Taylor series

Here is the summit of the course, where derivatives and series meet. A Taylor series rebuilds an entire function — sin, eˣ, ln — as an infinite polynomial, using the function's derivatives at a single point.

Better and better polynomials

Start at a point (here x = 0) and build a polynomial that matches the function there, then also matches its slope, then its concavity, then the next derivative, and so on. Each new term pins down one more derivative — and the polynomial hugs the true curve over a wider range:

• T₁ = x matches sin x only right near 0.
• T₃ = x − x³/6 follows it noticeably further.
• T₅ = x − x³/6 + x⁵/120 tracks a full hump.

Add infinitely many terms and the polynomial becomes the function.

The formula

Centered at 0 (the Maclaurin case), the pattern is:

f(x) = f(0) + f′(0)x + f″(0)/2! · x² + f‴(0)/3! · x³ + …

The nth term uses the nth derivative at the center, scaled by n!. Each derivative forces the polynomial to agree with f in one more way.

Three worth knowing

Those factorial denominators grow so fast that the terms shrink quickly, which is why just a few terms already approximate the function well.

Why this matters

Taylor series are how calculators and computers actually evaluate sin, eˣ, and logs — there's no other way to get them but to sum a series. They also let hard functions be replaced by simple polynomials in physics and engineering. It's the fitting close to the course: the derivative (where we began) and the series (where we end) together reconstruct any smooth function you like.