Ｃａｌｃｕｌｕｓ　１　ｌｅｃｔｕｒｅ　３．２

　A　Bｒｉｅｆ　Dｉｓｃｕｓｓｉｏｎ　Oｆ　Rｏｌｌｅ＇ｓ　Tｈｅｏｒｅｍ　ａｎｄ　Mｅａｎ－Vａｌｕｅ　ｔｈｅｏｒｅｍ

Ｒｏｌｌｅ＇ｓ　ｔｈｅｏｒｅｍ

• Introduction to Rolle's Theorem
  • Prerequisites:
    • The function must be continuous on the closed interval [𝓪, 𝓫].
    • The function must be differentiable on the open interval (𝓪, 𝓫).
    • The function must satisfy 𝒇(𝓪) = 𝒇(𝓫).
  • Implications:
    • These conditions guarantee that there exists at least one point 𝓬 in (𝓪, 𝓫) such that $𝒇'(𝓬)=0$.
    • Geometrically, this means that somewhere between 𝓪 and 𝓫, the function has a horizontal tangent line.
• [0:28]. Explanation of Rolle's Theorem
  • If a function 𝒇(𝒙) takes the same value at two endpoints of a closed interval, and is continuous and differentiable as stated above, then there must exist at least one point in between where the slope of the tangent line is zero.
  • A common visual example is when a function crosses the 𝒙-axis at two points.
    • If 𝒇(𝒙) = 0 at both endpoints and the function is continuous between them, then Rolle's Theorem guarantees a point in the middle where the slope is zero.
    • This represents a special case of Rolle’s Theorem when 𝒇(𝒂) = 𝒇(𝒃) = 0.
    • Graphical illustration of the concept.
  • Special case – Constant function (the trivial case of Rolle's Theorem):
    • If 𝒇(𝒙) = 𝒄 (a constant), then 𝒇 is continuous and differentiable everywhere, and 𝒇(𝓪) = 𝒇(𝓫) = 𝒄 for any [𝓪, 𝓫].
    • The derivative $𝒇'(𝒙)=0$ at every point in (𝓪, 𝓫), so Rolle’s Theorem is satisfied trivially at all points.
    • This is considered the trivial case of Rolle’s Theorem.
• [1:14]. Interpretation: At least one point between the two 𝒙-axis crossings will have a horizontal tangent.
• [1:25]. Transition to the Mean Value Theorem as a corollary of Rolle's Theorem.
• [3:18]. Formal definition of Rolle's Theorem.
  • Let 𝒇(𝒙) be a function satisfying the following conditions:
    • ⑴ Continuity: 𝒇(𝒙) is continuous on the closed interval [𝓪, 𝓫].
    • ⑵ Differentiability: 𝒇(𝒙) is differentiable on the open interval (𝓪, 𝓫).
    • ⑶ Equal Endpoint Values: 𝒇(𝓪) = 𝒇(𝓫).
  • i̲f̲ these conditions are met,
    t̲h̲e̲n̲ there exists at least one point 𝓬 ∈ (𝓪, 𝓫) such that $𝒇'(𝓬)=0$
• Practical Applications.
  • Useful in proving the existence of roots and understanding the behavior of functions.
  • Helps in establishing the existence of extrema within intervals.

