Owl: You need a quick estimate of ∛8.2, but you only "know" ∛8 = 2 by heart and you can find a slope at x=8. Zoom on the cube-root curve near 8 and it looks almost straight.
That straight stand-in is the tangent line: L(x) = f(a) + f'(a)(x − a). Nearby, f(x) ≈ L(x). Far away, the approximation can wander off badly.
Linearization is not exact — even "nearby" there's usually a little error. It's a controlled lie that gets better as you hug a.
What's the linearization of f(x)=∛x at a=8, and how would you use it for 8.2?