Linearization and Differentials

Calculus 1 · Axiom Academy

LESSON Linearization and Differentials When a curve is hard to evaluate, replace it near a point with the one line that matches it perfectly — its tangent. 1. Up Close, Every Smooth Curve Looks Straight Take a smooth curve y=f(x) and a point on it. The tangent line there touches the curve and shares its slope, the derivative f'(a) . Zoom in near that point and something remarkable happens: the curve and the tangent line become nearly impossible to tell apart. That is the whole secret — near a , the line is a stand-in for the curve. 2. Building the Linearization, Piece by Piece The tangent line is just a line, so we can write its equation. It passes through with slope f'(a) . Point-slope form gives the linearization of f at a : Read it as two pieces. Start at the height f(a) ; then walk a horizontal distance (x-a) and let the slope carry you up the rise f'(a)(x-a) . Where you land is L(x) — the line's height, our approximation for f(x) . The known starting height — the one value we can evaluate easily. Slope times the horizontal step (x-a) — how far the line climbs or drops. By construction L(a)=f(a) and L'(a)=f'(a) : the line matches the function's value and its slope at a . No other line agrees with f on both, which is exactly why this one hugs the curve. Let's cash in the formula on the opening question. Use with the anchor a=9 , since is exact and 9.1 is close by. Read straight off the tangent line at x=9.1 . The curve's height — about 0.00005 below the line.

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