epe3 - R 1 x 2 4 x 5 dx is completing the square If f is...

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Exam III Apr 1, 2003 MAC 2312 S Hudson Name Show all your work. Your method may count as much as your ﬁnal answer. Use the space provided, or leave a note. Don’t use a calculator or your own extra paper. 1) Find the arc length of the curve y = x 3 6 + 1 2 x from x = 1 to x = 3. 2) R tan 5 t sec 4 t dt 1

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3) R x 2 - 9 x dx 4) R 5 x - 4 x 2 - 4 x dx 5) R 2 x +5 x 2 +4 x +5 dx 2
6) R 0 xe - 3 x dx 7) Choose ONE, and explain thoroughly: a) State and prove the Divergence Test. b) Prove that lim n →∞ 1 /n = 0 (using the ± method). c) State and prove the M-test (you may quote theorems about sequences). 3

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8) CHOOSE ONE of these to do; A) R e x sin x dx B) Find the reduction formula for R tan n x dx using I.B.P. 9) (20 pts total) Answer True or False: A good ﬁrst step in solving R x 3 x 2 - 4 dx is a trig substitution. A good ﬁrst step in solving R tan - 1 x dx is to integrate by parts. A good ﬁrst step in solving
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Unformatted text preview: R 1 x 2 +4 x +5 dx is completing the square. If f is continuous and lim x →∞ f ( x ) = 0 then R ∞ 1 f ( x ) dx converges. If f is even and R ∞ f ( x ) dx converges, then so does R ∞-∞ f ( x ) dx . If a n converges, then lim a n-a n-1 = 0. The harmonic series converges to zero. If ∑ a n converges to S then the sequence of its partial sums converges to S . The series 1-1 + 1-1 + 1-1 + . . . converges to 0. Every increasing sequence that is bounded above converges. BONUS (5 pts) Compute the sum of the series 1 / 2+2 / 4+3 / 8+4 / 16+ . . . . Hint: it is not geometric or telescoping, so it is not easy. But it is related to the series 1 / 2+1 / 4+1 / 8+ . . . . 4...
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epe3 - R 1 x 2 4 x 5 dx is completing the square If f is...

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