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Derivatives are rates of change, and in the physical world that means things like velocity and acceleration. In fact, studying these quantities played a major role in the invention of Calculus.

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The above is the velocity-time graph of a runner. How far does this runner travel for \(16 \) seconds?

The figure’s vertical scaling is set by \( v_{s} = 8.0 \text{ m/s.} \)

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Sam throws a ball straight upward at a speed of \(30 \text{ m/s}\) from the edge of a cliff \(19 \text{ m}\) high, as shown in the above diagram. If the the velocity of the ball \(t\) seconds after the ball leaves his hand is \(v=30-10t\) (in \(\text{m/s}\)), what is the distance of the ball from the ground \(3\) seconds after the ball leaves his hand?

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