AP Calculus AB and BC glossary

Linearization

Also called: Tangent line approximation, Local linear approximation

Linearization approximates a function near a point by using its tangent line there. Because a smooth curve and its tangent line are nearly identical close to the point of tangency, the line gives a good estimate for inputs near that point.

L(x) = f(a) + f'(a)(x - a)

The whole method is choosing a nearby point $a$ where $f(a)$ and $f'(a)$ are easy. To estimate $\sqrt{4.1}$ , take $a = 4$ , where the value is 2 and the derivative is $\frac{1}{4}$ , giving $2 + \frac{1}{4}(0.1) = 2.025$ .

Concavity tells you the direction of the error. If the graph is concave down the tangent line sits above the curve, so the estimate is too large; if concave up, the estimate is too small.

The mistake

Choosing $a$ far from the input you care about. Accuracy falls off quickly with distance, and an inconvenient $a$ defeats the purpose of the method.

Appears in: Unit 4: Contextual Applications