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UTPA STEM/CBI Courses/Calculus/Mean Value Theorems

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Course Title: Mean Value Theorems for differentiable functions

Lecture Topic: Rolle's Mean Value Theorem.

Instructor: Dr. Rajendra P Rai

Institution: Department of Mathematics, University of Texas - Pan American.

Backwards Design

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Course Objectives

  • Primary Objectives- By the next class period students will be able to:
    • Identify the situation where Rolle's MVT can be applied and derive the conclusion if Rolle's MVT is applicable to the situation.
    • Better understand the proofs of other mean value theorems, viz., Lagrange, Cauchy and Taylor mean value theorems that are to follow subsequently and shortly.
  • Sub Objectives- The objectives will require that students be able to:
    • Get a good application of the concept of continuity of a function over a closed interval.
    • Get a good application of the concept of differentiability of a function over an open interval.
  • Difficulties- Students may have difficulty:
    • Understanding the concepts of closed itervals, open intervals and functions.
    • Grasping the concepts of continuity and differentiability of a function.
  • Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
    • Determining if a function has at least one horizontal tangent in a given open interval.
    • Determining if a function has a real zero in an open interval.

Model of Knowledge

  • Concept Map
    • On the the board, draw several variation of the function's graph that satisfy the conditions of Rolle's Theorem.
    • Help students realize that each one of the points along the function has at least one tangent parallel to the x-axis.
  • Content Priorities
    • Enduring Understanding
      • By looking at the pictures drawn in the concept map, students will realize that whatever the theorem states is possibly true often. But they will doubt whether the conclusion of the theorem is always true because the few pictures drawn on the board cover all possible situations. So, they would appreciate the proof that is to follow.
      • The proofs of other mean value theorem that follow can be given using the same technique. So, the proof that is about to be given would prepare them well for the proofs of the othere mean value theorems.
    • Important to Do and Know
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    • Worth Being Familiar with
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Assessment of Learning

  • Formative Assessment
    • In Class (groups)
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    • Homework (individual)
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  • Summative Assessment
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Legacy Cycle

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OBJECTIVE

By the next class period, students will be able to:

  • Check whether a given function satisfies the conditions of the Rolle's Thoerem and derive the conclusions desired.
  • Solve various application problems based upon the concept of the Rolle's MVT.
  • Be ready to understand the concepts in the subsequent mean value theorems of the differential calculus namely the Lagrange's Mean Value Theorem, the Cauchy's Mean Value Theorem and the Taylor's MVT. This will prepare the student to understand their proofs better.

The objectives will require that students be able to:

  • Check the Continuity of a function over a closed interval.
  • Check the differentiability of a function over an open interval.
  • Evaluate the value of a given function at a given point.


THE CHALLENGE

When can you say whether a given function defined over a closed interval has at least one horizontal tangent?

GENERATE IDEAS

Since we are talking about the slope of the tangent, should this function be differentiable?

Should this function be continuous?

What else would one need to make sure that there is at least one point in the domain where the tangent is parallel to the x axis?

MULTIPLE PERSPECTIVES

The above ideals will lead to multiple perspectives and to the statement of Rolle's Theorem.

RESEARCH & REVISE

The requirements of the Rolle's Theorem that have been missed out so far can be obtained through the sketches of functions that do not meet the requirement of the Rolle's theorem that have been missed out and do not have a horizontal tangent.

TEST YOUR METTLE

Synthesize the requirements for the functions obtained so far to determine whether they ensure the existence of a horizontal tangent. If that is so, then go to the next step.


GO PUBLIC

Give the statement of the Rolle's Theorem followed by its proof.

Pre-Lesson Quiz

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  1. What do you understand by a horizontal tangent?
  2. When do you say that a function is continuous over a closed interval?
  3. When do you say that a function is differentiable over an open interval?
  4. What conditions does a function need in ensure the existence of a horizontal tangent to that function?

Test Your Mettle Quiz

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  1. Let f(x) = 3x^2 + 6x + 2. Does this function satisfy the conditions of the Rolle's Theorem over the inter [-2, 0]? If yes, then find the point in the interval where tangent is horizontal.
  2. Show that the equation, x^3 - 9x + 10, has at least one root in the interval [0, 2].
  3. Show that x^3 + 12x + c = 0 has, at most, one real solution.
  4. Other similar questions will given from the textbook.
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