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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# We can say if yt is incr easing then y t is positi v e

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Unformatted text preview: nimum φ f , g = minim um (| θ f − θ g|, 180 o − | θ f − θ g|). m inim And, if f(x) and g(x) intersect at right angles at a, then : m f . mg = − 1 This is the Analytical geometr y point of view. From the Analysis point of view, we Analytical geometr eometry Analysis Anal Anal note that at the point of intersection a : tan f ’(a) = m f = tan θ f g ’(a) = mg = tan θ g tan So, to know if the two curves f(x) and g(x) intersect at right angles at a , all we have to do is verify if : f ’(a) . g’(a) = − 1 107 21. 21 . Increasing easing, Decreasing Incr easing , MAXIMUM, Decr easing VA M OT I VAT I O N You are standing on the ground with a height measuring instrument. Your friend is projected into the air in a module equipped with only a ver tical speedometer. Using your height measuring instrument you can measure the height of the module at any instant: so many meters above the ground. Your friend, by looking at the ver tical speedometer, can know his ver tical speed at any instant: climbing at so many meters per second or descending at so many meters per second. Without communicating with your friend and using only the information you have - the height at any instant, how can you know what your friend knows the ver tical speed of the module at any instant ? Can you tell when the ver tical speed is zero ? Vice versa, how can your friend, with only the information he has - his ver tical speed at any instant, find out what you know - his height at any instant ? Can your friend tell when he has reached a maximum height and exactly what is the maximum height ? 1 g t2 (0, 0) 2-Dimension y(t) 1-Dimension u Height Height y(t) y y(t) = u . s i n θ . t -- -2 θ Range x(t) (0, 0) x y(t1) h1 t1 T /2 T t VERTICAL POSITION POSITION 108 When an object is projected into the air with a given initial velocity u and angle of projection θ , there are two functions that describe its position. A VERTICAL function y(t) which describes its HEIGHT at any INSTANT and a HORIZONTAL function x(t) which describes its RANGE at any INSTANT. We know from Dynamics that : Range function x(t) = u. cos θ . t = u. sin θ . t -- -- g t 2 -- 1 2 Geometrically Geometricall...
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