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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Exercise 2 find an instant where fx x 2 x h as an

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Unformatted text preview: ititve posititve. -Over the interval [t 0 , t 3 ), from t 0 u pto but not including t 3 , we note that the tangent height y(t) is incr easing We also note that the SIGN of the tangent or increasing easing. equivalently the value of the f ir st deri v a ti v e y ’(t) evaluated at any instant irst deriv tiv over [t 0 , t 3 ) is positive. We can say : if y(t) is incr easing then y ’(t) is positi v e ( >0 ). increasing positiv Conversely : if y ’(t) is positi v e then y(t) must be incr easing. positiv increasing easing. At P 3 , where the object reaches its maximum height, the tangent to the cur ve is PARALLEL to the x-axis . Hence φ 3 = 0 . The f ir st deri v a ti v e of y(t) evaluated at time instant t 3 is irst deriv tiv y ’(t 3 ) = tan φ 3 = u . sin θ -- g t 3 = 0 -- Analytical geometr eometry tangent From the Analytical geometr y point of view, if we draw the tangent to the Anal cur ve at the point P 4 i t will intersect the x-axis at an obtuse angle φ 4 . The sign of tan φ 4 i s < 0, negative. The f ir st deri v a ti v e of y(t) evaluated at time instant t 4 irst deriv tiv y ’(t 4 ) = tan φ 4 = u . sin θ -- g t 4 must be negative negative. -Over the interval ( t 3 , t n ], from but not including t 3 t o t n, we note that the height y(t) is decr easing We also note that the SIGN of the tangent or decreasing easing. tangent equivalently the value of the f ir st deri v a ti v e y ’(t) evaluated at any instant irst deriv tiv over ( t 3 , t n ] i s negative negative. 111 We can say : if y(t) is decr easing then y ’(t) is ne g a ti v e (< 0). decreasing neg tiv easing. Conversely : if y ’(t) is ne g ati v e then y(t) must be decr easing. neg tiv decreasing In general, let f(x) be a function and a the point where f(x) is maximum maximum. At a maximum point a the value of the function is greater than the values around it. Let δ x be a small change in x. Then f (a -- δ x) < f(a) and f (a + δ x) < f(a) a a a a At a maximum point f ’(x) is b...
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