Linear Approximations and Differentials: Estimate Nearby Values

A linear approximation replaces a differentiable function near a convenient center with its tangent line, L(x)=f(a)+f'(a)(x-a). This lesson builds the linearization for the cube-root function at a=8, estimates the cube root of 8.05, and shows why the same line becomes unreliable far from its center. 1

You will also distinguish the actual change Δy=f(x+Δx)-f(x) from the differential dy=f'(x)dx, then compare them in a complete quadratic example and use differentials to estimate measurement error for a sphere. The closing checklist emphasizes center choice, derivative work, approximation notation, and units that earn exam credit. 2