[고려대 미적분학 기출] 2018년 2학기 미적분학1 -exam1

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)

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