## Unit 5 - Day 1

##### All Units
###### ​Learning Objectives​
• Justify conclusions about functions by applying the MVT over an interval.

###### ​Success Criteria
• I can verify that the conditions of the MVT have been met.

• I can use the MVT to justify a conclusion about a function’s average rate of change over an interval and the instantaneous rate of change at a point on that interval.

• I can interpret solutions to problems involving the MVT.

###### Overview

Unit 5 focuses on the analytical applications of differentiation: the MVT, using the First and Second Derivative tests on increasing/decreasing functions, the Candidate Test, concavity, and using derivatives to confirm extrema in context (optimization). These concepts might mean signal drudgery for students in a traditional calculus class, but today’s lesson provides an engaging and interesting launch to the Unit 5 content! Students should be able to work through all questions in the activity given sufficient time. Using the MVT as a justification is likely to be required, so we must insure students can write and use the theorem with fidelity.

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###### Teaching Tips

Reinforce the connection between mean and average and average rate of change so students understand why this is named the Mean Value Theorem. Include a graphical discussion of the relationship between the instantaneous ROC and the average ROC; have students draw functions that do (and do not) obey the MVT so they can visualize the results of this powerful theorem.

###### Exam Insights

Require students to confirm and state the hypotheses of the theorem in order to earn full points on the Free Response section of the AP Test. Using “MVT” on the AP Test is an acceptable abbreviation when referencing this theorem.

###### Student Misconceptions

As with the IVT and the EVT, stating that a function has met the hypotheses of the MVT is important and necessary. The MVT is an existence theorem guaranteeing a point on a differentiable function where the slope of the tangent line equals the slope of a secant line. You may discover your students are able to navigate the required calculus and algebra without actually knowing the meaning of their answer! Continuing to require an interpretation of results will help students toward better understanding.