Unit 4: Contextual Applications of Differentiation

Topic 4.1 fixes how you read any derivative in context: $f'(x)$ carries the units of $f$ divided by the units of $x$. If $V(t)$ is volume in liters and $t$ is in minutes, then $V'(t)$ is measured in liters per minute. Topic 4.3 extends this to applied rates that are not motion, such as how fast a tank drains or a population grows. When a problem hands you a context, read the units first: they tell you whether you need the function itself, its rate of change, or an accumulation.

Topic 4.2 connects position, velocity, and acceleration for an object moving along a line. Each is the derivative of the one before it, and speed is the absolute value of velocity. The object speeds up when velocity and acceleration have the same sign and slows down when their signs differ.

Related rates is the marquee application of the unit and a frequent free-response setup. The signal to watch for: a problem gives one rate of change and asks for another, with the two quantities tied together by a geometric or algebraic relationship. The engine is the chain rule (Topic 4.4 builds related rates on it). Write the equation relating the quantities, differentiate every variable with respect to time using $\frac{d}{dt}$, then substitute the known values. The product and quotient rules often appear when you differentiate. The habit that saves points: substitute numbers only after differentiating, never before.

Topic 4.6 uses the tangent line as a stand-in for the function near the point of tangency, since a differentiable curve looks linear up close (local linearity). The linearization at $x = a$ is $L(x) = f(a) + f'(a)(x - a)$, and you evaluate it at a nearby input to approximate $f(x)$. Whether the estimate runs high or low follows from concavity: the tangent line sits below a function that is concave up (an underestimate) and above one that is concave down (an overestimate).

Topic 4.7 evaluates limits that direct substitution turns into an indeterminate form. When $\lim_{x \to a} \frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, L'Hospital's Rule says the limit equals $\lim_{x \to a} \frac{f'(x)}{g'(x)}$, differentiating the top and bottom separately. The recognition step is the whole game: confirm the form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$ first, because applying the rule to a limit that is not indeterminate can give a wrong answer. Remember that $\frac{0}{0}$ is a label for an indeterminate form, not a number, and the exam does not test other forms such as $\infty - \infty$.