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a) In 10 years, I wish to remember: **Exponential functions are modelling reproducing process, and grows much faster than all Polynomials.**



b) I use **Analogy learning**, which means **finding connections of two different objects to explain certain phenomenon**. 

I connect the concept of **exponential functions** to **Polynomials** . On one hand,


$$
e^x=\lim_{n\rightarrow\infty}\left(1+\frac xn\right)^n.
$$
This explains how $e^x$ is like a polynomial of degree $\infty$. Since we have learned that **higher-degree polynomials grow faster than lower degree polynomials**. With **Analogy learning** that $e^x$ is a polynomial of degree $\infty$, I understand $e^x$ grows faster than any polynomials as $x\rightarrow \infty$.



c) I improve my habbits by **critical thinking**, specifically *I only belive my analogy when it could explain other phenomenons*. **For example, we know that the derivative of power function is proportional to a power function with degree less than it by 1,** for example
$$
\frac d{dx}x^{n}=
nx^{n-1}\propto x^{n-1}.
$$
To model reproduction process, we need a function with the derivative proportional to itself because a population grows proportionally to the number of individuals that are alive. So intuitively a polynomial of degre $\infty$ is used, which is not a polynomial but an exponential function $e^x$. By looking at some data (see below), the population of my hometown does grow faster than any polynomial.

![](https://i.imgur.com/HzDGapB.png)

`graph of population with a exponential line overlapping it and some polynomials`