AP Calculus 5.1 Mean Value Theorem

What are the two primary conditions a function must satisfy on a closed interval [a, b] for the Mean Value Theorem to apply?

• The function must be continuous on [a, b] and differentiable on (a, b).

• The function must be continuous on (a, b) and differentiable on [a, b].

• The function must be continuous on [a, b] and have equal function values at a and b (f(a) = f(b)).

• The function must be differentiable on (a, b) and have equal function values at a and b (f(a) = f(b)).

Geometrically, what does the Mean Value Theorem guarantee if its conditions are met for a function on an interval [a, b]?

• There is at least one point 'c' in (a, b) where the tangent line is perpendicular to the secant line connecting the endpoints.

• There is at least one point 'c' in (a, b) where the tangent line is parallel to the secant line connecting the endpoints.

• The function will have a local maximum or minimum within the interval (a, b).

• The function's average rate of change over the interval [a, b] is zero.

For which type of function on a closed interval [a, b] does the Mean Value Theorem apply?

• A function with a jump discontinuity at an interior point.

• A function with a sharp corner or cusp at an interior point.

• A function that is continuous on [a, b] and differentiable on (a, b).

• A function that is continuous on [a, b] but not differentiable at the endpoints.

If the conditions of the Mean Value Theorem are satisfied for a position function, what does the theorem tell us about the velocity of an object?

• The average velocity equals the instantaneous velocity at least once on the time interval.

• The instantaneous velocity is always greater than the average velocity.

• The average velocity is always greater than the instantaneous velocity.

• The instantaneous velocity is constant throughout the time interval.

If the derivative of a function f(x) is equal to 0 for all numbers in an open interval (a, b), what type of function is f(x) on the closed interval [a, b]?

• An increasing function

• A decreasing function

• A constant function

• A linear function