Borbin the 🐱

The Altas 7B shaper is a rare metal shaper dating back 70+ years. Even more rare is the optional indexer.
And a perfect new project to replicate this rotary indexer for my Atlas shaper!

While browsing eBay for some suitable round table with t-nuts I could use to get started for the project, I found this:

An original S7-425 rotary indexer table! Arrived a few days later:

With the top of the indexer covered, I needed material for the base. Lots of 3/4" thick 8" round plate available at eBay:

But a few mm to large for the lathe. To get the jaws a good grip and to pass the bed of the lathe, I had to cut out the four sides to make it fit.

Cutting out the hole for the plate:

This setup is for the engraving. With a insert for cutting threads, the degree ticks were done. A simple wood block ensured a constant length of the tick marks. Once done, the 5 and 10 degree marks were done with a slightly larger depth.

First part of the base is finished.

The final one:

To evaluate the limit, we apply standard trigonometric identities to decompose the expression into terms involving sin⁡(x)/x​ for which a limit for x→0 exists.
While algebraic manipulation to eliminate the denominator is sometimes effective, it is not applicable in this case.
Instead, recognizing and leveraging this fundamental limit allows for a straightforward and elegant solution.

Here is the third one:

Option c) leads to the solution. The expression
f(x) + f(y) + x²y + xy²
is part of the expansion of the binomial
(x+y)³
If f(x) corresponds to the first or last term of the binomial expansion, the equation can be solved.

Here is the second one:

Start by moving the x term to the right-hand side and setting x=0. This is the first equation for b.
Then with b in place, get the second equation. Isolate a and substitute 0 for x. This gets

I've been helping out with high school math lately, and some of the questions are quite interesting.
Check this one out:

The key is to derive two equations from the derivative. With a fourth-degree polynomial, we get three points where the slope is the same, but only two of them share a common tangent line.
After that, everything falls into place.