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# hw7. - Let M t denote the maximum value of | t-x |/x as x...

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University of Engineering and Technology Lahore Department of Electrical Engineering Calculus I Problem Set #7 Due date: December 30, 2010 R eading: Thomas’s Calculus Sections 4.4 (till Theorem 5), 4.5, 4.6. Spivak Chapter 13 (pages 250-259) 1. Assume that f is an odd function and f 0 exists everywhere. Show that for every positive number b there exists a c ( - b, b ) such that f 0 ( c ) = f ( b ) /b . 2. Find the following limits (if it exsits, or state that none exists) (a) lim x →∞ ln ln x x (b) lim x →∞ ln x sin πx (c) lim x 0 + sin x ln x (d) lim x →∞ x 1 /x (e) lim x 0 + x 1 - e 2 x 3. Let f ( x ) = e - 1 /x if x 6 = 0 and g ( x ) = x . Can you find lim x 0+ f ( x ) /g ( x )? 4. Prove that among all rectangles of a given area, the square has the smallest perimeter. 5. You are told that the number x [ a, b ], where a > 0. We wish to approximate x by another number t [ a, b ] so that the relative error | t - x | /x will be as small as possible.
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Unformatted text preview: Let M ( t ) denote the maximum value of | t-x | /x as x varies from a to b . (a) Prove that this maximum occurs at one of the endpoints of [ a,b ] (b) Prove that M ( t ) is smallest when t is such that 1 t = 1 2 ( 1 a + 1 b ) 6. Let f = 1 6 x 2 + 1 12 cos 2 x . Determine the intervals in which f is monotonic (i.e. increasing or decreasing) and intervals in which f is monotonic. 7. Show that x 2 = x sin x + cos x for exactly two real values of x . 8. A particle is constrained to move along a parabola whose equation is y = x 2 . (a) At what point on the curve are the abscissa and ordinate changing at the same rate? (b) Find this rate if the motion is such that at time t we have x = sin t and y = sin 2 t . 1...
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