Math 2B : Midterm # 2 Sample This exam consists of 5 questions and 90 total points. The point value of each problem is indicated. Read directions for each problem carefully. Please show all work needed to arrive at your solutions. Clearly indicate your final answers. 1.) Evaluate each of the following integrals. [10 pts. each] 𝑎) 𝑏) ln 𝑡 𝑑𝑡 𝑡! 𝑒 ! sin(3𝑥) 𝑑𝑥 sin! 𝜃 𝑑𝜃 𝑐) ! 𝑑) ! ! 1 𝑥 𝑥! − 4 𝑑𝑥 2.) Determine whether each improper integral below is convergent or divergent. Evaluate those that are convergent. [15 pts.] ! 𝑑𝑥 𝑥 ! ! ! 1 9 − 𝑥! ! ! 𝑧! 𝑑𝑥 𝑑𝑧 + 3𝑧 + 2 3.) Find the length of the curve 𝑓 𝑥 = 𝑥 ! + ! ! !"! on the interval , 2 . ! 4.) Determine whether each of the following statements is true or false. [10 pts.] i. If {𝑎! } is decreasing and 𝑎! > 0 for all 𝑛, then {𝑎! } is convergent. True False ii. If 𝑓(𝑥) ≤ 𝑔(𝑥) and iii. The integral iv. The integral v. ! !! 𝑒 𝑑𝑥 ! = ! ! ! !! ! 𝑔(𝑥)𝑑𝑥 ! diverges, then True ! 𝑓(𝑥)𝑑𝑥 also ! False 𝑑𝑥 can be solved by partial fractions. True False ! ! 𝑑𝑥 ! !! converges. ! !! 𝑒 𝑑𝑥 ! + ! !! 𝑒 𝑑𝑥. ! True False True False [10 pts.] diverges. 5.) Determine whether each of the following sequences is convergent or divergent. If a sequence is convergent, find its limit. [15 pts.] a.) 𝑎! = 𝑛 sin b.) 𝑎! = arcsin ! ! !! !!!! c.) 𝑎! = −5 + 0.9 d.) 𝑎! = 4 + −1 e.) 𝑎! = !! !!!!!" !!! ! !