dial 'c' for calculus
As of late, it seems Calculus is everywhere in my life. The uptick began when I ran into the tacoid a few months ago. In my first semester of college, I really enjoyed and aced calculus for a variety of reasons:
 I solved extra problems, not just those assigned
 I went to numerous TA office hours( hi Dunnah McMurray!)
 Great environment/students! ( hi Sandip Roy! )
In this post I would like to focus on the Calculus Perspective
, and demonstrate what an expression of Calculus is capable of:

The problem landscape, , can be expressed clearly without artifacts, such as those that would arise if the sequence required an index.

share a sudokulike equivalence
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A profound example with succinct clarity.
 Maxwell’s equations represent the full interaction, curl(dissimilar) and dotproduct(similar), between electromagnetic fields. Magnetic field rotation, described by the righthand rule, require a change of sign(artifact) to distinguish R/L.
 Clifford algebra, allows builtin sign change, using a bivector for . The geometric product multiplies the wedge product terms for curl with dot product terms for divergence.
 Thus 4 equations are expressed as one, with signflipping builtin.
In the examples that follow, I will highlight clear expression, the changing variable nature of the integral, vanishing infitesimals, Fundamental Law of Calculus..
The Tacoid
 problem
 Model a tortilla as a pliable disk radius=1, center at origin, and against surface \( z=f(x) \)
 arclength
s
of  is continuous, even, positive, and increasing from
 Can the volume (convex hull) be maximized with respect to all such ?
 construct an integral
 Changing variables:
ds
;V
between two surfaces length
along xaxis is 2xwidth
along yaxis between two surfaces is \( \sqrt{1s^2} \)height
of zaxis is \( dz = \dot{z} \, ds \) \[ Volume = \int_1^1 2x \, \sqrt{1s^2} \, \dot{z} \, \ast \, ds \]
 Changing variables:
 substitution
 There is then a substitution to remove the z, use of Eulerlagrange, and numerical solver rk4…
The important aspects to consider are the changing arc length along the edge of the tacods
Volume of the region bounded by the taco arc length. Alternatively, consider filling the taco from the bottom to top. From this perspective, moving up the zaxis, the integral is written as \[ Volume = \int 2x \, \sqrt{1z^2} \, \ast \,dz \].
See the link for more details.
Nicomachus Theorem
This is a case where Calculus is clearer than either visualization or induction.
Nicomachus, who was the son of Archimedis, discovered \[ 1^3 + 2^3 +…+ n^3 = (1+2+..+n)^2 \].

In
trying
to solve by induction, neither substitution using the seriessum form \( (n)(n+1)/2 )\, nor expansion of the cubes … er lead anywhere. 
This is a visual proof. Beyond vertigo
 Expressing the behaviour we are trying to prove, using Calculus, relies on the limiting precision of the
dx
term.
 The dx plays the role of the iterator, allowing us to express the \( \sum n^3 \) without worrying about series form or where possible substitutions, such as the sum of the series of odds \( n^2 \), might take place.
 Next, taking the sum of values is simply the integral, which we solve on both sides to show equality
\[ \int n^3 dn = \left(\int n dn \right)^2 \] \[ \frac{n^4}{4} = \left(\frac{n^2}{2} \right)^2 \] \[ \frac{n^4}{4} = \frac{n^4}{4} \]
find notes and write up:  Maxlikelihood estimator of German Tank Problem, both unbiased and meansquare error forms.  The change in boosting weights as logs, and their relation to the exponential loss function  Sorting in trees, inplace, and the binomial combinatorics.  Optimization minimax