I have a question that's always been bothering me. In calculus we are taught that most equations, when examined closely, enough are locally linear. This means that (in the case of derivatives) we can take a very small section of the graph and find the slope similarly to how we would for a straight line. The same is true for integrals, where we can take a small enough subsection of a slope graph and treat it as if it has a constant slope.
My question is why are functions locally linear as opposed to something else? This also may be a more physics related question, but When Newton was founding calculus (like with the example of velocity) he operated on the idea that position could be found by multiplying velocity by change in time (assuming velocity was constant. My question is why would he arrive at that conclusion? Is there some deeper mathematical reasoning for this?
The very definition of differentiability of a function forces it to be well approximated locally by a linear function. I guess you've seen the definition that $f: \Bbb{R} \to \Bbb{R}$ is differentiable at a point $a \in \Bbb{R}$ if \begin{align} \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h} \end{align} exists, in which case, we denote the limit to be $f'(a)$. Now, define \begin{align} R(h) := f(a+h) - f(a) - f'(a)h; \end{align} this is called the "first order remainder term". Then, we have \begin{align} f(a+h) -f(a) &= f'(a)\cdot h + R(h), \end{align} and \begin{align} \lim\limits_{h \to 0}\dfrac{R(h)}{h} = 0. \tag{$*$} \end{align}
In other words, by the very definition of a function being differentiable, we can approximate nearby values of a function (i.e $f(a+h) - f(a)$), by a linear term (i.e $f'(a) \cdot h$) plus a "small" remainder term, $R(h)$ (small in the sense of $(*)$, which means the remainder $R(h)$ goes to $0$ faster than a linear polynomial).
Since the remainder $R(h)/h \to 0$, it means to first order in $h$, we can approximate $f$ as: \begin{align} f(a+h) -f(a)&\approx f'(a) \cdot h, \end{align} provided that $h$ is sufficiently small in magnitude. Of course, this approximation $\approx$ is only good if $h$ is sufficiently small, and it is a good approximation only to first order in $h$.
By the way, not all functions can be locally approximated by linear ones. Only a certain collection of functions can be approximated as such, and these are precisely the differentiable functions, BY DEFINITION. So, really, there's nothing deep going on here. You just notice that linear functions are nice to work with, so you make a definition (differentiability) to capture the idea of "local approximation by a linear function", and then you investigate properties of such functions. That's in essence the main idea of differential calculus.