Calculus BC · CED Unit 5: Analytical Applications of Differentiation · 14 min read · Updated 2026-05-11
Determining Intervals Where a Function Is Increasing/Decreasing — AP Calculus BC
AP Calculus BC · CED Unit 5: Analytical Applications of Differentiation · 14 min read
1. Core Definitions and the Increasing/Decreasing Theorem★★☆☆☆⏱ 3 min
A function $f(x)$ is increasing on an interval $I$ if for any two points $x_1 < x_2$ in $I$, $f(x_1) < f(x_2)$. A function is decreasing on $I$ if $x_1 < x_2$ implies $f(x_1) > f(x_2)$. A function that is entirely increasing or decreasing on an interval is called *monotonic*, a term that may appear on the AP exam.
Exam tip: On AP FRQs, you must explicitly state that $f'(x) > 0$ (or $<0$) to justify your interval — you will lose points if you only give the interval without referencing the derivative sign.
2. Critical Points and Sign Chart Construction★★★☆☆⏱ 4 min
The sign of $f'(x)$ can only change at *critical points* and *domain breaks*. A critical point of $f$ is a point $x=c$ that is in the domain of the original function $f$ where $f'(c) = 0$ or $f'(c)$ is undefined. Domain breaks (points not in the domain of $f$ where $f'$ is undefined) do not count as critical points, but they still split the domain into separate intervals and must be included in your sign chart.
Find the domain of the original function $f$.
Compute and fully factor $f'(x)$.
List all critical points and domain breaks in order from left to right.
Split the domain into open intervals between consecutive ordered points.
Test the sign of $f'(x)$ in each interval, then assign increasing/decreasing based on the sign.
Exam tip: Always factor $f'(x)$ completely before building your sign chart. Factoring makes sign testing trivial by letting you evaluate the sign of each term separately, instead of recalculating the entire derivative for each test point.
3. Intervals for Parametric Functions★★★★☆⏱ 3 min
AP Calculus BC requires applying this method to parametric curves. For a parametric curve defined by $x(t)$ and $y(t)$, the slope of $y$ with respect to $x$ is $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. The same increasing/decreasing rule applies: $\frac{dy}{dx} > 0$ means the curve is increasing as a function of $x$, and $\frac{dy}{dx} < 0$ means it is decreasing.
If $x(t)$ is strictly monotonic (always increasing or decreasing), every interval of $t$ maps one-to-one to an interval of $x$, so you can convert your interval from $t$ to $x$ by substituting the endpoints of the $t$-interval into $x(t)$. If the question asks for intervals of $t$, you can leave your answer in terms of $t$.
Exam tip: Always read the question carefully to confirm whether it asks for intervals of $t$ or intervals of $x$. Forgetting to convert from $t$ to $x$ when required is a common mistake that costs points.
4. AP-Style Worked Practice Examples★★★☆☆⏱ 4 min
Common Pitfalls
Why: The definition of a critical point requires the point to be in the domain of the original function.
Why: Split points have $f'(x) = 0$ or undefined, so testing here gives no information about the sign of $f'$ on the interval.
Why: Critical points and domain breaks split the domain, and the derivative sign can change between non-adjacent intervals.
Why: AP exams require analytical justifications, not graphical intuition, for full credit.
Why: Any squared real term is non-negative, so it does not change the sign of the derivative.