Linear approximation
Also known as: tangent line approximation, linearization
Linear approximation is a calculus technique that estimates a function's value near a point by using its tangent line. The formula is L(x) = f(a) + f′(a)(x − a), where a is a nearby point where the function is easy to evaluate.
Linear approximation, also called tangent line approximation, uses the fact that a smooth curve looks nearly straight when you zoom in close enough. Near a point x = a, a differentiable function f is well approximated by its tangent line at a, given by the formula L(x) = f(a) + f′(a)(x − a).
To use it, pick a value a close to your target x where both f(a) and f′(a) are easy to compute. To estimate √4.1, let f(x) = √x and a = 4. Then f(4) = 2 and f′(4) = 1/(2√4) = 0.25, so L(4.1) = 2 + 0.25(4.1 − 4) = 2.025. The true value is about 2.0248 — the tangent line gets remarkably close with almost no work.
The approximation is only trustworthy near the point of tangency, and its error direction depends on concavity. If the curve is concave down (f″ < 0), the tangent line sits above the curve, so the approximation is an overestimate; if concave up (f″ > 0), it's an underestimate. Since √x is concave down, our 2.025 slightly overestimates √4.1, exactly as expected.
On the AP Calculus AB exam, linear approximation is a standard contextual application of the derivative. Expect to build L(x) from a table or formula, use it to estimate a function value, and determine whether the estimate is an overestimate or underestimate using the second derivative.
Key takeaways
- Linear approximation estimates f(x) near x = a using the tangent line: L(x) = f(a) + f′(a)(x − a).
- Choose a point a near the target where f(a) and f′(a) are easy to compute.
- Example: √4.1 ≈ 2 + 0.25(0.1) = 2.025 using f(x) = √x at a = 4.
- Concave down means the tangent line overestimates; concave up means it underestimates.
- AP Calculus AB tests building the linearization, estimating values, and judging over- versus underestimates.