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 point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Front

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

Back

The derivative is f'(x) = 6 / √(x).

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) = x² - 3x - 28?

Back

The derivative is f'(x) = 2x - 3.

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

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

Back

The derivative is f'(x) = 5 / (x+1)².

Front

What is the average rate of change of f(x) = x² + 5x + 5 over the interval [-2, 6]?

Back

The average rate of change is 9.

Front

What does it mean for a function to be continuous on an interval?

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 it mean for a function to be differentiable on an interval?

Back

A function is differentiable on an interval if it has a derivative at every point in that interval, meaning it is smooth and has no sharp corners.

Front

What is the significance of the point c in the Mean Value Theorem?

Back

The point c is where the instantaneous rate of change (the derivative) of the function equals the average rate of change over the interval [a, b].