Rolle's Theorem and Mean Value Theorem

Front

State Rolle's Theorem.

Back

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.

Front

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

Back

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.

Front

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

Back

π/2

Front

State the Mean Value Theorem.

Back

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].

Front

What is the significance of the Mean Value Theorem?

Back

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.

Front

Define continuity in the context of functions.

Back

A function is continuous on an interval if there are no breaks, jumps, or holes in the graph of the function on that interval.

Front

What does differentiable mean?

Back

A function is differentiable at a point if it has a defined derivative at that point, meaning the function has a tangent line at that point.

Front

Explain the concept of average rate of change.

Back

The average rate of change of a function over an interval [a, b] is given by (f(b) - f(a)) / (b - a).

Tags

CCSS.8.F.B.4

CCSS.HSF.IF.B.6

Front

What is the relationship between the derivative and the slope of the tangent line?

Back

The derivative of a function at a point gives the slope of the tangent line to the graph of the function at that point.

Front

Provide an example of a function that satisfies Rolle's Theorem.

Back

f(x) = x^2 on the interval [1, 1] satisfies Rolle's Theorem since f(1) = f(1) and it is continuous and differentiable.