2011_Tutorial_2 - on a straight shoreline, flies to a...

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

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online