<s>[INST] On various angular velocities of a straight line $y$ = $\frac{a}{100}\times x$, with $a$ varying from $-1000$ to $+1000$. Let a straight line be defined by $y$ = $\frac{a}{100}\times x$. 
Let $a$ vary from $-1000$ to $+1000$ 
Desmos : https://www.desmos.com/calculator/xs2sowsaal
When $a$ is big ( in absolute value) important changes in $a$ produce moderate angular velocities. 
But when $a$ is small ( say, between $-100$ and $+100$ ) the angular velocity is much bigger. 
That seems astonishing ( at first sight, at least to me). 
Is this observaton correct? And how to analyze this apparent fact? 
 [/INST] Consider the circunference $\gamma:\:x^2+y^2=1$ and let $A(x_A,y_A)$ the intersection of $\gamma$ and $t$ where $t: \: y=\frac{a}{10^2}x$.
Now $(x_A,y_A)$ are the solution to: $$\left\{\begin{matrix}
x^2+y^2=1
\\ y=\frac{a}{10^2}x
\end{matrix}\right.$$
So, I have: $$\left\{\begin{matrix}
x=\sqrt{\frac{10^4}{10^4+a^2}}=10^2\sqrt{\frac{1}{10^4+a^2}}
\\ y=\sqrt{1-\frac{10^4}{10^4+a^2}}=a\sqrt{\frac{1}{10^4+a^2}}
\end{matrix}\right.$$
Now, let $\theta$ the angle formed by $t$ with the $x$-axis; we have:$$\theta=tan^{-1}\left(\frac{y_A}{x_A}\right)=tan^{-1}\left(\frac{a\sqrt{\frac{1}{10^4+a^2}}}{10^2\sqrt{\frac{1}{10^4+a^2}}}\right)=tan^{-1}\left(\frac{a}{10^2}\right)$$
In this graph I skecth the trend of $\theta$ with $m=\frac{a}{\sqrt{10}}$ (because with $a=\frac{a}{10^2}$ it's very difficult to see the trend but the conclusion it's essentially the same) and we have: 

From her, it's very easy to see that if $a$ is small $\theta$ grows very fast: Whenever $a$ is bigger the $tan^{-1}\left(\frac{a}{10^2}\right)$ grows not so fast.
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