quiz04_s103_solns

# quiz04_s103_solns - y = 1/x dy/dx =-1/x 2 we obtain A = Z...

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Name: Math 1B Quiz 3 Solutions Section 103 September 26, 2009 1. Find the length of the curve x = ln(sec y ) for 0 y π/ 3. Since the function is given in x as a function of y , we may use the alternative length formula L = Z d c s 1 + ± dx dy ² 2 dy. First calculate dx dy = d dy (ln(sec y )) = 1 sec y sec y tan y = tan y. Then substitute L = Z π/ 3 0 p 1 + (tan y ) 2 dy = Z π/ 3 0 sec y dy = ln | sec y + tan y | π/ 3 0 = ln | 2 + 3 | - ln | 1 | = ln(2 + 3) 2. Gabriel’s horn is the curve y = 1 x for 1 x , rotated around the x axis. Show that the surface area of Gabriel’s horn is inﬁnite. The surface area formula for rotating around the x -axis is A = Z 2 πy ds where ds = p 1 + ( dy/dx ) 2 dx , so substituting with
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Unformatted text preview: y = 1 /x, dy/dx =-1 /x 2 , we obtain: A = Z ∞ 1 2 π 1 x s 1 + ±-1 x 2 ² 2 dx = Z ∞ 1 2 π x r 1 + 1 x 4 dx = Z ∞ 1 2 π x √ x 4 + 1 x 2 dx Now since √ x 4 +1 x 3 > 1 x for x ≥ 1, this integral must diverge by the Comparison Test. 3. Fill in the blank to complete the deﬁnition. A sequence { a m } has the limit c (and we write lim m →∞ a m = c, or a m → c as m → ∞ ) if for every ε > 0 there is a corresponding integer M such that if m > M then: | a m-c | < ε 1...
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