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The Calc Medic Ultimate Justifications Guide for AP Calculus

Updated: May 4, 2023

As your students are getting ready for the AP Calculus exam, one of the biggest things students need to review is how to give a proper justification to support an answer on a Free Response Question. Phrases like "give a reason", "explain", "justify" or "show the analysis that leads to your answer" all indicate that some kind of rationale is needed. In AP Calculus, these justifications need to be related to limits, derivatives, and integrals and make use of definitions and theorems.

Tips for justification:

  • Don't say too much! Short, succinct statements are preferred. Writing long sentences often means including additional incorrect information that you don't mean!

  • Relate your reasoning to the information given in the graph or table.

  • Use words and symbols to give explanations; don't rely on number lines, sign charts, or writing out calculator keys.

  • You may use common abbreviations and names (ex: IVT, abs. min., FTC, second derivative test)

  • Don't use vague phrases like "it", "the graph", or "the function". Be specific about which function or value you mean.

Some justifications are so prevalent you can expect to see them every single year. For example, in a particle motion problem being able to explain why the particle is changing direction, speeding up, moving away from the origin, etc. In graph analysis problems, you will need to know how to show that a function is increasing or decreasing, is concave up or concave down, or has a relative extrema, inflection point, or critical point.

We put all of these commonly seen justifications in one place on our Calc Medic Ultimate Justifications Guide.

The left column gives the statement and the right column gives the justification that is needed. You can download the blank version where students fill in the second column or the completed version with all justifications already given.

Blank version for students to fill-in:

Completed version:

How to Use the Calc Medic Ultimate Justifications Guide

  • Fill it out throughout the course: Give the blank version of the guide to students at the beginning of Unit 4. Have them complete the right column as they encounter each of the statements throughout the course.

  • Use it as an AP Exam review activity: Use one of your review days before the AP Exam to have students complete the right column of the blank version. Have students work individually first, then in pairs, then in small groups.

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Roberto Frontera
Roberto Frontera
Apr 04

This is my first time teaching this course and this is exactly what I was looking for to start my review process. Thank you!!


Lori Bouck
Lori Bouck
May 01, 2023

To justify that f is differentiable at x=a, your "state/show" column begins with "f is differentiable at x=a" instead of "f is continuous at x=a." Thought you'd want to correct it for next year. Thank you for all of the resources. They are really helpful!

Sarah Stecher
Sarah Stecher
May 05, 2023
Replying to

Great catch! The file has been updated!

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