alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

If f x is negative then fx is decreasing decreasing

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Unformatted text preview: efor zer neg tiv after. ero positi v e bef or e , z er o , ne g a ti v e after. What can you say about f ”(a ) ? a Exercise Exercis e 1: Given that you know the initial velocity u and the angle of projection θ , how long will it take to reach a maximum height ? Exercise 2: What is the duration of flight ? Exercise 3: What is the range ? 112 2 2. Decr easing, MINIMUM, Incr easing y P1 P3 P2 φ1 x1 x2 = 0 φ3 x3 x Consider the function y = x 2 dy ---- = y ’(x) = 2x dx The tang ent to the cur ve of y = x 2 a t P 1 w ill inter sect the x- axis tangent at obtuse angle φ 1 . The SIGN of tan φ 1 i s < 0, negative. The f ir st deri v a ti v e of y = x 2 e valuated at instant x 1 irst deriv tiv y ’(x 1 ) = tan φ 1 = 2 x 1 must be negative. negative. decreasing easing. When x < 0 we note that the function y = x 2 i s decr easing. We also note that the SIGN of the tangent or equivalently the value of the f ir st deri v a ti v e tangent irst deriv tiv y ’(x) evaluated at any instant x < 0 is negative. 113 We can say: if y(x) is decr easing then y ’(x) is ne g a ti v e (<0). decreasing neg tiv decreasing easing. Conversely : if y ’(x) is ne g a ti v e then y(x) must be decr easing neg tiv At P 2 , where y = x 2 i s minim um the tangent to the cur ve is PARALLEL to minimum um, tangent the x-axis. Hence φ 2 = 0 . So the f ir st deri v a ti v e of y = x 2 e valuated at irst deriv tiv instant x 2 is : y ’(x 2 ) = tan φ 2 = 2 x 2 = 0 The tang ent to the cur ve of y = x 2 a t P 3 w ill inter sect the x-axis tangent at acute angle φ 3 . positiv The SIGN of tan φ 3 i s > O, positi v e . The f ir st deri v a ti v e of y = x 2 e valuated at instant x 3 irst deriv tiv y ’(x 3 ) = tan φ 3 = 2 x 3 must be positive positive. increasing easing. When x > 0 we note that the function y = x 2 i s incr easing We also note irst deriv tiv that the SIGN of the tangent or equivalently the value of the f ir st deri v a ti v e y ’(x) evaluated at any instant x > 0 is positive positive. We can say : if y(x) is incr easing then y ’(x) is posi...
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