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Rolle's Theorem and the Mean Value Theorem

Rolle's Theorem

The French mathematician Michael Rolle proved Rolle's Theorem in 1691 but afterwards he became a vocal critical of the methods of his day and attacked calculus as being a "collection of ingenious fallacies". Let f be a function that satisfies the following three hypotheses:

f is continuous on the closed interval [a, b].
f is differentiable on the open interval (a, b).
f(a) = f(b)

Then there is a number c in (a, b) such that f '(c) = 0.

Geometrically, Rolle's Theorem guarantees that there is at least one point (c, f(c)) on the graph of the function f where the tangent is horizontal. View the presentation below illustrating how Rolle's Theorem can be applied.

image of airborne basketball Consider the situation when a ball is thrown into the air. The curve traced by the ball is continuous on a closed interval (beginning and end) and differentiable on an open interval. For every height above the ground f(a) on the way up, there is an equal height above the ground f(b)on the way down. Since all of the conditions of Rolle's Theorem are satisfied, then we can conclude that somewhere between a and b, the velocity of the ball is 0 at x = c.

Mean Value Theorem

Let f be a function that satisfies the following hypotheses:

f is continuous on the closed interval [a, b].
f is differentiable on the open interval (a, b).

Then there is a number c in (a, b) such that equation image indicator $LaTeX: f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ $f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

Geometrically, since f '(c) is the slope of the tangent line at the point (c, f(c)), the Mean Value Theorem guarantees that there is at least one point P(c, f(c)) on the graph where the slope of the tangent line is the same as the slope of the secant line through the points A(a, f(a)) and B(b, f(b)), i.e., there is a point P at which the tangent line is parallel to the secant line AB.

View the presentation illustrating a graphical interpretation of the Mean Value Theorem.

Recall that the average rate of change in a function f over [a, b] is the number equation image indicator $LaTeX: \frac{f\left(b\right)-f\left(a\right)}{b-a}$ $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ and the instantaneous rate of change is f '(c). Based on the Mean Value Theorem, at some value of c within the interval, the instantaneous rate of change must equal the average rate of change.

Rolle's Theorem and the Mean Value Theorem Practice

1. Verify Rolle's Theorem for equation image indicator $LaTeX: f\left(x\right)=x^2-2x$ $f\left(x\right)=x^2-2x$ on the interval [0,2].

Solution: $LaTeX: f\left(0\right)=f\left(2\right)=0\\ f'(x)=0 \: at \: x=1$ $f\left(0\right)=f\left(2\right)=0\\ f'(x)=0 \: at \: x=1$ text annotation indicator

2. Verify that the hypotheses of Mean Value Theorem are satisfied for f(x) = (x + 2)/x on [1, 2] and find all values of c that satisfy the conclusion.

Solution: f(x) is continuous on [1,2] and differentiable on (1,2) $LaTeX: c=\sqrt[]{2}$ $c=\sqrt[]{2}$ text annotation indicator

Rolle's Theorem and the Mean Value Theorem: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of Rolle's Theorem and the Mean Value Theorem.

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