NapkinCalc

Calculus 2B — Series & Taylor

Taylor series — polynomials impersonating functions

continues from lesson 2 — values defined earlier in the course stay live here

ELI5: near a point, any smooth function can be mimicked by a polynomial built from its derivatives. The first two terms of sin(x) are x − x³/6; it hugs the real sine until about |x| = 1.5, then drifts off.

x - x^3/6
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two-term Taylor approximation of sin(x)

✓ pass abs(0.5−0.536−sin(0.5))<0.001\mathrm{abs}\left(0.5 - \frac{0.5^{3}}{6} - \mathrm{sin}\left(0.5\right)\right) < 0.001 two terms already nail sin(0.5) to three decimals

Real-world hook: Taylor polynomials are how calculators and computers actually evaluate sin, cos, eˣ, and ln — it's polynomials all the way down.

Try it yourself: the two-term Taylor approximation of cos(x) is 1 − x²/2. Use it to estimate cos(0.3).

cosyou=1−0.322cos_{you} = 1 - \frac{0.3^{2}}{2} = 0.95500.9550 ✏️ Your turn: estimate cos(0.3) with 1 − x²/2 (x = 0.3). The check compares against the real cosine.
✓ pass abs(cosyou−cos(0.3))<0.01\mathrm{abs}\left(cos_{you} - \mathrm{cos}\left(0.3\right)\right) < 0.01 green when your estimate is close to cos(0.3)

Where next: Linear Algebra trades smooth curves for grids of numbers — and the machines that transform whole spaces.