Linearization, Differentials, and Over- or Under-estimates

The tangent line as a calculator

Near $x=a$ a differentiable function is well approximated by its tangent line:

Written with differentials, $dy=f'(x)\,dx$ estimates the change $\Delta y$ produced by a small change $dx$. This is the standard way AP asks about propagated measurement error: a measured radius with error $dr$ produces a volume error of about $\frac{dV}{dr}\,dr$.

Deciding whether the estimate is too big or too small

The sign of $f''$ settles it, and no other argument is accepted for full credit:

• $f''>0$ on the interval (concave up): the tangent line lies below the curve, so $L(x)too small.
• $f''<0$ (concave down): the tangent lies above, so the estimate is too large.

Choosing $a$ matters as much as the formula. It must be a point where $f(a)$ and $f'(a)$ are exactly computable, and it must be close to the target — the error of the approximation grows like $(x-a)^2$.