-
[고려대 미적분학 기출] 2018년 2학기 미적분학1 -exam1수학과 공부이야기 2022. 2. 24. 11:17반응형
1. Using the precise definition of limit, show that
\(\displaystyle \lim\limits_{x \rightarrow 1}\frac{x^2 +1}{x^3 +x}=1\). (12pts)
2. If \(\displaystyle f(x)\) and \(\displaystyle g(x)\) are differentiable functions such that \(\displaystyle f \left( g(x)\right)=\tan x\) and \(\displaystyle f'(x)=1+f^2 (x)\), \(\displaystyle \left ( -\frac{\pi}{2}<x<\frac{\pi}{2}\right) \), find the limit \(\displaystyle \lim\limits_{x \rightarrow 0}\frac{f(x)}{g(x)}\) . (13pts)
3. Find the value \(\displaystyle k\) at which the equation \(\displaystyle e^x =k \sin x,~(x>0)\) has exactly one solution. (13pts)
4.If the average of a continuous function \(\displaystyle g(x)\) on \(\displaystyle [0,~1]\) is \(\displaystyle k\), show that there exist a solution of equation \(\displaystyle g(x)-k=0\). (12 pts)
5.Find the \(\displaystyle \lim\limits_{x \rightarrow \infty} \left( \frac{3^{\frac{1}{x}}+ 3^{-\frac{1}{x}} }{2}\right)^x \) if it exists. If not, give reasons. (12pts)
6. If \(\displaystyle f(x)=\int_0^{x} \sec \sqrt t dt\), find \(\displaystyle \lim\limits_{x \rightarrow 0+} \frac{1}{tan^{-1}(x^3 )} \int_0^ {x^2} f(t) dt\) . (13pts)
7.Find the volume of the solid \(\displaystyle S\) where its base is the region enclosed by the parabola \(\displaystyle x=1-y^2\) and the \(\displaystyle y\)-axis, and its cross-sections perpendicular to \(\displaystyle x\)-axis are squares. (12pts)
8.If \(\displaystyle A\) is the triangular region bounded by the lines \(\displaystyle y=x+1\), \(\displaystyle x=0\), and \(\displaystyle y=0\), find the volume of the solid generated by revolving the region \(\displaystyle A\) about the line \(\displaystyle y=x\) using cylindrical shells. (13pts)
https://youtu.be/lnBNSXtkdCI반응형'수학과 공부이야기' 카테고리의 다른 글
[AP-Calculus] 고려대 미적분학 기출 (Spring, 2010) [더플러스수학] (0) 2022.03.12 [더플러스수학] 증가함수(또는 감소함수)의 역함수도 증가함수(또는 감소함수)이다. (0) 2022.03.05 [카이스트 미적분기출] 2008 Midterm Exam of Calculus I [더플러스수학] (0) 2022.02.20 [수학의 기초] 삼각부등식 [더플러스수학] (0) 2022.02.13 [카이스트 방학숙제2] winter 2022 assignment 2 [더플러스수학] (0) 2022.01.27