MIDTERM 2 CALCULUS 2 MATH 23 FALL 218 Moday, October 22, 5:15 PM to 6:45 PM. Name PRACTICE EXAM Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will be graded o the clarity of your expositio! Date: October 22, 218. 1
8 poits 1. (4 poits) Which of the followig itegrals gives the arc legth of the fuctio f (x) = 3 cos(x) from x = to x = π/4? You do ot eed to show ay work for this problem. (A). (B). π/4 π/4 1 + 9 cos 2 (x) dx (C). 1 + 9 si 2 (x) dx (D). π/4 π/4 1 3 si(x) dx 1 9 si 2 (x) dx 1 2. (4 poits) Which of the followig itegrals gives the average value of the fuctio g(x) = x l(x) betwee x = 1 ad x = 1? You do ot eed to show ay work for this problem. (A). (B). 1 1 1 1 x 2 l(x) dx x l(x) 1 dx (C). (D). 1 1 1 1 x l(x) 9 dx x l(x) dx 2
8 poits 3. Cosider the regio bouded by the x-axis, the y-axis, the lie x = 5, ad the curve y = xe x. 3.(a). (4 poits) Which of the followig expressios gives the x-coordiate of the ceter of mass of the regio? You do ot eed to show ay work for this problem. (A). xe x dx (C). x2 e x dx (B). x2 e 2x dx 2xe x dx e x dx (D). xe x dx xe x dx x2 e x dx 3.(b). (4 poits) Which of the followig epressios gives the y-coordiate of the ceter of mass of the regio? You do ot eed to show ay work for this problem. (A). x2 e 2x 5 dx (C). x2 e x dx 2xe x dx xe x dx (B). x2 e x dx xe x dx (D). x2 e x dx 2 dx 5 xe x 3 3
1 poits 4. The solid pictured below shows a tak filled with water. Set up a itegral that represets the work required to pump all of the water out of the the top of the tak. Use ρ for desity of water ad g for the acceleratio due to gravity. Show ay work that you use to arrive at your aswer. You do ot eed to evaluate the itegral. 4 4 meters 5 meters 3 meters 4
2 poits 5. Determie if each of the followig series coverges coditioally, coverges absolutely, or diverges. Recall that a series is coditioally coverget if it coverges but does ot coverge absolutely. Show all your work ad carefully ad fully justify your reasoig, icludig amig the covergece test. 5.(a). (1 poits) ( 1) + 2 5 5
2 5.(b). (1 poits)! 6
1 poits 6. (1 poits) Determie if the followig series coverges or diverges. Show all your work ad carefully justify your reasoig. If the series coverges, give the value of its sum. 3 2 +1 5 1 6 7
1 poits 7. (1 poits) The ifiite series ( 1) 2 coverges to some real umber s. We wat to estimate s by computig the partial sum N ( 1) s N = 2 = 1 + 1 4 1 9 + 1 16 + ( 1)N N 2 How may terms should we use i the partial sum if we wat the error to be less tha or equal to.1 = 1 4? 7 8
24 poits 8. (24 poits) For each of the followig series, determie if it coverges absolutely, coverges coditioally or diverges. Recall that a series is coditioally coverget if it coverges but does ot coverge absolutely. You do ot eed to show ay work. 8 ( 1) 8.(a). =2 l (A). coverges absolutely (B). coverges coditioally ( ) 1 8.(b). cos (A). coverges absolutely (B). coverges coditioally ( 1) 8.(c). 5 6 (A). coverges absolutely (B). coverges coditioally arcta( 8.(d). 2 ) (A). coverges absolutely (B). coverges coditioally 8.(e). 1 ( π) (A). coverges absolutely (B). coverges coditioally 8.(f). cos(π) (A). coverges absolutely (B). coverges coditioally ( ) 8.(g). 7 2 (A). coverges absolutely (B). coverges coditioally 9
3 5 + 3 8.(h). 2 2 + 3 (A). coverges absolutely (B). coverges coditioally 1
1 poits 9. (1 poits) Select the best method for determiig whether the followig series coverge or diverge. You do ot eed to show ay work. 9 9.(a). 1 l() (A). alteratig series test (B). divergece test (C). itegral test (D). ratio test 9.(b). 32 1 (2 + 1)! (A). p-series (B). divergece test (C). geometric series (D). ratio test 9.(c). 53 7 4 (A). alteratig series test (B). divergece test (C). ratio test (D). limit compariso test 9.(d). ( 1) 2 2 + 3 3 (A). alteratig series test (B). divergece test (C). itegral test (D). ratio test 9.(e). 1/2 (A). alteratig series test (B). divergece test (C). p-series (D). ratio test 11