Learning objectives
Interpret the value of a derivative in an applied context as a rate, stating the units as output units per input unit and writing the meaning as a complete sentence
Relate position, velocity and acceleration along a line, find when a particle is at rest or changes direction from the sign of velocity, and compute speed as the absolute value of velocity
Decide whether a particle is speeding up or slowing down by comparing the signs of velocity and acceleration, and justify the conclusion explicitly
Set up a related-rates problem by drawing a labeled diagram, writing an equation connecting the quantities, differentiating with respect to time, and substituting the given values only after differentiating
Solve the standard related-rates configurations, including expanding spheres, draining cones, sliding ladders, lengthening shadows and objects approaching along perpendicular paths, and report the answer with its sign and units
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