AP Calculus AB and BC · Exam guide
How to Justify Answers on AP Calculus FRQs: Theorems, Tests, and Sign Charts
Many calculus FRQ points are not for the answer but for the reason. This guide lists what to write when a question says justify, show the conditions of a theorem, or explain how you know. These habits apply to both Calculus AB and BC.
What to write for each argument
| Claim | What to show | How to write it |
|---|---|---|
| Intermediate Value Theorem (IVT) | \(f\) is continuous on \([a,b]\), and \(k\) is between \(f(a)\) and \(f(b)\). | "Since f is continuous on [a, b] and k is between f(a) and f(b), by the IVT there is a c in (a, b) with f(c) = k." |
| Mean Value Theorem (MVT) | \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\). | "Since f is continuous on [a, b] and differentiable on (a, b), by the MVT there is a c with f'(c) equal to the average rate of change." |
| Extreme Value Theorem (EVT) | \(f\) is continuous on a closed interval. | "Since f is continuous on [a, b], it has an absolute maximum and an absolute minimum on [a, b]." |
| Relative extremum | \(f'\) changes sign at the point. | "f' changes from positive to negative at x = c, so f has a relative maximum at x = c." |
| Absolute extremum | Compare values at critical points and endpoints (the candidates test). | "The candidates are f(a), f(c), and f(b). The greatest is f(c), so the absolute maximum is f(c)." |
| Concavity and inflection | The sign of \(f''\), and a sign change for an inflection point. | "f'' is positive on (a, b), so the graph of f is concave up there." For an inflection point, add that f'' changes sign. |
A worked example: the Mean Value Theorem
Let \(f(x) = x^3\) on \([0, 2]\). Show that the MVT applies, and find the value of \(c\) it guarantees.
The average rate of change is \(\dfrac{f(2) - f(0)}{2 - 0} = \dfrac{8 - 0}{2} = 4\). We need \(f'(c) = 3c^2 = 4\), so \(c^2 = \dfrac{4}{3}\) and \(c = \dfrac{2}{\sqrt{3}} \approx 1.155\), which is in \((0, 2)\).
It never states that f is continuous and differentiable, which are the conditions of the theorem.
It states the hypotheses, applies the theorem, and then solves.
Using a sign chart
A sign chart is a good scratch tool, but the grader reads your written work, not your chart. After you build one, write the conclusion in a sentence, such as "f' is negative on (-2, 1) and positive on (1, 5), so f has a relative minimum at x = 1." Be sure to say which function you are describing, since f, f', and f'' mean different things.
Common mistakes
- Saying "f'(x) = 0, so it is a maximum." A zero derivative does not prove a maximum or minimum.
- Using the sign of f' to decide concavity. Concavity depends on the sign of f''.
- Using a theorem without checking its conditions, especially continuity and differentiability.
- Giving only the interval where f' is positive and not saying what that means for f.
- Forgetting the endpoints when asked for an absolute extremum on a closed interval.
Frequently asked questions
Do I have to state the theorem by name?
Naming the theorem and showing its conditions is the safest way to earn the justification point. Be sure the conditions are stated for the function in the problem.
Is a sign chart enough justification?
A sign chart can help you, but write a sentence that states what the signs mean. Graders look for the reasoning in your written work.
Do I need to justify answers that use a calculator?
Yes, when the question asks you to justify. Write the setup and the reasoning, not only the calculator's output.
What if I forget the conditions of a theorem?
State the ones you remember, and note that the function is continuous and differentiable where it applies. Partial credit is often available for correct reasoning.
Are these rules the same for AB and BC?
Yes. The justification habits are the same on both exams.
Keep practicing
Every lesson in the AP Calculus courses ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:
- Free lesson 1.13: the Intermediate Value Theorem
- Free lesson 1.9: confirming continuity over an interval
See the AP Calculus AB and BC course Free lessons and lesson list See the AP Calculus BC course
Other free-response guides:
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