Unit 5: Analytical Applications of Differentiation

Executive Summary

Unit 5 uses the derivative to understand functions themselves: where they rise and fall, where they peak, and how they bend — plus the big existence theorems.

Mean Value Theorem

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, some $c$ in $(a,b)$ has $f'(c) = \frac{f(b)-f(a)}{b-a}$ — the instantaneous rate equals the average rate somewhere.

Extrema

Critical points: where $f' = 0$ or $f'$ is undefined. On a closed interval, the Candidates Test evaluates $f$ at critical points and endpoints — the largest and smallest win. The first-derivative test classifies local extrema by sign changes; the second-derivative test uses $f''$'s sign at critical points.

Concavity and inflection

$f'' > 0$: concave up; $f'' < 0$: concave down. Inflection points occur where concavity changes. $f$, $f'$, and $f''$ graphs interlock: where $f$ increases, $f'$ is positive; where $f'$ increases, $f''$ is positive.

Optimization

Translate the constraint into one variable, differentiate, find critical points, and justify with a sign analysis or the second-derivative test.

Exam traps

MVT requires continuity AND differentiability — check the interval. Absolute extrema on an open interval may not exist. An inflection point needs a concavity CHANGE, not merely $f'' = 0$. In optimization, verify the endpoint and boundary values too.