Skip to content
Marlbridge

AP® Calculus AB · Unit 5 · Topic 5.1

Using the Mean Value Theorem: Topic Checklist (Calculus AB 5.1)

Specific “I can…” statements for the Mean Value Theorem, each linked to the guide section and practice question that shows whether you can do it.

Work through the list without notes or a calculator. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.

Understanding

  • I can state the Mean Value Theorem, naming both conditions and the conclusion. (Guide: “The theorem, part by part”)
  • I can explain the conclusion as a tangent line parallel to a secant line, and as “instantaneous rate equals average rate”. (Guide: Figure 1; Practice Q7(a))
  • I can explain why continuity is needed on the closed interval but differentiability only on the open interval. (Guide: “The theorem, part by part”)
  • I can explain why a failed condition removes the guarantee but does not prove that no c exists. (Guide: Worked example 3; Practice Q4, Q5(b))

Skills

  • I can find every value of c the theorem guarantees, rejecting solutions outside (a, b). (Guide: Worked example 1; Practice Q1, Q5(a))
  • I can decide whether a function meets both conditions, spotting corners, cusps, vertical tangents and breaks. (Guide: Worked example 3; Practice Q2, Q4)
  • I can choose constants that make a piecewise function differentiable, then apply the theorem. (Practice Q6)
  • I can use a table of a differentiable function to show that the derivative takes a given value on a sub-interval. (Guide: Worked example 2; Practice Q3, Q7(a))

Reasoning

  • I can write a three-part justification: conditions with a reason, the average rate with numbers, and the theorem named. (Guide: “Writing a justification”; Practice Q7)
  • I can interpret the conclusion in context, with units. (Guide: Worked example 2; Practice Q7(a))
  • I can combine the Mean Value Theorem with the Intermediate Value Theorem applied to a continuous derivative. (Practice Q7(b))
  • I can explain why an average rate is not an upper or lower bound on the instantaneous rate. (Practice Q7(c))

All ticked? Move on to Topic 5.2, Extreme Value Theorem, Global Versus Local Extrema, and Critical Points, or return to the Calculus AB roadmap or the Calculus BC roadmap.

AP® and Advanced Placement® are trademarks registered by the College Board, which is not affiliated with, and does not endorse, this website. Marlbridge is an independent tuition provider. It is not an authorized AP course provider, an AP school or an exam centre, and these resources are not official College Board materials. Reading them or doing simulations does not meet the laboratory requirement of an AP science course.