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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 f that is:
    • Continuous on [a,b]
    • f(a)=f(b) There exists at least one c(a,b) where f(c)=0
  • Geometric Interpretation:
    • If a function starts and ends at the same y-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 c

Show that f(x)=x33x+2 satisfies Rolle’s Theorem on [1,2], and find all values of c that satisfy the conclusion.

The Mean Value Theorem and Its Meaning

The Main Idea 

  • Theorem Statement: For a function f that is:
    • Continuous on [a,b]
    • There exists at least one c(a,b) where: f(c)=f(b)f(a)ba
  • 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:
    • c is not necessarily unique
    • Theorem guarantees existence, not the exact value of c

Suppose a ball is dropped from a height of 200 ft. Its position at time t is s(t)=16t2+200. Find the time t when the instantaneous velocity of the ball equals its average velocity.

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

Corollaries of the Mean Value Theorem

The Main Idea 

  • Corollary 1: Functions with a Derivative of Zero
    • If f(x)=0 for all x in an interval I, then f(x) is constant on I
  • Corollary 2: Constant Difference Theorem
    • If f(x)=g(x) for all x in an interval I, then f(x)=g(x)+C for some constant C
  • Corollary 3: Increasing and Decreasing Functions
    • If f(x)>0 on latex[/latex], then f is increasing on [a,b]
    • If f(x)<0 on latex[/latex], then f is decreasing on [a,b]
  • 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