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# Mean value theorem review

Review your knowledge of the mean value theorem and use it to solve problems.

## What is the mean value theorem?

The mean value theorem connects the average rate of change of a function to its derivative. It says that for any differentiable function f and an interval open bracket, a, comma, b, close bracket (within the domain of f), there exists a number c within left parenthesis, a, comma, b, right parenthesis such that f, prime, left parenthesis, c, right parenthesis is equal to the function's average rate of change over open bracket, a, comma, b, close bracket.
Graphically, the theorem says that for any arc between two endpoints, there's a point at which the tangent to the arc is parallel to the secant through its endpoints.
A function is graphed. The x-axis goes from 0 to 9. The graph is a curve. The curve starts at a closed circle at (0, 0), moves upward until about (4.5, 8.2), moves downward, and ends with a closed point at about (6, 7.2). A secant line connects points (0, 0) and (6, 7.2). A tangent line is drawn parallel to the secant line, and touches the curve at certain point between x = 0 and x = 6.