Rolle's Theorem
What are the three conditions that must be met for Rolle's Theorem to apply?
1. The function must be continuous on the closed interval [a, b]
2. The function must be differentiable on the open interval (a, b)
3. The function must satisfy f(a)=f(b), meaning the values at the endpoints of the interval must be equal.
What is the derivative of f(x)= x^2-5x+4?
The derivative is
f'(x)=2x-5
What was Michel Rolle's main contribution to mathematics, and what does it help us understand?
Michel Rolle's main contribution was Rolle's Theorem, which is a fundamental result in calculus that helps us understand the behavior of continuous and differentiable functions.
Can you explain Rolle's Theorem in simple terms?
If a function is continuous and smooth (no breaks or sharp corners) on a closed interval and differentiable (able to take derivatives) on the open interval, and the values of the function at the endpoints are the same, then there must be at least one point where the derivative (slope) of the function is zero between those two points.
Find the critical points of f(x)=x(x-3)
f(x)= x^2-3x
f'(x)=2x-3
2x-3=0
x=3/2
The critical point is x=3/2
When was Michel Rolle born?
Michel Rolle was born in 1652
Does the function f(x)=x^2-5x+4 on the interval [1,4] meet the criteria for Rolle's Theorem? (State the criterias)
Yes,
The function is continuous on [1, 4] (Since it's a polynomial (Quadratic))
The function is differentiable on (1, 4)
f(1)=f(4)=0, so the values at the endpoints are equal.
So yes, Rolle's Theorem applies.
Is f(x)= x^2/3 -1 continuous on the interval [-8, 8]? Why or why not?
Yes, the function is continuous on [-8, 8]. Since the function is a polynomial, it is continuous everywhere, including the interval [-8, 8].
Which famous mathematical theorem is named after Michel Rolle?
The famous theorem named after Michel Rolle is Rolle's Theorem.
Why does f(x) = x^2/3 -1, [-8, 8] fail the conditions for Rolle's Theorem?
It is not differentiable at x=0 because a derivative doesn't exist there.
Rolle's Theorem does not apply
x^2-5x+4, [1, 4]
f(1)=(1)^2-5(1)+4=1-5+4=0
f(4)=(4)^2-5(4)+4=16-20+4=0
What condition from Rolle's Theorem applies here?
f(a)=f(b)
Why was Michel Rolle's work controversial in his time?
Michel Rolle's work was controversial because his proof of Rolle's Theorem lacked the rigor that would be expected by modern standards. He was one of the first to formulate and apply such ideas, but his work was not initially widely accepted by other mathematicians, who questioned the validity of his proofs.