[AP-Calculus] 고려대 미적분학 기출 (Spring, 2010) [더플러스수학]

2010학년도 봄, 고려대 미적분학 기출 Calculus Ⅰ Exam 1(Spring, 2010)

 

1. (15 pts) Let

$\displaystyle f ( x)= {\begin{cases}x ^{2} +1, ~&  \rm {if}~ x \leq 0\\x ^{3} +1, ~&\rm{if}~ x>0\end{cases}}$

Use the '$\displaystyle \epsilon - \delta$ argument' to show that $\displaystyle f$ is continuous on the real line.

 

 

 

 

 

 

 

 

 

 

2. (15 pts) Let $\displaystyle f$ be a polynomial and let {$\displaystyle x _{0} ,x _{1} , \cdots , x _{n}$} be the set of distinct real roots of the equation $\displaystyle f ( x)=0$ on $\displaystyle [0,~1]$. Prove that there exists $\displaystyle a \in [0,1~]$ such that $\displaystyle f ^{[n]} ( a)=0$, where $\displaystyle f ^{ ( n)}$ is the $\displaystyle n$-th derivative of $\displaystyle f$.

 

 

 

 

 

 

 

 

 

 

 

3. (14 pts) Find the linearization $\displaystyle L ( x)$ of $\displaystyle f ( x)$ at $\displaystyle x=1$ when $\displaystyle f ( x)$ is defined as

$\displaystyle e ^{x-1} -x ( f ( x)) ^{3} - ( x-1) ^{3} f ( x)=0$.

 

 

 

 

 

 

 

 

 

 

4. (14 pts) Let $\displaystyle f \,:\, \mathbb{R} \longrightarrow \mathbb R$  be defined by

$\displaystyle f ( x)= \int _{\cos x} ^{\sin x} {e ^{t ^{2}}} dt$

Find $\displaystyle f ' ( 0)$ and $\displaystyle f '' ( 0)$.

 

 

 

 

 

 

 

5. (14 pts) Let $\displaystyle f$ be a continuous function on the real line satisfying $\displaystyle f ( x+2)=f ( x)$, $\displaystyle f ( x)>0$ when $\displaystyle 0 < x < 1$, and $\displaystyle f ( x) < 0$ when $\displaystyle -1 < x < 0$. Define $\displaystyle F$ as

$\displaystyle F ( x)= \int _{0} ^{x} {f ( t)} dt$

Find all local maxima and local minima. Find the condition where the function $\displaystyle F$ has at least one absolute maximum on the real line.

 

 

 

 

 

6. (14 pts) Let $\displaystyle V ( a)$ be the volume of the solid generated by revolving the region bounded by $\displaystyle y=e ^{-ax}$, $\displaystyle y=0$, $\displaystyle x=f ( a)$, and $\displaystyle x=g ( a)$ about the $\displaystyle x$-axis, where $\displaystyle f$ and $\displaystyle g$ are continuous functions with $\displaystyle f ( a) < g ( a)$ for any real number $\displaystyle a$. Find $\displaystyle V ( a)$ and $\displaystyle \lim\limits_{a \rightarrow 0} {V ( a)}$.

 

 

 

 

 

 

7. (14 pts) Set $\displaystyle f ( x)=x ^{2}$ and $\displaystyle x _{0} =3$. Find $\displaystyle x _{4}$ in Newton's method. Describe the procedure.

 

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