On the AP Calculus AB and BC exams, an "increasing and decreasing functions" prompt asks a candidate to identify where a function rises, where it falls, and — most often — to justify that judgement with derivative sign. The question type carries its own dedicated language inside the scoring guidelines: words like increasing, decreasing, interval, open, and closed are not interchangeable, and a single misplaced endpoint can cost an entire rubric row. This article walks through what the AP Calculus reader is actually looking for, the first derivative test as a working tool, the difference between a sign analysis and a value claim, and the way BC-only items push the same skill into particle motion and related rates.
The rubric language behind "increasing" and "decreasing"
Before a candidate ever touches a derivative, the AP Calculus scoring guidelines define two conditions that a reader enforces word-for-word. A function f is increasing on an interval I if, for any a < b in I, f(a) < f(b). It is decreasing on I if, for any a < b in I, f(a) > f(b). These are not informal statements; the rubric row is satisfied only when the candidate's answer communicates both the algebraic idea and the appropriate interval notation. A student who writes "f is increasing where f'(x) > 0" without naming the interval loses the row even if the underlying analysis is perfect.
In practice, the AP Calculus scoring guidelines award a typical FRQ row — call it the first of three in a standard increasing-decreasing prompt — only when the candidate states the interval in the correct form. Open intervals written as (a, b) are the safe default because the definition only requires the inequality to hold inside the interval. Closed intervals or square-bracket notation are technically acceptable when the function is monotonic across the boundary, but most trained readers treat a stray closed bracket as a flag, not a fatal error, while an unspecified interval is fatal. The student who writes "increasing on the interval where f'(x) > 0" has answered a different question than the one asked.
The second rubric row in a typical increasing-decreasing FRQ is the justification row. The reader needs to see, somewhere on the page, a sign analysis of the derivative. The cleanest justification is a sign chart: a small table or a list that shows f'(x) evaluated at sample points in each sub-interval, with a + or − and a verbal conclusion. The justification can also take the form of a written sentence — "f'(x) changes from negative to positive at x = 2, so f is decreasing on (−∞, 2) and increasing on (2, ∞)" — but the candidate must connect the derivative's sign to the function's behaviour explicitly. A correct interval paired with no derivative reasoning will score the interval row but not the justification row.
The third rubric row in many prompts is a consequence row: a request for a local minimum, a local maximum, or a global statement. To earn this row, the candidate typically must identify a critical point, evaluate the function there, and use the surrounding derivative sign — or, less commonly, the second derivative — to classify it. A frequent mistake is to claim a maximum or minimum without naming the point, or to confuse a critical point (where f' = 0 or f' is undefined) with an extremum (where f actually changes direction). On the AP Calculus exam, a critical point that is not an extremum still receives partial credit for being correctly identified, but the extremum row is only awarded when the candidate has demonstrated the direction change.
First derivative test versus second derivative test on the exam
For most AP Calculus increasing-decreasing prompts, the first derivative test is the working tool, and the second derivative test appears as an alternative or a follow-up question. The first derivative test works by sign: find where f' is positive, where f' is negative, and read off the intervals. The second derivative test works by concavity: a positive f'' at a critical point indicates a local minimum, a negative f'' a local maximum, and f'' = 0 leaves the test inconclusive.
In the AB syllabus, the first derivative test is the workhorse. BC candidates see additional prompts where the second derivative test saves time — for instance, in problems where finding f' and factoring is straightforward, but reading the sign of f' requires evaluating awkward expressions at multiple sample points. Most trained BC readers, in my experience, prefer a hybrid: use the first derivative test to identify intervals of increase and decrease, then use the second derivative test to classify extrema when the prompt explicitly asks for concavity information or when f'' has a clean sign at the critical point. The hybrid is faster than a pure second-derivative approach and more rigorous than a pure graphical approach.
The trap to watch for is using the second derivative test where it doesn't apply. If f'' is zero at the critical point, the test is inconclusive, and the candidate must fall back to the first derivative test or to the actual values of f on either side. On the AP Calculus exam, this situation appears often enough that a candidate who reflexively writes "f''(c) < 0, so local max" without checking the value of f'' will lose the row on roughly one prompt out of every three. The defensive habit is: before applying the second derivative test, evaluate f'' at the critical point; if the value is zero or undefined, switch tools.
A second trap is sign chart construction. The AP Calculus sign chart is a small but distinct skill. The candidate should mark the critical points of f' on a number line, choose one test point inside each resulting sub-interval, and write the sign of f' at that test point. Common errors include: picking test points that are themselves critical points, evaluating f' algebraically but writing the sign in the wrong row of the chart, and forgetting to mark points where f' is undefined. For piecewise functions, undefined interior points behave exactly like zeros of f' on the sign chart — they are sign-change candidates, not skipped entries.
Reading a graph for intervals of increase
About a third of AP Calculus multiple-choice items, and a meaningful share of FRQ setup lines, give the candidate a graph rather than a formula. The reading strategy has three steps that, when followed in order, eliminate most careless errors.
Step one is to read the function's behaviour, not the function's derivative. The candidate scans the curve from left to right and marks, on a rough sketch, every point at which the curve switches from rising to falling or vice versa. These are the local extrema, and they are the boundaries of the intervals the candidate will eventually name. A common mistake is to mark the inflection points instead — points where the curve changes concavity but not direction. An inflection point is not a boundary of an increasing or decreasing interval, and a candidate who confuses the two will write an interval with a stray interior point.
Step two is to identify any flat or near-flat sections. A horizontal tangent is not, by itself, evidence of a sign change in f'. If the curve is increasing, briefly levels off, and continues increasing, the candidate should record a single increasing interval across the level section, not two separate intervals separated by an artificial extremum. On a typical AP Calculus graph prompt, this distinction is worth at least one rubric point.
Step three is to handle endpoints. The function is increasing or decreasing on the visible portion of the graph, but the rubric definition requires the candidate to name the interval explicitly. If the graph is drawn for x in [0, 5] and the function is rising throughout, the answer is "increasing on [0, 5]" (or, more precisely, on the open interval, but most readers accept the closed interval when the endpoints are visible). A candidate who writes only "increasing" without an interval loses the row. This is the most common single error I see on graph-based increasing-decreasing items.
For BC candidates, the graph-reading step also includes a check for asymptotes and removable discontinuities. A vertical asymptote is a hard boundary; the function cannot be increasing across a vertical asymptote in any rigorous sense, because the limit does not exist. A removable discontinuity is softer; the function can be increasing through a hole if the underlying rule is increasing and the missing point does not change the order. The AP Calculus reader expects the candidate to handle these cases with the language of limits, not with hand-waving.
Connecting derivative sign to function behaviour in writing
The transition from "f'(x) > 0" to "f is increasing" is a logical step that the AP Calculus rubric treats as a distinct claim. The candidate must show both pieces: the derivative sign on a specified interval, and a sentence that ties that sign to monotonicity. The justification row is awarded for the second piece, not the first.
A model response reads: "f'(x) > 0 on the interval (1, 4), so f is increasing on (1, 4) by the definition of the first derivative test." The candidate does not need to repeat the definition of "increasing" in full; the rubric accepts a citation of the test by name, or a one-line explanation, or even a parenthetical aside. What the rubric will not accept is the bare statement "f is increasing on (1, 4)" with no supporting derivative work, even if the interval is correct. The justification row is independent of the interval row, and a candidate who answers both correctly receives credit for both, regardless of the order in which they appear on the page.
For the second derivative test, the language is similar but the citation is different. A clean response reads: "f''(c) < 0, so by the second derivative test f has a local maximum at x = c." If the second derivative test is inconclusive — f''(c) = 0 or undefined — the candidate must write that explicitly and then fall back to the first derivative test. A candidate who writes "f''(c) = 0 so we use the first derivative test" and then does the first derivative analysis correctly will earn the row. A candidate who writes only "f''(c) = 0 so the test is inconclusive" and stops will not.
The verbal scaffolding matters because the reader is scoring a written response under time pressure. Most AP Calculus readers spend between 30 and 90 seconds per FRQ row, and the rubric is applied as a checklist. A response that hits the checklist in any order, with any reasonable wording, scores the row. A response that hits the checklist only implicitly — by writing the right answer to a later row without ever stating the derivative sign — typically scores that later row but loses the earlier one. The defensive habit is to write the derivative sign, the interval, the test citation, and the conclusion, in that order, even if it feels redundant.
BC-only extensions: particle motion and related rates
AP Calculus BC candidates see increasing-decreasing reasoning pushed into two specific contexts. The first is particle motion: the function is the position of a particle on a line, the first derivative is the velocity, and the second derivative is the acceleration. A typical prompt asks the candidate to identify when the particle is moving to the right, when it is at rest, and when it is speeding up or slowing down. The increasing-decreasing vocabulary maps directly: the particle moves to the right when velocity is positive, to the left when velocity is negative, and is at rest at critical points where velocity is zero.
The second context is related rates. A related-rates prompt might ask when a quantity is increasing at an increasing rate, or when two quantities are both increasing at a particular instant. The first derivative test still applies, but the function being analysed is no longer a single closed-form expression; it is a relation between variables, and the candidate must differentiate implicitly before applying the test. The sign analysis is then performed on the implicit derivative, not on the original relation.
For both contexts, the same rubric rows apply: interval, justification, consequence. A particle-motion prompt that asks "when is the particle moving to the right and what is its maximum velocity in the first 5 seconds" is, at its core, an increasing-decreasing problem with a maximum-value follow-up. The interval row is "moving to the right on (a, b)", the justification row is "v(t) > 0 on (a, b)", and the consequence row is the velocity value at the relevant critical point. Most BC candidates who struggle on these prompts are losing the consequence row, not the interval or justification row, because the second step — translating the position function into a velocity function and then into a critical-point calculation — is a chain of three operations, and a slip in any one of them drops the row.
Common pitfalls and how to avoid them
The increasing-decreasing question type is one of the highest-yield categories on the AP Calculus exam, and most point losses are avoidable with a short list of habits. The first habit is to write the interval. A correct monotonicity claim with no interval is a half-answer; the rubric row is for the interval, not for the bare adjective. The second habit is to cite the test. The rubric does not require the candidate to name "first derivative test" by label, but the logical link from derivative sign to function behaviour must be visible on the page. The third habit is to handle endpoints and asymptotes. The increasing-decreasing definition breaks at vertical asymptotes, and the candidate must either exclude them or treat them as interval boundaries; an answer that lumps two pieces of a function on either side of an asymptote into a single interval is technically incorrect and typically loses a row.
The fourth habit is to keep the sign chart and the interval list parallel. After the sign chart is drawn, the candidate should be able to read off every increasing interval and every decreasing interval by scanning the chart left to right. If the chart shows + + − − + + but the candidate's answer lists four intervals of increase and only one of decrease, the chart and the answer are out of sync, and one of them is wrong. The defensive move is to write the answer after the chart, not before, and to make sure each row of the answer corresponds to a sign region of the chart.
The fifth habit is to verify critical points. A critical point is where f' = 0 or f' is undefined, and the candidate should mark both kinds on the sign chart. An answer that misses a critical point because f' was undefined there will misstate every subsequent interval. The sixth habit, specific to BC, is to check whether the second derivative test is conclusive before applying it. f''(c) = 0 is not a license to claim a local extremum; it is a license to switch back to the first derivative test.
The seventh habit is to read the function's behaviour, not the function's derivative, when the prompt gives a graph. The graph is a picture of f, not of f'. The candidate should mark the extrema of f, not the zeros of f', and then translate the extrema into intervals of monotonicity. A candidate who marks the zeros of f' on a graph of f will draw the boundaries of increasing-decreasing intervals in the wrong places.
Worked example: a typical AB FRQ row
Consider a function f defined on [0, 6] with f'(x) = (x − 1)(x − 3). The candidate is asked to find the intervals on which f is increasing and decreasing, and to identify any local extrema. The first step is to find the critical points: f'(x) = 0 at x = 1 and x = 3. Both are interior points of the domain, and f' is defined and continuous everywhere, so the only critical points are 1 and 3. The sign chart has three regions: (0, 1), (1, 3), and (3, 6). Choosing test points x = 0.5, 2, and 4.5, the candidate computes f'(0.5) = (−0.5)(−2.5) = +1.25 (positive), f'(2) = (1)(−1) = −1 (negative), and f'(4.5) = (3.5)(1.5) = +5.25 (positive). The sign chart is therefore +, −, +.
