Mean Value Theorem

Front

What is the Mean Value Theorem?

Back

The Mean Value Theorem states that 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 c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

What does the Mean Value Theorem imply about the slope of a function?

Back

It implies that there is at least one point in the interval where the instantaneous rate of change (slope of the tangent) equals the average rate of change over the interval.

Front

How do you find the average rate of change of a function over an interval?

Back

The average rate of change of a function f(x) over the interval [a, b] is calculated as (f(b) - f(a)) / (b - a).

Front

What is the derivative of f(x) = 12√(x)?

Back

f'(x) = 6 / √(x).

Front

What is the derivative of f(x) = x² + 5x + 5?

Back

f'(x) = 2x + 5.

Front

What is the derivative of f(x) = (x-4)/(x+1)?

Back

f'(x) = 5 / (x+1)².

Front

What is the significance of the derivative in the context of the Mean Value Theorem?

Back

The derivative represents the instantaneous rate of change of the function, which is compared to the average rate of change over an interval.

Front

If f(x) is continuous and differentiable, what can we conclude about its behavior on an interval?

Back

We can conclude that there exists at least one point where the slope of the tangent (derivative) matches the average slope over that interval.

Front

What is the average rate of change of f(x) = 12√(x) over the interval [1, 16]?

Back

The average rate of change is 12 / 5.

Front

How do you determine the x-value where the function has the same slope as the average rate of change?

Back

Set the derivative equal to the average rate of change and solve for x.