alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

The sign of tan 1 i s 0 negative the f ir st deri v a

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Unformatted text preview: y , by looking at the picture that describes the flight path, we can see that the height increases, reaches a maximum and then decreases. Height function y(t) Analyticall ytically Anal ytically , by looking at the expression of the HEIGHT function y(t), how can we tell when the height is increasing, decreasing and at a maximum ? With a little thinking you can infer that when the height is maximum the ver tical speed must be zero. Conversely, your friend, by closely watching the ver tical speedometer, can infer that when the ver tical speed is zero (while he is still in the air) he has reached the maximum height. Anal yticall y , can you and your friend tell at which instant in time the Analyticall ytically maximum height is reached ? Analyticall ytically Anal ytically , can your friend find out the maximum height ? 109 Height p2 p1 h1 p3 = maximum h3 p4 h2 h4 φ1 (0, 0) t0 φ4 t1 t2 t3 t4 tn Time The expression for the INSTANTANEOUS RATE OF CHANGE in height is d y(t) ------- = y ’(t) = u . sin θ -- g t -dt We know from obser vation that the INSTANTANEOUS RATE OF CHANGE in height or ver tical speed i s not the same at different INSTANTS t 1, t 2 , t 3 ertical and t 4 i n time and this is reflected in the expression. When we evaluate this expression for the INSTANTANEOUS RATE OF CHANGE in height at any par ticular instant ti w e get a numerical value = tan φi . We tan shall now learn how to interpret the SIGN of this value. We shall make use of the SIGN of the tan φi in the 4 quadrants with the angle φi tan measured in the counter-clockwise (positive) direction. sin all and the fact that tan cos 110 y ,(ti ) = tan φi Analytical geometr eometry From the Analytical geometr y point of view, if we draw the tangent to the tangent Anal cur ve at the point P 1 i t will intersect the x-axis at an acute angle. Let us call this angle φ 1 . The sign of tan φ 1 i s > 0, positive. The f ir st deri v a ti v e of y(t) evaluated at time instant t 1 irst deriv tiv y ’(t 1 ) = tan φ 1 = u . sin θ -- g t1 must be pos...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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