Due Monday Sep. 17 by 4:00 pm MA 301 Calculus III (Fall 2012) Homework 2 Find th

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Due Monday Sep. 17 by 4:00 pm MA 301 Calculus III (Fall 2012) Homework 2 Find the arc length of the graph y = 9 - x2 over the interval [0,3]. Find the area of the surface formed by revolving the graph of y = 2e-x on the interval [0, infinity) about the x-axis. Write the first five terms of the sequence. an = (-1)n+1(2/n), (b)an = 4n/n!, (c)a1 = 4,akappa+1 = (kappa+1/2)a kappa. Find the limit of the sequence. (a)an = 5n/ n2+3, (b)an = cos 2/n, (c)an = n sin 1/n. Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. (a)an = l + (-1), (b)an = (2/3)n, (c)an=sin n/n. Determine whether the sequence with the given nth term is monotonic. Discuss the boundedness of the sequence. (a)an =n/n+1(b)an = (3/2), (c) an = (-1)n(1/n). Use Theorem 9.5 (Bounded and monotonic sequence must be converge) to show that the sequence with the given nth term converges. (a)an = 3 + 1/n, (b)an = 5-2/n.

Explanation / Answer

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1. y = (9-x^2)

dy/dx = -2x/(2(9-x^2)) = -x/(9-x^2)

(dy/dx)2 = x2/(9-x2)

1+(dy/dx)2 = 1 + x2/(9-x2) = 9/(9-x2)

(1+(dy/dx)2 ) = 3/(9-x2)

Arc length = integral from 0 to 3 (1+(dy/dx)2 )

= integral of 3/(9-x2)

= 3 sin-1(x/3) with limits 0 to 3

= 3 [ sin-1(1) - sin-1(0)] = 3 [ /2 - 0 ] = 3

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