“Inflection Point” is when the derivative is at an extreme, (equivalently, when the second-derivative crosses 0, or where the concavity/complexity of the graph inflects), not when the total is at an extreme
On the main theme, would a paycheck and banking analogy really be more interesting to kids than physical motion (as in sports and rocket ships)?
I remember when I was in high school, I openly mocked my mother for reading the Business section of the newspaper.
(Nowadays I resent the paper for printing the Sports section.)
@Michael: Great point – I’ll have to keep that in mind when writing about the binomial expansion :).
@Mike: Doh! Yes, my mistake – I removed reference.
I’m being a little facetious, but I think the idea of a bank account / income / raise is more approachable. Most physics examples I saw tended to be “Let’s track the trajectory of a ball” and inevitably require more new concepts to get going (What is force? Gravity? Acceleration? Velocity?). But fun may be in the eye of the beholder :).
The thing about physics is that it’s more appropriate for describing continuous variations. Money is a discrete process and thus a case of non-continuous variations…
But I agree it is a nice view to explain it, and maybe link it to the boring use made in physics (probably because it’s always presented in the same way…).
it’s true that a lot of people get bored by the direct physics application of calculus if its introduced too soon
money seems like a natural way to understand it
of course… it might not be able to so easily give an intuitive understanding of the more grainy aspects of calculus, but whatever, thanks!
@Johann: Thanks for dropping by, and great point about continuous vs. discrete. The funny thing is that many physicists treat the formulas as “discrete” (i.e. using infinitesimal dx, dy, dz quantities to make a 3d cube, for example) and then “let it disappear” to make it continuous again. The neat thing is that using discrete quantities really shows how the error margin is there (the difference between the actual sum of squares and 1/2 * x^2) and how limits / Riemann sum help us shrink this.
I agree though, that physics would be cool if it were shown to be an example of these general principles (and not the “definition” as is often seen).
@Prudhvi: Yep, there’s always details that you can’t get to when you make analogies. But you have to start somewhere :).
[…] post que explica los conceptos básicos del cálculo usando como metáfora los salarios y la cuenta del banco. Me hubiese gustado que me lo explicasen así. Las matemáticas pueden ser divertidas y entendidas […]
If someone had outright told me at any point in Calc I or Calc II that the ‘+ C’ can be thought of as an initial condition, I might have actually remembered to tack it to the end of integrals, instead of considering it an arbitrary annoyance that has little context.
That makes so much sense (and yet is so, so, so, painfully obvious), that it’s not even funny.
I love your belief that 5th grade is the time to introduce these concepts. These basic ideas are way too powerful to relegate to the few who ever make it into calculus. To go one further…why isn’t the concept of the circle and its rotation (sins and cosines) also taught at 5th grade the instant students learn the concept of the square root? (leading of course to that unique number pattern that gives you the values on the unit circle at 0, pi/6, pi/4, pi/3 and pi/2.
You are a genius, and an inspiration to me. your posts are of immense help in my CFA studies.
I can really relate with calculus and understand these financial concepts in depth (compared to just memorizing d formulae).
If possible, could you in one of your posts incorporate more examples connecting calculus (mainly concepts of slope, inflection point)and your favorite and most releveant subject- Money (maybe i could broaden the horizon here calling money as finance)
Thanks again…and looking forward to more insightful posts!
@Ramm: Thanks for suggestion and kind words! I’d love to do some follow-ups on calculus – I think the key is getting the basic metaphor in your head (raise/interest/bank account) and then seeing how specific ideas (inflection points, slope, etc.) can be understood. Appreciate the comment!
Thanks for your wonderful website! I am teaching myself maths, and it is so useful to have a resource which takes the intuitive approach. Ideas which are inscrutable when disguised in dry formulas and textbooks become clear with your explanations and analogies, and seeing how maths can be explained (i.e. in the way you do it) it amazes me that your style of explanation isn’t the norm. Thanks!