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미적분학 및 연습 I - 생과대 (MATH 161) 제1차 시험 (2016년도 2학기)
학과 : 학번 : 이름 :모든 풀이 과정은 알아볼 수 있도록 정성을 들여 작성한다. 칸이 부족한 경우, 각 문제 바로 뒷면에 이어 작성한다. 1. [12점] Use the definition of a limit to prove the statement:
lim
→
.
2. [13점] Let be a one-to-one differentiable function on such that
and tan .
If is the differentiable inverse function of , find the linearization of at . Give reasons for your answer.
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학과 : 학번 : 이름 :
3. [12점] Find the limits. (각 답안 경계를 확실히 구분할 것) (a)
lim
→ sinh tanh cot (b)lim
→ tanh
sin 4. [13점] Two runners
and
start a race at the same time on a straight road. If the positions of
and
after time are
and
, respectively, exactly how manytimes will they meet during the race? Give reasons for your answer. (One cannot use the shapes of functions to explain answer.)
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학과 : 학번 : 이름 :
5. [13점] Let be a differentiable function and let
with the properties that
and ″ . If , Determine . Give reasons for your answer.
6. [13점] Let be a continuous function on and let
∈ be a constant. Prove that there is a ∈ such that
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학과 : 학번 : 이름 :
7. Evaluate the integrals. (각 답안 경계를 확실히 구분할 것) (a) [8점]
tan tan (b) [8점]
