A particle moves along the $x$-axis so that its velocity at time $t$, for $0 \leq t \leq 6$, is given by a differentiable function $v$ whose graph is shown above. The velocity is 0 at $t = 0 , t = 3$, and $t = 5$, and the graph has horizontal tangents at $t = 1$ and $t = 4$. The areas of the regions bounded by the $t$-axis and the graph of $v$ on the intervals $[ 0,3 ] , [ 3,5 ]$, and $[ 5,6 ]$ are 8, 3, and 2, respectively. At time $t = 0$, the particle is at $x = - 2$.
(a) For $0 \leq t \leq 6$, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer.
(b) For how many values of $t$, where $0 \leq t \leq 6$, is the particle at $x = - 8$ ? Explain your reasoning.
(c) On the interval $2 < t < 3$, is the speed of the particle increasing or decreasing? Give a reason for your answer.
(d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer.

For $0 \leq t \leq 12$, a particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v ( t ) = \cos \left( \frac { \pi } { 6 } t \right)$. The particle is at position $x = - 2$ at time $t = 0$.
(a) For $0 \leq t \leq 12$, when is the particle moving to the left?
(b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time $t = 0$ to time $t = 6$.
(c) Find the acceleration of the particle at time $t$. Is the speed of the particle increasing, decreasing, or neither at time $t = 4$ ? Explain your reasoning.
(d) Find the position of the particle at time $t = 4$.

A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the road and the edge of the water was changing during the storm is modeled by $f ( t ) = \sqrt { t } + \cos t - 3$ meters per hour, $t$ hours after the storm began. The edge of the water was 35 meters from the road when the storm began, and the storm lasted 5 hours. The derivative of $f ( t )$ is $f ^ { \prime } ( t ) = \frac { 1 } { 2 \sqrt { t } } - \sin t$.
(a) What was the distance between the road and the edge of the water at the end of the storm?
(b) Using correct units, interpret the value $f ^ { \prime } ( 4 ) = 1.007$ in terms of the distance between the road and the edge of the water.
(c) At what time during the 5 hours of the storm was the distance between the road and the edge of the water decreasing most rapidly? Justify your answer.
(d) After the storm, a machine pumped sand back onto the beach so that the distance between the road and the edge of the water was growing at a rate of $g ( p )$ meters per day, where $p$ is the number of days since pumping began. Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water.

Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function $B$ models Ben's position on the track, measured in meters from the western end of the track, at time $t$, measured in seconds from the start of the ride. The table above gives values for $B ( t )$ and Ben's velocity, $v ( t )$, measured in meters per second, at selected times $t$.
(a) Use the data in the table to approximate Ben's acceleration at time $t = 5$ seconds. Indicate units of measure.
(b) Using correct units, interpret the meaning of $\int _ { 0 } ^ { 60 } | v ( t ) | d t$ in the context of this problem. Approximate $\int _ { 0 } ^ { 60 } | v ( t ) | d t$ using a left Riemann sum with the subintervals indicated by the data in the table.
(c) For $40 \leq t \leq 60$, must there be a time $t$ when Ben's velocity is 2 meters per second? Justify your answer.
(d) A light is directly above the western end of the track. Ben rides so that at time $t$, the distance $L ( t )$ between Ben and the light satisfies $( L ( t ) ) ^ { 2 } = 12 ^ { 2 } + ( B ( t ) ) ^ { 2 }$. At what rate is the distance between Ben and the light changing at time $t = 40$ ?

A passion fruit producer uses a water tank with volume $V$ to feed the irrigation system of his orchard. The system draws water through a hole at the bottom of the tank at a constant flow rate. With the water tank full, the system was activated at 7 a.m. on Monday. At 1 p.m. on the same day, it was found that 15\% of the water volume in the tank had already been used. An electronic device interrupts the system's operation when the remaining volume in the tank is 5\% of the total volume, for refilling.
Assuming that the system operates without failures, at what time will the electronic device interrupt the operation?
(A) At 3 p.m. on Monday.
(B) At 11 a.m. on Tuesday.
(C) At 2 p.m. on Tuesday.
(D) At 4 a.m. on Wednesday.
(E) At 9 p.m. on Tuesday.

