Applications of Derivatives The motivation behind applying differentiation lies in the units of the derivative. For example, if s ( ) gives the height of an object in meters, where / is in seconds, then the derivative, s'(1 ) , has units of meters per second, m / s . With a little bit of familiarity with physics, we can identify this as a velocity function. A second derivative introduces a second denominator power, so that s" (( ) has units m / S , indicating an acceleration function. Aside from motion being a puzzling physical phenomenon in philosophy (see, Zeno's paradoxes), the physical problem of calculating position, velocity, and acceleration was a driving force behind the development of calculus. Exercises 10. Suppose a leaf is kicked into the air for 12 seconds so that its height, in feet above the ground, is given by the position function s (1) = -0.03/*+0.56' -3.361 +7.681, where / is seconds after it is kicked. Determine the velocity and acceleration functions and state the units of each function. (10 points) 11. Determine the position, velocity, and acceleration of the leaf in the instant the leaf is kicked. (6 points) 12. Note that s(12) =0. Interpret this fact briefly in the context of the problem. (3 points) 13. Use the critical points and points of inflection of the position function to sketch its graph on its domain on a separate page with printed gridlines. Classify any local extrema using mathematical justification. You may round values to two decimal places. (13 points)