Math XXXXXXXXXX)
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 co
ect answer without any work will receive little or no credit.
• Please write neatly. Illegible answers will be assumed to be inco
ect.
• This final exam is worth 100 points.
• Good Luck!
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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 ove
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
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