2011_Tutorial_2

# 2011_Tutorial_2 - on a straight shoreline ﬂies to a point...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 2 1. For each of the following functions: (a) y = x + 1 x 2 + 1 , x [ - 3 , 3] (b) y = ( x - 1) 3 x 2 , x ( -∞ , ) determine (i) the critical points; (ii) the intervals where it is increasing and decreasing; (iii) the local and absolute extreme values. Ans. (a) local min. - 1 2( 2+1) at x = - 1 - 2 and 2 5 at x = 3; local max. 1 2( 2 - 1) at x = - 1 + 2 and - 1 5 at x = - 3. (b) local min. - 3 5 ( 2 5 ) 2 / 3 at x = 2 5 ; local max. 0 at x = 0. 2. Ornithologists have determined that some species of birds tend to avoid ﬂights over large bodies of water during daylight hours. It is believed that more energy is required to ﬂy over water than land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point B
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Unformatted text preview: on a straight shoreline, ﬂies to a point C on the shoreline, and then ﬂies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart. If it takes 1.4 times as much energy to ﬂy over water as land, ﬁnd the distance between B and C. Ans. 5.1 km. 3. Use L’Hopital’s rule to ﬁnd the following limits. (a) lim x → π/ 2 1-sin x 1 + cos2 x (b) lim x → ln(cos ax ) ln(cos bx ) , a, b > (c) lim x →∞ x tan 1 x (d) lim x → 0+ x a ln x , a > (e) lim x → 1 x 1 1-x (f) lim x → + x sin x (g) lim x → ± sin x x ¶ 1 x 2 Ans. (a) 1 4 (b) a 2 b 2 (c) 1 (d) 0 (e) e-1 (f) 1 (g) e-1 / 6...
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