AP Calculus BC

Rates of Change in Different Forms

Average rate of change

The average rate of change of $f$ on $[a, b]$ is the slope between the endpoints:

$\frac{f(b) - f(a)}{b - a}$

For a linear function, this is the same on every interval, its slope. For any other function, it depends on which interval you pick.

Instantaneous rate of change

The instantaneous rate of change at $x = a$ is the limit of the average rate of change as the interval shrinks to a single point:

$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

For $f(x) = x^2$, this simplifies to $2a$: expand $(a+h)^2$, cancel the $h$ that's left in every term, and let $h \to 0$.

Reconstructing and combining functions

A function given by a table of values can be reconstructed if you know its type. A linear function has a constant rate of change between any two points. A quadratic function, written as $a(x - h)^2 + k$, has a vertex you can often spot from repeated or symmetric table values. Once you have a rule, you can evaluate it anywhere, find where it is 0, or combine it with another function.

Optimization with a vertex

A quadratic that models area, revenue, or height is maximized (or minimized) at its vertex, $x = -\dfrac{b}{2a}$. Some setups need an extra step first, like substituting a constraint to get the quadratic in the first place.