Approximating √26 Using Linearization

Calculus 1 · Axiom Academy

EXAMPLE Approximating Using Linearization Estimate a square root by hand with the tangent line to Use linearization (the tangent line approximation) to estimate by hand — no calculator. We will build the tangent line to at a nearby point and read off its value at x=26 . Nice work — you approximated a square root by hand using a tangent line. The pieces worth keeping: Linearization formula: — the tangent line at x=a . Pick the base point well: choose an a near your target where f(a) is exact and easy. Here a=25 gives for free. Closer is better: the estimate is most accurate when x is near a . Since 26 is one unit from 25 , the error is tiny. Result: , versus the true — an error under 0.001 . Why it overshoots: , so is concave down and sits just below its tangent — the line slightly overestimates. Linearization is the seed of differentials and of Newton's method — the same "replace the curve with its tangent line" idea, used over and over.

This is the written version of the interactive lesson above. See the full Calculus 1 course.