linear approximation
Stand close enough to any smooth curve and it looks straight — the same way the horizon looks flat even though the Earth is round. Linear approximation cashes in on that: near a point you understand well, you replace the messy curve with its tangent line and read values off the line instead. Want to estimate sqrt(4.1)? You already know sqrt(4) = 2 exactly, so you nudge along the tangent line a little and get a very good answer with no calculator.
The formula is the equation of the tangent line at x = a: f(x) ~ f(a) + f'(a)(x - a) for x near a. The known value f(a) is your anchor, and f'(a) is the slope telling you how fast the output changes as you step away. The same idea is written compactly with differentials: dy = f'(x) dx, where dx is a small change in the input and dy is the resulting straight-line estimate of the change in output, an approximation of the true change in y.
The honest caveat is that this is an estimate, best only near the anchor point. The further x drifts from a, the more the real curve peels away from the tangent line, and the error grows. How fast it grows depends on the curvature — the second derivative f''. For sqrt(4.1) the estimate is excellent; for sqrt(9) starting from a = 4 it would be hopeless. Linear approximation is the foundation of Newton's method, error analysis, and the first term of every Taylor series.
With f(x) = sqrt(x), a = 4: sqrt(4.1) ~ 2 + (1/4)(0.1) = 2.025, very close to the true 2.0248.
Linear approximation is just the first-order piece of the Taylor series; it ignores curvature, so the smaller the step (x - a) and the gentler the bend, the more trustworthy the estimate.