The Mean Value Theorem

Rolle's theorem

The mean value theorem

Consequences for monotonicity

Bounding differences

Which trio of hypotheses makes the conclusion of Rolle's theorem valid on [a, b]?

The mean value theorem applied to f on [a, b] guarantees a point c in (a, b) satisfying which equation?

A highway patrol photographs a car at mile 0 and again at mile 120 exactly 2 hours later. A mean value theorem argument supports what conclusion?

A differentiable function f satisfies f prime of x = 0 for every x in an interval. What does the mean value theorem imply about f on that interval?

On an open interval, f prime of x is positive at every point. Which conclusion follows from the mean value theorem?

The standard proof of the mean value theorem reduces it to Rolle's theorem by which construction?

For f(x) = x^2 on [1, 3], the secant slope (f(3) - f(1))/(3 - 1) equals what number? Answer with a number.

For f(x) = x^2 on [1, 3], the mean value theorem point c satisfies f prime of c = 2c = 4. What is c? Answer with a number.

The mean value theorem requires continuity on the closed interval and differentiability on the open one. How many distinct hypotheses are there? Answer with a number.

f(0) = 3 and |f prime of x| is at most 5 everywhere. By the mean value theorem bound, |f(2) - 3| is at most what number? Answer with a number.