Point P starts from the origin and moves on a number line. The velocity $v ( t )$ of point P at time $t ( 0 \leqq t \leqq 5 )$ is as follows. $$v ( t ) = \begin{cases} 4 t & ( 0 \leqq t < 1 ) \\ - 2 t + 6 & ( 1 \leqq t < 3 ) \\ t - 3 & ( 3 \leqq t \leqq 5 ) \end{cases}$$ For a real number $x$ with $0 < x < 3$, let $f ( x )$ be the minimum among:

• the distance traveled from time $t = 0$ to $t = x$,
• the distance traveled from time $t = x$ to $t = x + 2$,
• the distance traveled from time $t = x + 2$ to $t = 5$.

Which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
$\langle$Remarks$\rangle$ ㄱ. $f ( 1 ) = 2$ ㄴ. $f ( 2 ) - f ( 1 ) = \int _ { 1 } ^ { 2 } v ( t ) d t$ ㄷ. The function $f ( x )$ is differentiable at $x = 1$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ

There is a point P that starts from the origin at time $t = 0$ and moves on a number line. For a real number $k$, the velocity $v ( t )$ of point P at time $t$ ($t \geq 0$) is $$v ( t ) = t ^ { 2 } - k t + 4$$ Which of the following in are correct? [4 points]
ᄀ. If $k = 0$, then the position of point P at time $t = 1$ is $\frac { 13 } { 3 }$. ㄴ. If $k = 3$, then the direction of motion of point P changes once after departure. ㄷ. If $k = 5$, then the distance traveled by point P from time $t = 0$ to $t = 2$ is 3.
(1) ᄀ
(2) ᄀ, ᄂ
(3) ᄀ, ᄃ
(4) ㄴ, ㄱ
(5) ᄀ, ᄂ, ᄃ

180. The equation of motion of a body in SI is $x = 2t^3 - 6t^2 + 6t$ . In the time interval zero to 2 seconds, which statement is correct?

1. [(1)] Average acceleration is equal to zero.
2. [(2)] The direction of motion changes once.
3. [(3)] The motion is first decelerating and then accelerating.
4. [(4)] The motion is initially in the direction of the $x$-axis and then opposite to the $x$-axis.