[10 points] A spider starts at the origin and runs in the first quadrant along the graph of $y = x^{3}$ at the constant speed of 10 unit/second. The speed is measured along the length of the curve $y = x^{3}$. The formula for the curve length along the graph of $y = f(x)$ from $x = a$ to $x = b$ is $\ell = \int_{a}^{b} \sqrt{1 + f'(x)^{2}}\, dx$. As the spider runs, it spins out a thread that is always maintained in a straight line connecting the spider with the origin. What is the rate in unit/second at which the thread is elongating when the spider is at $\left(\frac{1}{2}, \frac{1}{8}\right)$?
You should use the following names for variables. At any given time $t$, the spider is at the point $\left(u, u^{3}\right)$, the length of the thread joining it to the origin in a straight line is $s$ and the curve length along $y = x^{3}$ from the origin till $\left(u, u^{3}\right)$ is $\ell$. You are asked to find $\frac{ds}{dt}$ when $u = \frac{1}{2}$. (Do not try to evaluate the integral for $\ell$: it is unnecessary and any attempt to do so will not get any credit because a closed formula in terms of basic functions does not exist.)

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

The velocity $v ( t )$ at time $t ( t \geq 0 )$ of a point P moving on a number line is $$v ( t ) = - 2 t + 4$$ What is the distance traveled by point P from $t = 0$ to $t = 4$? [3 points]
(1) 8
(2) 9
(3) 10
(4) 11
(5) 12

The position $x$ at time $t$ ($t \geq 0$) of a point P moving on a number line is $$x = - \frac { 1 } { 3 } t ^ { 3 } + 3 t ^ { 2 } + k \quad ( k \text{ is a constant} )$$ When the acceleration of point P is 0, the position of point P is 40. Find the value of $k$. [4 points]

Two points P and Q move on a number line. Their positions $x _ { 1 } , x _ { 2 }$ at time $t ( t \geq 0 )$ are $$x _ { 1 } = t ^ { 3 } - 2 t ^ { 2 } + 3 t , \quad x _ { 2 } = t ^ { 2 } + 12 t$$ Find the distance between points P and Q at the moment when their velocities are equal. [4 points]

The velocity $v ( t )$ of a point P moving on a number line at time $t ( t \geq 0 )$ is $$v ( t ) = 2 t - 6$$ If the distance traveled by point P from time $t = 3$ to time $t = k$ ($k > 3$) is 25, what is the value of the constant $k$? [4 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

The velocity $v ( t )$ and acceleration $a ( t )$ of a point P moving on a number line at time $t$ ($t \geq 0$) satisfy the following conditions. (가) When $0 \leq t \leq 2$, $v ( t ) = 2 t ^ { 3 } - 8 t$. (나) When $t \geq 2$, $a ( t ) = 6 t + 4$.
Find the distance traveled by point P from time $t = 0$ to $t = 3$. [4 points]

At time $t = 0$, two points P and Q simultaneously depart from the origin and move on a number line. Their velocities at time $t$ ($t \geq 0$) are respectively $$v_1(t) = t^2 - 6t + 5, \quad v_2(t) = 2t - 7$$ Let $f(t)$ denote the distance between points P and Q at time $t$. The function $f(t)$ increases on the interval $[0, a]$, decreases on the interval $[a, b]$, and increases on the interval $[b, \infty)$. Find the distance traveled by point Q from time $t = a$ to time $t = b$. (Here, $0 < a < b$) [4 points]
(1) $\frac{15}{2}$
(2) $\frac{17}{2}$
(3) $\frac{19}{2}$
(4) $\frac{21}{2}$
(5) $\frac{23}{2}$

A point P starts at time $t = 0$ and moves on a number line. At time $t$ ($t \geq 0$), its position $x$ is given by $$x = t^{3} - \frac{3}{2}t^{2} - 6t$$ What is the acceleration of point P at the time when its direction of motion changes after starting? [4 points]
(1) 6
(2) 9
(3) 12
(4) 15
(5) 18

158- The equation of motion of a particle in SI is $x = \dfrac{2}{3}t^3 - 6t^2 + 20t$. What is the minimum speed (in meters per second) that this particle reaches along its path?
(1) zero (2) $1$ (3) $2$ (4) $4$

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.

159. According to the position–time graph below, the motion of the particle is uniform. The speed of the particle at moment $t = 8\,\text{s}$ is how many meters per second?
\begin{minipage}{0.45\textwidth} [Figure: $x$(m) vs $t$(s) graph. The curve starts at $x=0$, goes to a minimum near $t=4\,\text{s}$, then rises to $x=12$ and continues increasing in a parabolic (uniform acceleration) shape.] \end{minipage} \begin{minipage}{0.45\textwidth}

• [(1)] $3$
• [(2)] $4$
• [(3)] $6$
• [(4)] $12$

\end{minipage}

45-- The position--time equation of a particle in SI units is $x = 2t^2 - 12t + 8$, at moment $t = 0$. How many seconds after $t = 0$ is the particle's distance from the origin less than or equal to 8 meters?
(1) $2$ (2) $3$ (3) $4$ (4) $6$
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The co-ordinates of a moving particle at any time '$t$' are given by $x = \alpha t^{3}$ and $y = \beta t^{3}$. The speed of the particle at time '$t$' is given by
(1) $3t\sqrt{\alpha^{2} + \beta^{2}}$
(2) $3t^{2}\sqrt{\alpha^{2} + \beta^{2}}$
(3) $t^{2}\sqrt{\alpha^{2} + \beta^{2}}$
(4) $\sqrt{\alpha^{2} + \beta^{2}}$

The relation between time $t$ and distance x is $\mathrm{t} = a\mathrm{x}^2 + \mathrm{bx}$ where a and b are constants. The acceleration is
(1) $-2abv^2$
(2) $2bv^3$
(3) $-2av^3$
(4) $2av^2$

A particle located at $x = 0$ at time $t = 0$, starts moving along the positive $x$-direction with a velocity '$v$' that varies as $v = \alpha \sqrt{x}$. The displacement of the particle varies with time as
(1) $t^{3}$
(2) $t^{2}$
(3) $t$
(4) $t^{1/2}$

An object, moving with a speed of $6.25 \mathrm{~m}/\mathrm{s}$, is decelerated at a rate given by :
$$\frac{\mathrm{dv}}{\mathrm{dt}} = -2.5\sqrt{\mathrm{v}}$$
where $v$ is the instantaneous speed. The time taken by the object, to come to rest, would be:
(1) 2 s
(2) 4 s
(3) 8 s
(4) 1 s

A particle is moving with speed $v = b \sqrt { x }$ along positive $x$ - axis. Calculate the speed of the particle at time $t = \tau$ (assume that the particle is at origin at $t = 0$ )
(1) $b ^ { 2 } \tau$
(2) $\frac { b ^ { 2 } \tau } { \sqrt { 2 } }$
(3) $\frac { b ^ { 2 } \tau } { 2 }$
(4) $\frac { b ^ { 2 } \tau } { 4 }$

A particle is moving with a velocity $\vec { v } = K y \hat { i } + x \hat { j }$, where $K$ is a constant. The general equation for its path is:
(1) $y ^ { 2 } = x +$ constant
(2) $x y =$ constant
(3) $y = x ^ { 2 } +$ constant
(4) $y ^ { 2 } = x ^ { 2 } +$ constant

A ball is thrown upward with an initial velocity $V _ { 0 }$ from the surface of the earth. The motion of the ball is affected by a drag force equal to $\mathrm { m } \gamma v ^ { 2 }$ (where m is mass of the ball, $v$ is its instantaneous velocity and $\gamma$ is a constant). Time taken by the ball to rise to its zenith is:
(1) $\frac { 1 } { \sqrt { \gamma g } } \ln \left( 1 + \sqrt { \frac { \gamma } { g } } \mathrm {~V} _ { 0 } \right)$
(2) $\frac { 1 } { \sqrt { \gamma g } } \tan ^ { - 1 } \left( \sqrt { \frac { \gamma } { g } } \mathrm {~V} _ { 0 } \right)$
(3) $\frac { 1 } { \sqrt { \gamma \mathrm {~g} } } \sin ^ { - 1 } \left( \sqrt { \frac { \gamma } { \mathrm {~g} } } \mathrm {~V} _ { 0 } \right)$
(4) $\frac { 1 } { \sqrt { 2 \gamma g } } \tan ^ { - 1 } \left( \sqrt { \frac { 2 \gamma } { g } } \mathrm {~V} _ { 0 } \right)$