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 are the conditions for the Mean Value Theorem to apply?

Back

1. The function must be continuous on the closed interval [a, b]. 2. The function must be differentiable on the open interval (a, b).

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 over 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 has a defined slope.

Front

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

Back

The average rate of change is (f(6) - f(-2)) / (6 - (-2)) = (77 - 1) / 8 = 9.5.

Tags

CCSS.8.F.B.4

CCSS.HSF.IF.B.6

Front

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

Back

Set the derivative f'(x) equal to the average rate of change and solve for x.

Tags

CCSS.8.F.B.4

CCSS.HSF.IF.B.6

Front

What is the derivative of f(x) = x^2 + 5x + 5?

Back

f'(x) = 2x + 5.

Front

For the function g(x) = x^3 - 3x + 2, does the Mean Value Theorem apply on [-2, 5]?

Back

Yes, g(x) is continuous and differentiable on [-2, 5].

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 (slope of the tangent) equals the average rate of change over the interval.

Front

If f(x) = x^2 - 3x - 28, what is the average rate of change over the interval [-4, 7]?

Back

The average rate of change is (f(7) - f(-4)) / (7 - (-4)) = (1 - 37) / 11 = -3.27.

Tags

CCSS.8.F.B.4

CCSS.HSF.IF.B.6