Help Needed - Taylor Series

Hi, I've been tasked to create a PID controller for a single tank system which will keep the water level at a desired level. I'd worked out a mathematical model for the system and my lecturer wrote the process for linearizing this model using Taylor series as shown below.

I have little idea as to how he obtained these values or what they state; my guess as of right now is that he took the current water height 'h' as the center point for the Taylor polynomial and the desired water height 'hd' as the first order derivative, thus approximating between those points.

If someone could explain what the process below means and how it was obtained, that would be amazingly appreciated.

https://i.redd.it/8rff6ndc4k221.png

The below picture also shows the process to which the mathematical model was obtained and a lil illustration of the single-tank system including the variables used.

https://i.redd.it/1ibx3iec4k221.png

Thanks for any help you can offer!