What is calculus about?

W Gilbert Strang, a really excellent lecturer who sees teaching as central to his purpose in life, has made his complete calculus course available for free here. He has also summarised the basics of calculus, using simple words and examples, boiling the subject down to 2.5 hours here.

Calculus is not all there is to post-secondary mathematics. But it does symbolise “difficult! complicated! advanced!” for a lot of people who didn’t devote ≥hundreds of hours of their adult lives to mathematics. If you felt like you never “got” maths in school, watching and understanding 5 half-hour videos about a “complicated! scary!” subject with Sir Isaac Newton’s name attached to it could give you a huge boost in self-confidence.

 

I’ll try to say what I think calculus is about in even fewer words.

  • Calculus is about zooming in on things.
  • In some sense the subject is infinity. You zoom in to an infinitesimal level.
  • It can feel downright spiritual.
  • When you zoom in on a smooth curve to an infinite degree, you find that, “under the microscope”, the curve is actually straight.
  • It’s like what would happen if you kept hitting “zoom” on a picture of Y=sin(X) on a graphing calculator.

     
  • Straight things are really easy to analyse mathematically. Like a straight-line projection is much simpler than a noisy, seasonal, highly variable projection.
     
  • The most surprising thing to come out of the rigorous study of infinitesimally zooming in is that area is the opposite of slope.
  • In some sense this finding is like multiplication is the opposite of division.
    • multiplication ↔ area ↔ ”integral” ∫ ↔ “tabulate” or “tally up” , and
    • division ↔ slope ↔ ”derivative” ∂ ↔ “pulse”
  • Saying that calculus is about derivatives ∂ and integrals ∫ is a little misleading. Saying that ∂ is about “slope” is not quite right either. Saying ∫ is about “area” is not quite right either! But just like the solar system model of the atom, I am lying to you at the start because the real story is too complicated.
  • Have fun!
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About isomorphismes

Argonaut: someone engaged in a dangerous but potentially rewarding adventure.
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