Linear Approximations

A linear approximation is simply using a straight line in place of a curve, because lines are easy to compute with and curves often aren't. All we're really saying is: if you know where you are and which way you're heading, you can predict where you'll be a short step later. The formula has three pieces: $f(a)$ (where you start, a value you know exactly), $f'(a)$ (the slope there, how fast $f$ is changing) and $(x - a)$ (how far you step away). Starting value plus slope times step: that's the whole thing.

Notice that this is just the tangent line from the derivatives unit, wearing a new name. The point-slope equation of the tangent at $a$ is $y = f(a) + f'(a)(x - a)$. The only new idea is using it to estimate values of $f$ you can't easily compute, like $\sqrt{4.1}$, from values you can, like $\sqrt4 = 2$.

How to use it. Pick a base point $a$ close to where you want a value, with $f(a)$ and $f'(a)$ easy to compute. Then $f(x) \approx f(a) + f'(a)(x - a)$.

Accuracy. The error grows with the distance $|x - a|$ and with how much the graph bends (the second derivative). It makes sense: a curve that barely bends stays close to its tangent for a long way.

Over or under? If $f$ is concave up near $a$, the tangent line lies below the graph and underestimates; concave down means it overestimates.

Famous linearizations at $0$: $\sin x \approx x$, $e^x \approx 1 + x$, $\ln(1 + x) \approx x$, $(1 + x)^k \approx 1 + kx$.

Physicists linearize constantly: $\sin\theta \approx \theta$ makes the pendulum solvable, and engineers linearize nonlinear circuits and control systems around an operating point to design them.