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Linear Approximation

Zoom in on any smooth curve and it flattens into a line — its tangent. That's local linearity, and it's the whole trick: to estimate a hard value, ride the easy straight line instead.
Near a point you know, a function barely differs from its tangent line. So replace the curve with that line — the linearization L(x) = f(a) + f′(a)(x−a) — and read off an estimate. It's exact at a and drifts as you move away; concavity tells you which way it misses.

Where the formula comes from

The linearization looks like a new formula to memorize. It isn't — it's a line you already know, used in a new way. Here's the whole idea, built from the ground up, one step at a time.

Ride the tangent

Now watch it live. Here's a curve and its tangent at a. Slide a to move the tangent point; slide zoom to close in. Watch two things: as you zoom in, the curve becomes the line (local linearity), and the estimate L(x) snaps toward the true f(x). The gap between them is the error — and its direction is set by which way the curve bends.

the curve f(x) tangent line L(x) the estimate at x
tangent point a
zoom in →

The linearization, part by part

Every linear approximation is this one formula. Read it as a set of directions: start where you know the value, then walk along the tangent's slope for however far you've stepped.

L(x) = f(a) + f′(a) · (x − a)
f(a) — the known height at the anchor point  ·  f′(a) — the tangent's slope there  ·  (x − a) — how far you step from the anchor. The step is (x − a), never just x — anchoring at a is the whole point, and the classic slip is writing f′(a)·x. Small step, good estimate; big step, the curve escapes the line.

The ideas everything else builds on

Local linearity

Zoom far enough into a smooth point and the curve is indistinguishable from a straight line — its tangent. That's what makes the estimate work at all.

The linearization

The tangent line written as a function: L(x) = f(a) + f′(a)(x−a). Use it in place of f near a. Exact at a, approximate nearby.

Over or under — read the bend

Concave up: the tangent lies below, so L under-estimates. Concave down: tangent above, L over-estimates. The sign of f″(a) decides.

Differentials

The same idea in change-language: dy = f′(x)\,dx estimates the output change Δy for a small input change dx — the tangent's rise over that step.

Your turn

Six problems — build a linearization, estimate with it, and call the over/under. Try each before revealing.

Quick quiz

Six fast checks. Pick an answer to see whether it's right and why — then work all six.

Question 1 of 6
Score: 0 / 0

Common mistakes & exam tips

Writing f′(a)·x instead of f′(a)(x−a)

The step is measured from the anchor. Forgetting the −a shifts the whole line and wrecks the estimate. Always L(x)=f(a)+f′(a)(x−a).

Estimating far from a

The tangent only hugs the curve near the anchor. Use the value of a closest to your target — √4.1 from a=4, not a=1 — or the error balloons.

Flipping over- and under-estimate

Concave up means the tangent is below the curve, so L under-estimates. It feels backwards — draw the bend and the tangent to be sure.

Calling the estimate exact

L(x) equals f(x) only at x = a. Everywhere else it's an approximation; say "approximately" and know there's an error.

Confusing dy with Δy

dy = f′(x)dx is the tangent-line change (the estimate); Δy = f(x+dx)−f(x) is the true change. dy approximates Δy for small dx.

Using the wrong sign of the slope

If f is decreasing at a, f′(a) < 0 and the linearization slopes down. Compute f′(a) with its sign and keep it.

On the AP exam

You can now trade any smooth curve for its tangent and estimate with it — the first taste of replacing a hard function with a simple one (Taylor series, in Unit 10, is this idea taken to the limit). Next: what to do when a limit refuses to resolve — continue to L’Hôpital’s Rule →, or head back to the Unit 4 Guide.