@Jon: Thank you for the kind words! Yes, I too wish more teachers gave intuitive explanations… I have a few ideas I’ll be announcing soon to make this easier ;).
[…] 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 […]
At the risk of sounding like an idiot, you lost me with the first chart. I’m not American, so maybe there’s a difference with terminology.
Would you mind clarifiying the following?
“With a quick subtraction, we can figure out his weekly paycheck. Turns out Joe is making a steady $100/week.” What’s the subtraction?
The first chart, “Bank Account by week”, what does that show? $0 for 5 weeks? He’s not earning anything, in the next chart, he’s earning $100 a week.
Sorry, but I just don’t get it.
@Stu: No worries! Having questions usually means the explanation wasn’t clear enough, and it’s a signal to improve :).
The very first chart is supposed to be a tongue-in-cheek joke that Joe doesn’t have any activity in his bank account (“Ack. Clearly, not much happened – Joe isn’t earning anything”). The next diagram is another scenario where he is actually earning $100 per week.
I should clarify this point in the article, I appreciate the feedback!
Amazing article Kalid, thanks! But I didn’t understand how linear increase in salary could represent a triangle, and how the area of the triangle would equal the sum of payments.
If there is more detail in the book on that, just let me know. I have the original PDF version.
@Sachin: Thanks for the comment! I’m not sure I understand the question, so let me know if this doesn’t clear things up:
The linear increase in salary ($100/week, $200/week, $300/week) can be shown as an increasing line on a graph (a “triangle” which goes 0, 100, 200, 300).
Now, the question is: what is the total amount of all these payments? It’s 100 (the first week) + 200 (the second week) + 300 (the third week) and so on. We are really breaking the triangle into chunks of 1 week each and adding it up.
Normally, this will result in a blocky “staircase” pattern. If we imagine our salary changing day by day, hour by hour, or second by second, then suddenly we have to be careful for how we calculate the total payment. We can graph the salary (second-by-second), and then compute the salary earned at each second. On the graph, this looks like the area underneath: a series of tiny rectangles where the width is “one second” and the height is "how much you earned in that second. Come to think of it, I should probably do an article on this :).
@Don: Thanks, glad you liked it.
interesting using a bank account as an example
Your arrticle is very explanatory and helps a lot but I still struggle about concept derivative at a point = slope of tangent. Any ideas?
@rdamle: Great question – I’d like to do a follow-up article on derivatives to really understand the nature of change. The high-level insight is that the slope of the tangent and the derivative are two ways to describe how much you are changing. However, “slope of tangent” isn’t a really intuitive way to see how things are changing on a graph. I’d like to do a follow up on this!
@Ed: Thanks for the note!
Yes, that’s a great point – because the weeks are so “chunky” the actual curve may not line up exactly: we are trying to model a discrete process (100, 300, 600…) with a smooth curve.
I’d definitely like to do a follow-up on how these approximations work and get closer – this is one of the hearts of calculus. I’m still thinking of really good analogies here… the closest I can come to is “eventually, if you make the pixels small enough, the jagged squares can look like a smooth image”.
Appreciate the support!
Kalid - Fantastic posting again !
With regard to the bank account accumulation from the salary(income) lodgements,
I wondered on the following points for some clarification -
" ■Linear increase in salary (100, 200, 300, 400) which leads to a…
■Quadratic (something * n^2) increase in bank account (100, 300, 600, 1000… you see it curve!) "
Is it more accurate that the bank account increase as derived from a granting of a salary uplift represented by a linear relationship ( 100*n( week no) ) the actual
curve is 50n^2 + C. Under those conditions the cumulative bank account balances at any given week would differ slightly to the above chart representation.
On the discrete/continuous debate, could you consider writing up some more Calculus
intuition that deals with the Error margin issue (the difference between the actual sum of squares and 1/2 * x^2 and Limits / Riemann sums) and some of your insights on
Love the work here!
Hey, just wanted to let you know that I translated this great calculus article to Russian.