Unit 4: Contextual Applications of Differentiation

Executive Summary

Unit 4 applies derivatives to the real world: interpreting rates in context, motion along a line, related rates, and linear approximations.

Interpreting derivatives

$f'(x)$ carries units of "output per input" — dollars per item, meters per second. $f'(a) > 0$ means the quantity is increasing at $x = a$; the sign and magnitude together tell the story.

Straight-line motion

Position $s(t)$, velocity $v(t) = s'(t)$, acceleration $a(t) = v'(t) = s''(t)$. Speed is $|v(t)|$. A particle changes direction when velocity changes sign; speed increases when $v$ and $a$ share a sign, decreases when they oppose.

Related rates

Differentiate an equation connecting quantities with respect to time, then substitute the given rates. Classic setups: expanding circles, balloon/sphere volumes, sliding ladders, draining cones, and shadows.

Linear approximation and L'Hôpital's Rule

The tangent line approximates: $f(x) \approx f(a) + f'(a)(x-a)$. L'Hôpital's Rule resolves $0/0$ and $\infty/\infty$: $\lim \frac{f}{g} = \lim \frac{f'}{g'}$ (apply repeatedly if needed).

Exam traps

Related rates: substitute constant values only AFTER differentiating. Speed is always nonnegative — do not confuse with velocity. L'Hôpital applies only to $0/0$ and $\infty/\infty$ forms — check first.