Calculus×Computing
ES

Exponential functions

Level FundamentalDifficulty ★★★★★Concept⌖ Open in the map

What is it?

axa^x, and above all exe^x: the only function (up to a factor) that is its own derivative. It models anything whose rate of change is proportional to its size — populations, interest, radioactive decay, and exponential algorithms.

Intuition

Adding a constant to the input multiplies the output by a constant: ex+h=exehe^{x+h} = e^x e^h. Doubling time is constant. That is why exponential growth always wins in the end: every nkn^k is eventually overtaken by 2n2^n.

Formulas

e=lim⁡n→∞(1+1n)n≈2.71828e = \lim_{n\to\infty}\left(1 + \frac1n\right)^n \approx 2.71828
ddxex=ex,ea+b=eaeb\frac{\dd}{\dd x} e^x = e^x, \qquad e^{a+b} = e^a e^b
y′=ky  ⟺  y(t)=y0ekty' = ky \iff y(t) = y_0 e^{kt}
growth proportional to size

Where it shows up in computing

  • Population and epidemic models★★★★★fundamentalPhysics and simulation

    Early epidemic growth and unconstrained populations are exponential, N(t)=N0ertN(t) = N_0 e^{rt}.

  • Algorithm analysis and complexity★★★★★frequentScientific computing and algorithms

    Brute-force search over nn bits takes 2n2^n steps: exponential time is the signature of intractability.

Where it shows up in AI

  • Activation functions★★★★★frequentAI and machine learning

    Sigmoid, softmax and tanh are all built from exe^x.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

ℒ AI and machine learning

⚙ Robotics and control

⚛ Physics and simulation

∿ Signals, media and vision

⇄ Optimization and systems

What depends on it

Exercises

1Applied

A dataset doubles every 18 months. How long until it is 100 times larger?

Solution

2t/1.5=100⇒t=1.5log⁡2100≈1.5⋅6.64≈102^{t/1.5} = 100 \Rightarrow t = 1.5 \log_2 100 \approx 1.5 \cdot 6.64 \approx 10 years.

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