[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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