.

Other Posts In This Series

1. A Gentle Introduction To Learning Calculus
2. Understanding Calculus With A Bank Account Metaphor
3. Prehistoric Calculus: Discovering Pi
4. A Calculus Analogy: Integrals as Multiplication
5. Calculus: Building Intuition for the Derivative
6. How To Understand Derivatives: The Product, Power & Chain Rules
7. How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms
8. An Intuitive Introduction To Limits
9. Intuition for Taylor Series (DNA Analogy)
10. Why Do We Need Limits and Infinitesimals?
11. Learning Calculus: Overcoming Our Artificial Need for Precision
12. A Friendly Chat About Whether 0.999... = 1
13. Analogy: The Calculus Camera
14. Abstraction Practice: Calculus Graphs
15. Quick Insight: Easier Arithmetic With Calculus
16. How to Add 1 through 100 using Calculus
17. Integral of Sin(x): Geometric Intuition