Hi Ava,

Good question. Try breaking down the values week by week, starting with a small amount:

Week 0: Bank ($0), Income ($0), Raise ($0), Raise Raise ($1)
Week 1: Bank ($0), Income ($0), Raise ($1), Raise Raise ($1)
Week 2: Bank ($0), Income ($1), Raise ($2), Raise Raise ($1)
Week 3: Bank ($1), Income ($3), Raise ($3), RR ($1)
Week 4: Bank ($4), Income ($6), Raise ($4), RR ($1)
Week 5: Bank ($10), Income ($10), Raise ($5), RR ($1)
Week 6: Bank ($20), Income ($15), Raise ($6), RR ($1)
Week 7: Bank ($35), Income ($21), Raise ($7), RR ($1)
Week 8: Bank ($56), Income ($30), Raise ($8), RR ($1)

Here, the raise is growing the same amount each week ($1, $2, $3, etc.)
The income is growing to the 2nd power, and is roughly: 1/2 x^2.

The bank balance is growing to the 3rd power, and is roughly: 1/6 x^3

See https://instacalc.com/52176

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Can we keep going “down” (taking derivatives) beyond the raise?

Well, why not? If the raise is $100/week, if we take the difference again we see it drops to 0 (there is no “raise raise”, aka the raise is always steady). But, we can imagine the case where the raise itself is raising (week1 raise = 100, week2 raise = 200). Using our intuition: if the “raise raise” is constant, the raise is linear (something * n), the income is quadratic (something * n2) and the bank account is cubic (something * n3). And yes, it’s true!
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But how does this work? I’ve been trying for hours but I don’t understand. The only way I could think of for the quadratic one was:

(100 * n^2) – C = income.

But with this I’m not even sure which 100 I’m using (raise?).

Then there’s the one with the bankaccount and ^3 and I’m just completely lost. I’ve tried everything. Please explain!

Edit: Also, what is the “something” in something * n/^2/^3

Thanks Michael — yep, many textbook give you the how but not the why. Computers can crunch the equations, but we want to learn to read and interpret them ourselves. Hope you enjoy the site!

The problem is that nearly all my efforts to learn math have been a waste of time because no one seems to actually explain it to you. It’s the same reason I hated math in school, and it seems like no one is breaking it down in the way they do with other complex topics, designed to be understood without being practiced.

I don’t care that some equation or formula exists, and I don’t want to memorize it. I’m not a trained monkey who wants to know how to repeat things. I want to UNDERSTAND it. I want to know WHY we do things in a certain order. WHY these shorthand formulas have come to represent something that is obviously true.

I don’t actually care about using math to find out the answer to equations, I just want to understand math. How it works. Why it works.

When someone uses calculus to describe the transformation of a probabilistic wavefunction, I don’t need to be able to compute it, or to prove it, I just want to understand what it actually means.
When someone uses calculus to show the relationship between wages and employment, I don’t need to be able to see a numerical outcome, I just want to know roughly how an outcome might change when a variable is changed.

It seems to me it should be possible to get to this level without memorizing equations and spending years learning how to compute complex math. It’s just that no one seems to be able to separate math as a tool for computation from math as a representation of an underlying truth.

I’m looking forward to reading more of your content, because it seems you might understand this idea.