The Mean Value Theorem: Fresh Take

  • Use Rolle’s theorem and the Mean Value Theorem to show how functions behave between two points
  • Discuss three key implications of the Mean Value Theorem for understanding function behavior

Rolle’s Theorem

The Main Idea 

  • Theorem Statement: For a function [latex]f[/latex] that is:
    • Continuous on [latex][a,b][/latex]
    • [latex]f(a) = f(b)[/latex] There exists at least one [latex]c \in (a,b)[/latex] where [latex]f'(c) = 0[/latex]
  • Geometric Interpretation:
    • If a function starts and ends at the same [latex]y[/latex]-value, it must have at least one horizontal tangent line between those points
  • Key Conditions:
    • Continuity over the closed interval
    • Differentiability over the open interval
    • Equal function values at endpoints
  • Importance:
    • Foundation for the Mean Value Theorem
    • Useful in proofs and theoretical analysis
  • Limitations:
    • Does not apply if the function is not differentiable at even one point
    • Does not guarantee uniqueness of [latex]c[/latex]

Show that [latex]f(x) = x^3 - 3x + 2[/latex] satisfies Rolle’s Theorem on [latex][-1,2][/latex], and find all values of [latex]c[/latex] that satisfy the conclusion.

The Mean Value Theorem and Its Meaning

The Main Idea 

  • Theorem Statement: For a function [latex]f[/latex] that is:
    • Continuous on [latex][a,b][/latex]
    • There exists at least one [latex]c \in (a,b)[/latex] where: [latex]f'(c) = \frac{f(b) - f(a)}{b - a}[/latex]
  • Geometric Interpretation:
    • At some point, the instantaneous rate of change (slope of tangent line) equals the average rate of change (slope of secant line)
  • Relation to Rolle’s Theorem:
    • Mean Value Theorem is a generalization of Rolle’s Theorem
    • Can be proved using Rolle’s Theorem
  • Applications:
    • Velocity and displacement problems
    • Proving mathematical inequalities
    • Establishing properties of functions
  • Key Points:
    • [latex]c[/latex] is not necessarily unique
    • Theorem guarantees existence, not the exact value of [latex]c[/latex]

Suppose a ball is dropped from a height of [latex]200[/latex] ft. Its position at time [latex]t[/latex] is [latex]s(t)=-16t^2+200[/latex]. Find the time [latex]t[/latex] when the instantaneous velocity of the ball equals its average velocity.

A car travels [latex]240[/latex] miles in [latex]4[/latex] hours. Prove that at some point during the trip, the car’s instantaneous speed was exactly [latex]60[/latex] mph.

Corollaries of the Mean Value Theorem

The Main Idea 

  • Corollary 1: Functions with a Derivative of Zero
    • If [latex]f'(x) = 0[/latex] for all [latex]x[/latex] in an interval [latex]I[/latex], then [latex]f(x)[/latex] is constant on [latex]I[/latex]
  • Corollary 2: Constant Difference Theorem
    • If [latex]f'(x) = g'(x)[/latex] for all [latex]x[/latex] in an interval [latex]I[/latex], then [latex]f(x) = g(x) + C[/latex] for some constant [latex]C[/latex]
  • Corollary 3: Increasing and Decreasing Functions
    • If [latex]f'(x) > 0[/latex] on latex[/latex], then [latex]f[/latex] is increasing on [latex][a,b][/latex]
    • If [latex]f'(x) < 0[/latex] on latex[/latex], then [latex]f[/latex] is decreasing on [latex][a,b][/latex]
  • Implications:
    • Corollary 1 proves the converse of “the derivative of a constant function is zero”
    • Corollary 2 shows that antiderivatives differ by at most a constant
    • Corollary 3 connects the sign of the derivative to the behavior of the function