The Speed of Money
In Problem A, calculus described a spacecraft: where it was (position), how fast it moved (velocity), whether it sped up or slowed down (acceleration).
Money moves too — your savings account, and the entire economy's money supply. Same three ideas: level, velocity, acceleration. Calculus is the language of change, and once you can read it, you can read it anywhere.
Part 1 — Meet the Federal Reserve
Before any interest rate math, it helps to know where interest rates come from. The Federal Reserve ("the Fed") is the central bank of the United States. Its dual mandate from Congress is to pursue stable prices (keep inflation low and predictable) and maximum employment (keep as many people working as the economy can sustain without overheating).
About eight times a year, a committee called the FOMC (Federal Open Market Committee) sets a target for the federal funds rate — the interest rate banks charge each other for overnight loans of reserves. That sounds narrow, but it's the foundation everything else is built on: when the Fed moves this one rate, banks adjust the rates they charge on mortgages, car loans, and credit cards, and the rates they pay you on savings accounts and CDs, all in roughly the same direction.
As of April 2026, the effective federal funds rate is
3.64% (data: FRED, series
FEDFUNDS).
Federal funds rate, 2015–present
Notice the near-zero stretches (2015–2018 recovery, the 2020 pandemic response) and the rapid climb in 2022–2023 as the Fed fought inflation.
That federal funds rate is essentially the real-world value of r in the interest formulas below. When the Fed moves it, every r in every loan and savings account in the economy shifts a little too — which is exactly why "what did the Fed just do?" is front-page news.
Part 2 — Algebra: Simple vs. Compound Interest
You deposit P = $1,000 at an annual interest rate of r = 5% for t = 10 years.
How much do you have after 10 years — with simple interest, vs. with interest compounded annually?
Simple interest only ever pays interest on the original principal — the amount you started with:
$$ A = P(1 + rt) $$
$$ A = 1000(1 + 0.05 \times 10) = 1000(1.5) = \$1{,}500 $$
Compound interest pays interest on your principal and on the interest you've already earned — interest earns interest:
$$ A = P\left(1 + \frac{r}{n}\right)^{nt} $$
$$ A = 1000\left(1 + \frac{0.05}{1}\right)^{1 \times 10} = 1000(1.05)^{10} \approx \$1{,}628.89 $$
Same rate, same time, but $128.89 more — and that gap grows every year. Compounding is why "start saving early" is such a big deal: time is doing real, exponential work for you.
Try it yourself
Compare simple and compound growth for any principal, rate, time, and compounding frequency.
$1,000 at 5% — simple vs. compound vs. continuous
Part 3 — Calculus: Continuous Compounding
Part 2's compound-interest formula has an n in it — the number of times per year interest is compounded. What happens if a bank compounded monthly? Daily? Every second? For the same $1,000 at 5% over 10 years:
| Compounding | n / year | After 10 years |
|---|---|---|
| Annually | 1 | $1,628.89 |
| Quarterly | 4 | $1,643.62 |
| Monthly | 12 | $1,647.01 |
| Daily | 365 | $1,648.66 |
| Hourly | 8,760 | $1,648.72 |
| Continuous | → ∞ | $1,648.72 |
The result keeps growing but levels off — it converges to a limit. That limit has a name:
$$ \lim_{n\to\infty} \left(1 + \frac{r}{n}\right)^{nt} = e^{rt} $$
That's where the famous constant e ≈ 2.71828 comes from — it's not arbitrary, it's what compounding approaches when you compound continuously. The continuous-compounding formula is:
$$ A = Pe^{rt} $$
Take the derivative of $A = Pe^{rt}$ with respect to time:
$$ \frac{dA}{dt} = \frac{d}{dt}\left(Pe^{rt}\right) = rPe^{rt} = rA $$
In words: the rate at which your money grows is proportional to how much you already have. This is the defining property of exponential growth — the same equation form describes population growth, radioactive decay (with a negative rate), and, as you'll see in Part 4, the money supply itself.
Running the formula backwards with the natural log
(ln) answers a different question — how long until
your money doubles?
$$ 2P = Pe^{rt} \;\Longrightarrow\; \ln 2 = rt \;\Longrightarrow\; t = \frac{\ln 2}{r} \approx \frac{0.693}{r} $$
At 5%, that's about 13.86 years to double — close to the quick mental-math "Rule of 72" estimate of 72÷5 ≈ 14.4 years.
Part 4 — The Velocity and Acceleration of Money
Now let's go back to the Problem A playbook. A spacecraft has a position (how far it's traveled), a velocity (the derivative of position — how fast that's changing), and an acceleration (the derivative of velocity — whether it's speeding up or slowing down).
An economy has something analogous: M2, a measure of the total money supply — cash, checking accounts, savings accounts, and similar "spendable" money, added up across the whole economy. M2 is a level, just like position. And just like position, we can ask: how fast is it changing (velocity), and is that change speeding up or slowing down (acceleration)?
M2 money supply, 2015–present (trillions of USD)
M2's "velocity" here is its annualized growth rate — roughly, the same kind of percentage-per-year number as an interest rate, but describing how fast the entire money supply is growing or shrinking month to month:
$$ \text{velocity}(t) = \left(\frac{M2(t)}{M2(t-1\text{mo})}\right)^{12} - 1 $$
Velocity of M2 — annualized growth rate (%)
And M2's "acceleration" is the month-to-month change in that growth rate — the second derivative, exactly like the spacecraft's acceleration was the second derivative of its position:
$$ \text{acceleration}(t) = \text{velocity}(t) - \text{velocity}(t-1\text{mo}) $$
Acceleration of M2 (percentage-point change in growth rate)
In spring 2020, M2 grew at an annualized rate of roughly 80% — by far the fastest pace on record — as emergency spending poured new money into the economy. That's a huge spike in velocity.
Look at the acceleration chart around mid-2021: it dips negative — the growth rate was still high, but it had started decelerating. Then look at the velocity chart around May 2022: that's when the growth rate itself finally crossed below zero — M2 was actually shrinking, something that hadn't happened in decades.
Notice the order: acceleration turned negative about a year before velocity did. That's the same relationship as a rocket that's still climbing but has started decelerating — the acceleration tells you something is changing before the velocity (let alone the altitude) makes it obvious. Economists watch indicators built this way — sometimes called a "credit impulse" — for exactly this reason: the second derivative can be a leading indicator.
Methodology note: M2 is smoothed with a 3-month moving average
before taking derivatives (real-world data is noisy — smoothing
before differentiating is standard practice, the same idea as
fitting a smooth curve through scattered data points). Data:
FRED, series M2SL and FEDFUNDS.