- 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)=x3−3x+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)b−a
- 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
- 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