ws_6 - 5. Prove that the function f ( x ) = x 101 + x 41 +...

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Mathematics for Engineers I (Math103). .. student-intranet/faculties/math/ 2009 - Fall/math103 Class Work Sheet 6 1 . 1. Using L’Hospital rule, evaluate each of the following limits: a) lim x 1 x 9 - 1 x 2 - 1 b) lim x π 2 + cos x 1 - sin x c) lim t 0 e 3 t - 1 t d) lim x →∞ e x x e) lim x →∞ ln(ln x ) x f) lim x 0 e x - 1 - x x 2 g) lim x 0 1 - cos x x 2 h) lim x 0 x tan - 1 (4 x ) i) lim x 0 + x ln x j) lim x 0 + sin x ln x k) lim x →∞ x tan( 1 x ) l) lim x 0 ( 1 x - csc x ) m) lim x →∞ x 2 + x - x n) lim x →∞ ( x e (1 /x ) - x ) o) lim x 1 ( 1 ln x - 1 x - 1 ) p) lim x 0 + ( x ) x 2 q) lim x 0 + ( tan2 x ) x r) lim x 0 ( 1 - 2 x ) 1 x s) lim x →∞ ( x ) ln 2 1+ln x t) lim x →∞ ( e x + x ) 1 x u) lim x 0 ( cos3 x ) 5 x 2. Prove that: lim x →∞ e x x n = for any positive integer n . 1 prepared by: Hany El-Sharkawy using “L A T E X2 ε 1
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3. Find the absolute maximum and absolute minimum values of each of the following functions in the given intervals: (i) f ( x ) = 3 x 3 - 12 x + 5; [0 , 3]. (ii) f ( x ) = x 4 - x 2 ; [ - 1 , 2] (iii) f ( x ) = sin x + cos x ; [0 , π 3 ] (iv) f ( x ) = x - 2cos x ; [ - π,π ] (v) f ( x ) = x - e x ; ( -∞ , ) (vi) f ( x ) = e x x ; x > 0 4. Show that x = 5 is a critical point for f ( x ) = 2 + ( x - 5) 3 but it is not a point of local extreme value.
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Unformatted text preview: 5. Prove that the function f ( x ) = x 101 + x 41 + x + 3 has neither a local maximum nor a local minimum. 6. Find two positive numbers whose product is 100 and whose sum is minimum. 7. A farmer wants to fence an area of 1 . 5 million square feet in a rect-angular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? 8. Find the point on the line 6 x + y = 9 that is closest to the point (-3 , 1). 9. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r . 10. Where should the point P be chosen on the line segment AB so as to maximize the angle θ ? Θ P A B 3 2 5 2...
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This note was uploaded on 01/08/2010 for the course MATH Math103 taught by Professor Dr.prof.mostafaamer during the Winter '09 term at German University in Cairo.

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ws_6 - 5. Prove that the function f ( x ) = x 101 + x 41 +...

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