alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

83 17 units of measure a function may be dependent on

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online