From the sign chart, the intervals of monotonicity are: f is increasing on (0, 1) and (3, 6), and decreasing on (1, 3). The justification row is satisfied by citing the sign of f' on each interval. The extrema: f has a local maximum at x = 1 and a local minimum at x = 3, because f' changes from positive to negative at 1 and from negative to positive at 3. The consequence row is satisfied by naming the points and stating the direction change.
The full rubric row, in the order a trained reader expects to see it, reads: "f is increasing on (0, 1) ∪ (3, 6) and decreasing on (1, 3); f' > 0 on (0, 1) and (3, 6), f' < 0 on (1, 3); f has a local maximum at x = 1 and a local minimum at x = 3." This single response, in roughly four lines of writing, would earn the full three-row credit on a standard AP Calculus prompt.
Preparation strategy: how to drill this question type to a 5
For most candidates, the increasing-decreasing question type is a high-yield category because the underlying algebra is bounded and the rubric rows are stable. A focused six-week preparation plan looks like this. Weeks one and two: sign-chart drills on quadratic and cubic polynomials, building the habit of writing the chart before the answer. Weeks three and four: piecewise and rational functions, where undefined interior points complicate the chart. Week five: trigonometric and exponential functions, where the test points need to be chosen carefully (for example, π/2 for cosine to verify a sign change at π). Week six: full FRQ practice under timed conditions, with self-scoring against the released scoring guidelines.
The most efficient use of released scoring guidelines is to read the sample student responses sorted by score. The AP Calculus program publishes responses at the 1, 2, 3, 4, 5, 6, 7, 8, and 9 marks on each FRQ, and a candidate who studies the difference between a 6 and a 7 on an increasing-decreasing row will see the exact wording the reader is looking for. The jump from a 6 to a 7 on a typical increasing-decreasing row is almost always a missing interval or a missing test citation, not a wrong algebraic manipulation.
For BC candidates, the same six-week plan is followed by a two-week overlay on particle motion and related rates. The BC scoring guidelines tend to be more lenient on particle-motion prompts because the language ("moving to the right", "at rest") is more familiar than the calculus vocabulary, but the rubric rows are identical in structure. A candidate who has internalised the three-row pattern on the AB side will transfer it to the BC side with minimal extra work.
Scoring and exam format: where this topic sits
Increasing and decreasing functions appear on the AP Calculus exam in two places: as multiple-choice items in Section I, and as a recurring sub-task inside the FRQs in Section II. On the multiple-choice side, typical items give the candidate a graph or a derivative and ask for the interval of increase, the location of an extremum, or a true-false statement about monotonicity. These items are usually 2 or 3 points out of the typical 45 multiple-choice points in a single section, depending on whether the candidate is taking AB or BC. On the FRQ side, increasing-decreasing reasoning is a near-universal sub-task in the first two FRQs, which together account for 18 of the 54 points in Section II.
The exam format rewards consistency. A candidate who scores the interval row, the justification row, and the consequence row on the first FRQ can apply the same three-row discipline to the second FRQ, the third, and (for BC) the fourth. The cumulative effect is meaningful: across a full AP Calculus exam, the increasing-decreasing rubric rows account for a noticeable share of the total available points, and a candidate who has drilled the question type to automaticity will harvest those rows without burning time on the more open-ended differential-equation or series prompts.
Conclusion and next steps
AP Calculus increasing and decreasing functions is a question type where the rubric is more important than the calculus. A candidate who can find critical points, build a sign chart, write the intervals in correct notation, cite the first derivative test, and name the extrema has earned the full credit, regardless of whether the function was a polynomial, a trigonometric expression, or an implicit relation. The path to a 5 is to drill the three-row pattern, study the released scoring guidelines sorted by score, and apply the pattern to AB and BC variants in turn. AP Courses' one-to-one AP Calculus programme runs timed FRQ drills on the increasing-decreasing rubric rows and reviews each candidate's sign-chart discipline against the released scoring guidelines, turning a 5 target into a concrete preparation plan.