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7-51.

The point from problem 7-19 travels along the -axis so that at time its position is given by , where . Calculate the average velocity of the point. At what time(s) during the interval was the point traveling at this average velocity?

This is an application of the Mean Value Theorem of (given ).
In other words, where on the interval is the slope of the tangent the same as the slope of the secant?

Determine where IROC AROC.

real velocity average velocity
slope of tangent slope of secant
Solve for to find out WHERE the equation above holds true.

The Mean Value Theorem

The Mean Value Theorem for Integrals

If is continuous on , then there exists at least one point in such that .

The Mean Value Theorem for Derivatives

If is continuous on and differentiable on , then there exists at least one point in such that .