Rolle's Theorem and Mean Value Theorem

State the Mean Value Theorem.

• If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the interval (a, b) where the instantaneous rate of change (derivative) of the function is equal to the average rate of change of the function over the interval [a, b].

• The Mean Value Theorem states that the derivative of a function is equal to the slope of the tangent line to the function at a specific point.

• The Mean Value Theorem states that the derivative of a function is equal to the instantaneous rate of change of the function at a specific point.

• The Mean Value Theorem states that the derivative of a function is equal to the average rate of change of the function over a closed interval.

What is the significance of the Mean Value Theorem?

• The Mean Value Theorem states that the derivative of a function is always equal to the average rate of change of the function over an interval.

• The Mean Value Theorem states that there exists at least one point in an interval where the derivative of a function is equal to the average rate of change of the function over that interval.

• The Mean Value Theorem states that the derivative of a function is equal to the slope of the tangent line at any point on the function.

• The Mean Value Theorem states that there exists at least one point in an interval where the derivative of a function is equal to zero.

Find the value of c that satisfies the Mean Value Theorem for the function f(x) = x^2 - 4x + 3 on the interval [1, 3].

• 2

• 5

• 4

• 0

State Rolle's Theorem.

• Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and the function values at the endpoints are equal, then there exists at least one point c in the open interval (a, b) where the derivative of the function is zero.

• Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and the function values at the endpoints are negative, then there exists at least one point c in the open interval (a, b) where the derivative of the function is zero.

• Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and the function values at the endpoints are not defined, then there exists at least one point c in the open interval (a, b) where the derivative of the function is zero.

• Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and the function values at the endpoints are unequal, then there exists at least one point c in the open interval (a, b) where the derivative of the function is zero.

What is the difference between Mean Value Theorem and Rolle's Theorem?

• Mean Value Theorem states that there exists at least one point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over the interval [a, b], while Rolle's Theorem states that there exists at least one point c in (a, b) where the derivative of the function is not defined.

• Mean Value Theorem states that there exists at least one point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over the interval [a, b], while Rolle's Theorem states that there exists at least one point c in (a, b) where the derivative of the function is positive.

• Mean Value Theorem states that there exists at least one point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over the interval [a, b], while Rolle's Theorem states that there exists at least one point c in (a, b) where the derivative of the function is zero.

• Mean Value Theorem states that there exists at least one point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over the interval [a, b], while Rolle's Theorem states that there exists at least one point c in (a, b) where the derivative of the function is negative.

Find the value of c that satisfies Rolle's Theorem for the function f(x) = x^3 - 3x^2 + 2x on the interval [0, 2].

• c = 3

• c = 0

• Rolle's Theorem is satisfied for this function on the interval [0, 2].

• c = 1

Find the average rate of change of the function f(x) = 2x^2 - 5x + 3 on the interval [1, 4].

• -5

• 10

• 2x^2 - 5x + 3

• 5

Tags

CCSS.8.F.B.4

CCSS.HSF.IF.B.6

Determine whether the function f(x) = x^3 - 3x^2 + 2x satisfies the conditions of Rolle's Theorem on the interval [0, 2].

• Yes

• Not sure

• Maybe

• No

Find the value of c that satisfies the Mean Value Theorem for the function f(x) = sin(x) on the interval [0, π/2].

• 2π

• π

• π/2

• 0