Calculus is the mathematics of approximations. At some point the approximations are ‘close enough’.................
That’s why designers, engineers and architects put tolerances on drawings/models.
I just came upon a drawing of Newton with the following verbiage overlaid:
“Think you’re bored? Sir Isaac Newton invented calculus during the plague. Do you have any idea of how bored you must be to invent calculus?”
Wish I had seen this years ago when I was teaching some grad math. ‘Twould have added some humor to “Calculus on Manifolds”.
“Calculus is the mathematics of approximations. “
Differential Calculus is the mathematics of approximations.
I was in my final semester of Calculus and still trying to figure out exactly why I needed to take it, when I was faced with a real-world problem at work that was similar to the equations we were studying at that time in Calc.
I was asked to solve the volume of a berm of garbage at the local landfill. I was essentially two different sized trapezoids separated by some length. I took it to my math prof, and he said he would solve it over the weekend. In the meantime I used the average end-area method.
On Monday evening at class the Prof said that was a quite the problem and it took him several hours and several sheets of paper to get the answer. When he showed it to me, I showed him my answer using the much simpler process. They were close enough that a few yards of garbage here or there were meaningless.