Ｍｅａｎ　ｖａｌｕｅ　ｔｈｅｏｒｅｍ

• [1:34]. Introduction to the Mean Value Theorem.
  • Prerequisites:
    • The function must be continuous on the closed interval [𝓪, 𝓫].
    • The function must be differentiable on the open interval (𝓪, 𝓫).
  • Implications:
    • The theorem guarantees that the average rate of change over [𝓪, 𝓫] is attained by the instantaneous rate of change at least once within (𝓪, 𝓫).
    • This is a fundamental theorem in calculus that bridges the gap between average and instantaneous rates of change.
• [1:41]. Explanation of the Mean Value Theorem:
  • Draw a secant line between two points (A and B) on a curve.
  • If the function 𝒇(𝒙) is differentiable between A and B, there must exist at least one point on the curve where the slope of the tangent is equal to the slope of the secant.
  • Graphical illustration of the concept, showing the tangent parallel to the secant.
• [2:23]. Interpretation: At least one point 𝓬 within the interval (𝓪,𝓫) will have a tangent line with 𝓪 slope equal to the slope of the secant line that connects the endpoints (𝓪, 𝒇(𝓪)) and (𝓫, 𝒇(𝓫)).
• [2:42]. Relationship with Rolle's Theorem.
  • Rolle's Theorem is a special case of the Mean Value Theorem in which the function values at the endpoints are equal (𝒇(𝓪) = 𝒇(𝓫)), so the slope of the secant line is zero, and the conclusion becomes $𝒇'(𝓬)=0$.
  • Although Rolle’s Theorem is more specific, it is often proved first and used to derive the more general Mean Value Theorem, which is why the MVT is sometimes introduced as a corollary of Rolle’s Theorem.
  • Formal relationship between MVT and Rolle’s Theorem:
    • Mean Value Theorem (more general):
      • i̲f̲ 𝒇 is continuous on [𝓪, 𝓫] and differentiable on (𝓪, 𝓫),
        t̲h̲e̲n̲ there exists 𝓬 ∈ (𝓪, 𝓫) such that:
        $𝒇'(𝓬)=\dfrac{𝒇(𝓫)-𝒇(𝓪)}{𝓫-𝓪}$
    • Rolle’s Theorem (special case of MVT):
      • i̲f̲, in addition, 𝒇(𝓪) = 𝒇(𝓫),
        t̲h̲e̲n̲ $𝒇'(𝓬)=\dfrac{𝒇(𝓫)-𝒇(𝓪)}{𝓫-𝓪}=0$ ⇒ $𝒇'(𝓬)=0$
• [3:57]. Formal definition of the Mean Value Theorem:
  • i̲f̲ a function 𝒇 satisfies the following conditions:
    ⑴ Continuity: 𝒇 is continuous on the closed interval [𝓪, 𝓫].
    ⑵ Differentiability: 𝒇 is differentiable on the open interval (𝓪, 𝓫).
    t̲h̲e̲n̲ there exists at least one point 𝓬 ∈ (𝓪, 𝓫) such that:
    $𝒇'(𝓬)=\dfrac{𝒇(𝓫)-𝒇(𝓪)}{𝓫-𝓪}$
• [4:26]. Interpretation of the formula: The slope of the tangent at point 𝓬 is equal to the slope of the secant connecting
  the points (𝓪,𝒇(𝓪)) and (𝓫, 𝒇(𝓫)).
• Practical Applications.
  
  • Applied in various fields such as physics for motion analysis, in economics for cost and revenue analysis,
    and in engineering for system behavior.
  • Foundation for proving other theorems in calculus like Taylor's Theorem and Lagrange's Remainder Theorem.

Ｖｅｒｉｆｉｃａｔｉｏｎ　ｏｆ　ｔｈｅ　ｔｈｅｏｒｅｍｓ

• [5:20]. How to verify Rolle's Theorem:
  
  • Verify that the function is continuous on [𝓪, 𝓫] and differentiable on (𝓪, 𝓫).
  • Check that 𝒇(𝓪) = 𝒇(𝓫).
  • Differentiate the function and solve $𝒇'(𝒙)=0$ to find critical point(s).
  • Make sure that the solution(s) lie within the open interval (𝓪, 𝓫).
• [5:37]. How to verify the Mean Value Theorem:
  
  • Verify that the function is continuous on [𝓪, 𝓫] and differentiable on (𝓪, 𝓫).
  • Calculate the slope of the secant line: $\dfrac{𝒇(𝓫)-𝒇(𝓪)}{𝓫-𝓪}$.
  • Differentiate the function and solve $𝒇'(𝒙)=\dfrac{𝒇(𝓫)-𝒇(𝓪)}{𝓫-𝓪}$ to find the value(s) of 𝒙 where 𝒙 = 𝓬.
  • Ensure that the solution(s) lie within the open interval (𝓪, 𝓫).
• [6:04]. Simple summary: – [image]