## Thinking “Infinitesimally”

1. SO I’d like to reiterate the intuitive picture one should have when working with calculus. We should think of a differential $\mathrm{d}x$ as a “really small” change in $x$…well, it’s the “smallest” possible change!

The reason I bring this up, well, just to reiterate the point when we consider $y=f(x)$ graphed in a neighborhood $x_{0}-\Delta x\leq x\leq x_{0}+\Delta x$, as $\Delta x$ gets smaller…the curve more closely resembles a line. This is the tangent line to the point $(x_{0},f(x_{0}))$.

Mixing notations, if $y=f(x)$, we obtain the differential quantity $\mathrm{d}y = f'(x)\,\mathrm{d}x$. The intuition should be: when I look at the “microscopic” scale, the curve “behaves linearly.”

2. Example. Lets consider the sine function. The linear approximation near $x=0$ would be $t(x)=x$. So if we plot the sine function in light blue, the approximation dashed in red, for values of $x$ between $-\pi/4\leq x\leq\pi/4$, we have the following graph:

Observe the linear approximation begins to differ from the actual function near the boundaries of the domain.

Look, we can even consider the area between these curves! We see this is the cumulative error, and it is
(1)$\displaystyle E = \int^{\pi/4}_{0}\bigl(\sin(x)-x\bigr)\,\mathrm{d}x = \left.-\cos(x)-\frac{x^{2}}{2}\right|^{\pi/4}_{0}$
Evaluating the limits gives us
(2)$\displaystyle E = \frac{-\sqrt{2}}{2}-\frac{\pi^{2}}{32}+1\approx -0.0155319187$
This is just for the values of $x\geq0$. We double this quantity to get the total error of our approximation, in the sense that this describes the area between the two curves.

3. If we “zoom in” more, considering the domain $-\pi/8\leq x\leq\pi/8$, then our graph becomes

The cumulative error in this case becomes
(3)$\displaystyle E = -\cos(\pi/8)-\frac{\pi^{2}}{128}+1\approx -0.000985817$
Observe our linear approximation now works better than before! Since the error is greatest at the boundary, we see
(4)$\displaystyle \sin(\pi/4)-\pi/4\approx 0.078291$ and $\displaystyle \sin(\pi/8)-\pi/8\approx 0.010015$
The punchline is: as we zoom in closer and closer to the curve, it appears to more and more resemble a line.

4. Microscopic versus Macroscopic. If we consider quantities $\mathrm{d}x$ as “microscopic”, then what’s a “macroscopic” quantity? It’s simply a finite number, i.e., one without “$\mathrm{d}$”s of any sort.