MVT Flashcard

Front

What is the Mean Value Theorem (MVT)?

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 are the conditions required for the Mean Value Theorem to apply?

Back

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

Front

How do you verify the Mean Value Theorem for a given function?

Back

To verify the MVT, calculate the average rate of change of the function over the interval [a, b] and find the derivative of the function. Then, solve for c where the derivative equals the average rate of change.

Front

What is the significance of the value 'c' in the Mean Value Theorem?

Back

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

Front

What is the Extreme Value Theorem?

Back

The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then it attains both a maximum and a minimum value at least once in that interval.

Front

Can a function be discontinuous and still satisfy the Mean Value Theorem?

Back

No, a function must be continuous on the closed interval [a, b] to satisfy the Mean Value Theorem.

Front

What is an example of a function that is discontinuous on the interval [-1, 1]?

Back

An example is y = log(x), which is undefined for x ≤ 0, making it discontinuous on the interval [-1, 1].

Front

How do you find the value of 'c' guaranteed by the Mean Value Theorem for f(x) = x^2 on [0, 3]?

Back

Calculate the average rate of change: (f(3) - f(0)) / (3 - 0) = (9 - 0) / 3 = 3. Then set f'(c) = 2c = 3, solving for c gives c = 3/2.

Front

What is the derivative of f(x) = x^3 + 2x?

Back

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

Front

How do you apply the Mean Value Theorem to the function f(x) = (x-4)(x-6) on the interval [4, 10]?

Back

First, find f(4) and f(10), then calculate the average rate of change. Next, find f'(x) and solve for c where f'(c) equals the average rate of change.