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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: 5쪽 큰 문항수: 9문항
1. [10pts] Use the
definition of a limit to show thatlim
→
2. [12pts] Evaluate the integral.
ln
2
3. [11pts] Suppose that a differentiable function
satisfies
and
. Find the linearization
of
at
. 4. [12pts] Find
by implicit differentiation. (a)cosh
ln
tanh
(b)sin
ln
3
5. [12pts] Give answers to the following questions. (a) Let
and
be real numbers such that
. Show that there exist real numbers
and
such that
cosh
.(b) Use 5-(a) to find the hyperbolic function which is equals to
.6. [10pts] Give answers to the following questions. (a) Find the partial fraction decomposition of the function
(b) Use 6-(a) to evaluate the integral
4
7. [11pts] Find the limit, if it exists. If the limit does not exist,
explain why.
lim
→
arctan
tanh
8. [11pts] Suppose that a differentiable function
satisfies
and ′ ′
for any ≤
. Use the Mean Value Theorem to prove that
on the interval∞
.[Be careful NOT to use the inequality
′′
to prove the statement because we do NOT know whether ′′
exists.]5
9. [11pts] Determine if the following improper integral converges:
