Florida B.E.S.T. MA.912.C.2.10
The Standard
Apply the Mean Value Theorem.
Florida B.E.S.T. Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students check that a function is continuous on the closed interval and differentiable inside it. They calculate the secant slope, then find or justify an interior point where the derivative equals that slope.
What Mastery Looks Like
- Given a formula, graph, or context, a student decides whether the theorem applies and supports the decision with interval conditions. The student finds all valid values of c and explains how they connect average and instantaneous rates.
Common Misconceptions
- Students may skip checking continuity and differentiability before using the theorem. They may solve f(c) equal to the secant slope instead of f'(c), include endpoints, or assume there is only one valid c.
How to Assess It
- Exit ticket: For f(x) = x² on [1, 3], verify the required conditions, compute the average rate of change, and find every valid c.
Lesson moves
Ways to Teach It
Give students a printed curve, ruler, and movable tangent strip; have them find an interior tangent parallel to the endpoint secant.
Ask students to compare a smooth graph, a jump graph, and a corner graph, then explain which intervals meet the hypotheses.
Run a card match with function, interval, secant slope, and valid c cards; teams must justify each completed set.
Use a smooth distance-time graph for a car trip; students calculate average speed and locate a time with matching instantaneous speed.
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Related Standards
- MA.912.C.2.1
State, understand and apply the definition of derivative. Apply and interpret derivatives geometrically and numerically.
- MA.912.AR.6.2
Explain and apply the Remainder Theorem to solve mathematical and real-world problems.
- MA.912.C.4.4
Evaluate definite integrals by using the Fundamental Theorem of Calculus.
- MA.912.C.1.11
Apply the Intermediate Value Theorem and the Extreme Value Theorem.
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