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Calculus I - College of LS&B (MATH 161) Test 2 (Fall 2019)
Division/Department: ID Number: Name:모든 풀이 과정은 알아볼 수 있도록 정성을 들여 작성하여라. 칸이 부족한 경우, 해당 문제 바로 뒷면에 이어 답안을 작성하여라. 전체 쪽수: 단면 4쪽 큰 문항수: 8문항
1. [12pts] Consider the parametric curve
cos cos
, cos sin
( ≤ ≤
).(a) Sketch the graph of the curve
.(b) Find the points on the curve
where the tangent line is vertical.2. [11pts] Find
if
. (Use the factorial notation if necessary.)2 Name: 3. [12pts] Let
, …
.(a) Use the inequality
(for ≠
) to show that
. (Hint:
)(b) Recall the binomial series
⋯
⋯
⋯
for a positive integer
. Show that
⋯
≤
for any positive integer
.(c) Use the result of part (b) to show that
≤
for any positive integer
.(d) Explain why the sequence
is convergent.4. [10pts] Determine the value of
for which the integral is convergent.
∞
3
Name:
5. [10pts] Determine the interval of convergence of the power
series.
∞
6. [12pts] Answer the questions and explain your reasons.
(a) Find the Maclaurin series of
.
(b) Find the sum of the series
∞
.4
Name:
7. [Short-answer questions] Write answers in the corresponding boxes. No need to explain your answer. No credits for empty box or box with incorrect answer. No partial credits.
ⓐ [3pts]
ln
ⓑ [3pts] Find all points of intersection of the polar curves
sin
and sin
.ⓒ [3pts] Find all convergent sequences.
ln
ln
,
,
arctan
,
ⓓ [6pts] Find the exact value of the sum of the series if it
converges. If the series diverges, say so.
∞
∞
8. [Short-answer questions] Mark the appropriate box with a check mark(þ). No credits for empty box or box with incorrect answer. No partial credits. [18pts]
ⓐ Determine whether the statement is true or false.
[ ※ =True, =False ] ▶
∞
is divergent for any real value of
. ▶ If
is continuous on ≥
and
∞
is convergent, then
∞
is convergent. ▶ If
is divergent, then
is divergent. ▶ If
for all
and
is convergent, then
is convergent. ⓑ Determine the convergence of series.
[※ =Absolutely convergent,
=Conditionally convergent, =Divergent] ▶
∞
: ▶
∞
: ▶
∞
: ⓒ Determine the convergence of the integrals. [※ =Convergent, =Divergent] ▶
