Math XXXXXXXXXX)Calculus II Final ExamFall, 2022UMGCInstructions:• The deadline for this exam is 11:59 PM (ET) Tuesday 12/13/2022. Your exam must besubmitted through LEO platform.• You...

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Math 141 (6380) Calculus II Final Exam Fall, 2022 UMGC Instructions: • The deadline for this exam is 11:59 PM (ET) Tuesday 12/13/2022. Your exam must be submitted through LEO platform. • You may use your textbook on this exam. • No collaboration of any sort is allowed. • In order to receive full credit, you must show your work and carefully justify your answers. The correct answer without any work will receive little or no credit. • Please write neatly. Illegible answers will be assumed to be incorrect. • This final exam is worth 100 points. • Good Luck! *********************** 1. (30 points) Evaluate the following integrals. (a) ∫ 2/3 √ 2/3 dx x5 √ 9x2 − 1 (b) ∫ sin5 t cos4 t dt (c) ∫ sec3 θ dθ (d) ∫ x2 (x2 + 2x+ 5)(x2 − 1) dx (e) ∫ ∞ 0 x arctanx (1 + x2)2 dx 2. (4 points) Let f(x) = 3 + x2 + tan (πx 2 ) , where −1 < x="">< 1. find (f−1)′(3). 3. (10 points) find the exact length of graph of y = √ x− x2 + arcsin( √ x), where x varies over the entire domain of the function. 4. (10 points) find the exact area of the surface obtained by rotating the curve y = 1 4 x2− 1 2 lnx, with 1 ≤ x ≤ 2, about the y-axis. 5. (10 points) find the volume of the solid obtained by rotating the region bounded by y2 = x and x = 2y about the y-axis. sketch the bounded region. 6. consider the sequence { an }∞ n=1 = {√ 2, √ 2 + √ 2, √ 2 + √ 2 + √ 2, √ 2 + √ 2 + √ 2 + √ 2, · · · } . notice that this sequence can be recursively defined by a1 = √ 2, and an+1 = √ 2 + an for all n ≥ 1. (a) (5 points) show that the above sequence is monotonically increasing. hint: you can use induction. (b) (5 points) show that the above sequence is bounded above by 3. hint: you can use induction. (c) (2 points) apply the monotonic sequence theorem to show that limn→∞ an exists. (d) (5 points) find limn→∞ an. (e) (3 points) determine whether the series ∞∑ n=1 an is convergent. 7. (16 points) determine whether each of the following series is absolutely convergent, condition- ally convergent, or divergent. (a) ∞∑ n=1 ( 1− 1 n )n2 (b) ∞∑ n=1 (−1)n e1/n n ************************************************************************* 1.="" find="" (f−1)′(3).="" 3.="" (10="" points)="" find="" the="" exact="" length="" of="" graph="" of="" y="√" x−="" x2="" +="" arcsin(="" √="" x),="" where="" x="" varies="" over="" the="" entire="" domain="" of="" the="" function.="" 4.="" (10="" points)="" find="" the="" exact="" area="" of="" the="" surface="" obtained="" by="" rotating="" the="" curve="" y="1" 4="" x2−="" 1="" 2="" lnx,="" with="" 1="" ≤="" x="" ≤="" 2,="" about="" the="" y-axis.="" 5.="" (10="" points)="" find="" the="" volume="" of="" the="" solid="" obtained="" by="" rotating="" the="" region="" bounded="" by="" y2="x" and="" x="2y" about="" the="" y-axis.="" sketch="" the="" bounded="" region.="" 6.="" consider="" the="" sequence="" {="" an="" }∞="" n="1" =="" {√="" 2,="" √="" 2="" +="" √="" 2,="" √="" 2="" +="" √="" 2="" +="" √="" 2,="" √="" 2="" +="" √="" 2="" +="" √="" 2="" +="" √="" 2,="" ·="" ·="" ·="" }="" .="" notice="" that="" this="" sequence="" can="" be="" recursively="" defined="" by="" a1="√" 2,="" and="" an+1="√" 2="" +="" an="" for="" all="" n="" ≥="" 1.="" (a)="" (5="" points)="" show="" that="" the="" above="" sequence="" is="" monotonically="" increasing.="" hint:="" you="" can="" use="" induction.="" (b)="" (5="" points)="" show="" that="" the="" above="" sequence="" is="" bounded="" above="" by="" 3.="" hint:="" you="" can="" use="" induction.="" (c)="" (2="" points)="" apply="" the="" monotonic="" sequence="" theorem="" to="" show="" that="" limn→∞="" an="" exists.="" (d)="" (5="" points)="" find="" limn→∞="" an.="" (e)="" (3="" points)="" determine="" whether="" the="" series="" ∞∑="" n="1" an="" is="" convergent.="" 7.="" (16="" points)="" determine="" whether="" each="" of="" the="" following="" series="" is="" absolutely="" convergent,="" condition-="" ally="" convergent,="" or="" divergent.="" (a)="" ∞∑="" n="1" (="" 1−="" 1="" n="" )n2="" (b)="" ∞∑="" n="1" (−1)n="" e1/n="" n="">