Determining intervals where a function is increasing/decreasing — AP Calculus AB
1. Core Definitions and Exam Context ★★☆☆☆ ⏱ 3 min
This foundational skill makes up 15-18% of the total AP Calculus AB exam weight, connecting the graphical shape of a function to the sign of its first derivative. It appears in MCQ (identifying correct intervals from functions or derivative graphs) and FRQ (justifying behavior for curve sketching and optimization problems).
Exam tip: Always justify your interval conclusions by referencing the sign of the first derivative on FRQ questions to earn full credit.
2. The First Derivative Rule for Monotonicity ★★☆☆☆ ⏱ 4 min
The relationship between derivative sign and function behavior is formalized by the Mean Value Theorem. For a function $f$ continuous on closed interval $[a,b]$ and differentiable on open interval $(a,b)$:
- If $f'(x) > 0$ for all $x \in (a,b)$, then $f$ is increasing on $[a,b]$
- If $f'(x) < 0$ for all $x \in (a,b)$, then $f$ is decreasing on $[a,b]$
- If $f'(x) = 0$ for all $x \in (a,b)$, then $f$ is constant on $[a,b]$
Intuitively, the derivative equals the slope of the tangent line at a point: positive slope means the function rises from left to right, and negative slope means it falls.
3. Partitioning the Domain with Critical Points and Domain Breaks ★★★☆☆ ⏱ 3 min
To correctly test the sign of $f'(x)$, you first split (partition) the domain of $f$ into intervals where $f'(x)$ is entirely positive or entirely negative. The derivative can only change sign at two types of points:
- **Critical points**: Points $x=c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ does not exist
- **Domain breaks**: Points $x=c$ not in the domain of $f$, where $f'(x)$ is also undefined
Derivatives satisfy the Intermediate Value Theorem, so they cannot change sign inside a partitioned interval, meaning you only need one test per interval. A common early mistake is forgetting to include domain breaks as partition points.
4. Sign Analysis of the First Derivative ★★★☆☆ ⏱ 4 min
Once you have your ordered partition points, you perform sign analysis to determine if $f'(x)$ is positive or negative in each interval, which tells you if $f$ is increasing or decreasing. There are two common, AP-accepted methods:
Both methods are fully acceptable on the AP exam, but the factor method is faster for the factored derivatives you will encounter on most exam questions.
Test-point Method
Pick any value inside the interval, plug it into $f'(x)$, and check the resulting sign of the output.
Factor Method
For a factored derivative, count the number of negative linear factors. An even number of negative factors gives a positive product, an odd number gives a negative product.
A key shortcut for the factor method: any linear factor raised to an even power is always non-negative (it can never be negative), so it does not affect the sign of $f'(x)$ and can be ignored during sign analysis.
Exam tip: If you use the test-point method, pick easy integer values for testing to avoid arithmetic errors.
Common Pitfalls
Why: Students confuse domain breaks with critical points in the domain, incorrectly extending intervals to include points not in $f$'s domain
Why: Students confuse the derivative at a single point with monotonicity over an interval
Why: Students forget that squaring removes the negative sign from any real number
Why: Students confuse zero derivative at an isolated point with zero derivative over an entire interval
Why: Students rush on exam questions and mix up which function determines increasing/decreasing behavior
Why: Students are used to combining same-sign intervals and forget that monotonicity requires the inequality to hold across the entire interval