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83 17 units of measure a function may be dependent on

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Unformatted text preview: peed, i.e. ver tical acceleration, which in our case is g for gravity. height y(t) = u . s i n θ . t -- 1 g t 2 -2 (0,0) T T/ 2 t y ’(t) = u . s i n θ -- g t -- g = gravity y ”(t) = -- g (the minus sign indicates the downward direction of gravity) d2 d 2y 12 = dt2 (u . s i n θ . t -- -- g t ) = -- g 2 dt opera As with any calculation there are two par ts: the operation p ar t and the oper measure units of measur e p ar t. Here the height is measured in [meters] and time t is measured in [secs]: Vertical acceleration : y ”(t) = [meters] y = vertical position or height [meter s] [meters sec] y ’ = s peed [meter s / sec] sec [meters sec y ” = a cceleration [meter s / sec 2 ] se 81 Deriv tiv inverse Deri v a ti v e of the inv er se function inverse Let y = x2 . Then the inver se function is : x = y ½. We may now differentiate the in inverse inver se function x with respect to y to get dx/dy = 1/ 2y ½. Alternatively, we know that dy/dx = 2x. Hence: 1 dx 1 = = dy/ dx 2x dy inverse Now we may substitute x = y ½ to get the derivative of the inver se function x with in respect to y : dx/ dy = 1/ 2y ½ . So the derivative of the inver se function x is: inverse in dx 1 = dy/ dx dy Diff erentia entiation parametric or orm Dif f er entia tion in par ametric ffor m We have now learned to find the INSTANTANEOUS RATE OF CHANGE of a function. The reader should ask the question: what about the INSTANTANEOUS RATE OF CHANGE of one function with respect to another function of the same parameter ? A plane flying horizondally at a speed of 600 kms / hour descends at the rate of 1200 feet / minute. What is the glide ratio ? glide horizontal speed u’(t) = 600 kms / hour ≅ 500 feet / sec speed downward speed v’(t) = 1200 feet / minute = 20 feet / sec speed glide ratio = horizontal speed u’(t) speed = downward speed v’(t) speed 500 feet / sec = 25 20 feet / sec That is to say, for every 25 feet forward the plane looses height (descends) by 1 foot. Notice the absence of the time...
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