Researchers on a boat are investigating plankton cells in a sea. At a depth of $h$ meters, the density of plankton cells, in millions of cells per cubic meter, is modeled by $p ( h ) = 0.2 h ^ { 2 } e ^ { - 0.0025 h ^ { 2 } }$ for $0 \leq h \leq 30$ and is modeled by $f ( h )$ for $h \geq 30$. The continuous function $f$ is not explicitly given.
(a) Find $p ^ { \prime } ( 25 )$. Using correct units, interpret the meaning of $p ^ { \prime } ( 25 )$ in the context of the problem.
(b) Consider a vertical column of water in this sea with horizontal cross sections of constant area 3 square meters. To the nearest million, how many plankton cells are in this column of water between $h = 0$ and $h = 30$ meters?
(c) There is a function $u$ such that $0 \leq f ( h ) \leq u ( h )$ for all $h \geq 30$ and $\int _ { 30 } ^ { \infty } u ( h ) d h = 105$. The column of water in part (b) is $K$ meters deep, where $K > 30$. Write an expression involving one or more integrals that gives the number of plankton cells, in millions, in the entire column. Explain why the number of plankton cells in the column is less than or equal to 2000 million.
(d) The boat is moving on the surface of the sea. At time $t \geq 0$, the position of the boat is $( x ( t ) , y ( t ) )$, where $x ^ { \prime } ( t ) = 662 \sin ( 5 t )$ and $y ^ { \prime } ( t ) = 880 \cos ( 6 t )$. Time $t$ is measured in hours, and $x ( t )$ and $y ( t )$ are measured in meters. Find the total distance traveled by the boat over the time interval $0 \leq t \leq 1$.

The height of a tree at time $t$ is given by a twice-differentiable function $H$, where $H ( t )$ is measured in meters and $t$ is measured in years. Selected values of $H ( t )$ are given in the table below.

\begin{tabular}{ c } $t$

(years)

& 2 & 3 & 5 & 7 & 10 \hline

$H ( t )$

(meters)

& 1.5 & 2 & 6 & 11 & 15 \hline \end{tabular}
(a) Use the data in the table to estimate $H ^ { \prime } ( 6 )$. Using correct units, interpret the meaning of $H ^ { \prime } ( 6 )$ in the context of the problem.
(b) Explain why there must be at least one time $t$, for $2 < t < 10$, such that $H ^ { \prime } ( t ) = 2$.
(c) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the average height of the tree over the time interval $2 \leq t \leq 10$.
(d) The height of the tree, in meters, can also be modeled by the function $G$, given by $G ( x ) = \frac { 100 x } { 1 + x }$, where $x$ is the diameter of the base of the tree, in meters. When the tree is 50 meters tall, the diameter of the base of the tree is increasing at a rate of 0.03 meter per year. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the time when the tree is 50 meters tall?

An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function $r$, where $r ( t )$ is measured in centimeters and $t$ is measured in days. The table below gives selected values of $r ^ { \prime } ( t )$, the rate of change of the radius, over the time interval $0 \leq t \leq 12$.

\begin{tabular}{ c } $t$

(days)

& 0 & 3 & 7 & 10 & 12 \hline

$r ^ { \prime } ( t )$

(centimeters per day)

& - 6.1 & - 5.0 & - 4.4 & - 3.8 & - 3.5 \hline \end{tabular}
(a) Approximate $r ^ { \prime \prime } ( 8.5 )$ using the average rate of change of $r ^ { \prime }$ over the interval $7 \leq t \leq 10$. Show the computations that lead to your answer, and indicate units of measure.
(b) Is there a time $t , 0 \leq t \leq 3$, for which $r ^ { \prime } ( t ) = - 6$ ? Justify your answer.
(c) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of $\int _ { 0 } ^ { 12 } r ^ { \prime } ( t ) d t$.
(d) The height of the cone decreases at a rate of 2 centimeters per day. At time $t = 3$ days, the radius is 100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time $t = 3$ days. (The volume $V$ of a cone with radius $r$ and height $h$ is $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.)

The temperature of coffee in a cup at time $t$ minutes is modeled by a decreasing differentiable function $C$, where $C(t)$ is measured in degrees Celsius. For $0 \leq t \leq 12$, selected values of $C(t)$ are given in the table shown.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 3 & 7 & 12 \hline

$C(t)$

(degrees Celsius)

& 100 & 85 & 69 & 55 \hline \end{tabular}
(a) Approximate $C'(5)$ using the average rate of change of $C$ over the interval $3 \leq t \leq 7$. Show the work that leads to your answer and include units of measure.
(b) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the value of $\int_{0}^{12} C(t)\, dt$. Interpret the meaning of $\frac{1}{12} \int_{0}^{12} C(t)\, dt$ in the context of the problem.
(c) For $12 \leq t \leq 20$, the rate of change of the temperature of the coffee is modeled by $C'(t) = \frac{-24.55 e^{0.01t}}{t}$, where $C'(t)$ is measured in degrees Celsius per minute. Find the temperature of the coffee at time $t = 20$. Show the setup for your calculations.
(d) For the model defined in part (c), it can be shown that $C''(t) = \frac{0.2455 e^{0.01t}(100 - t)}{t^2}$. For $12 < t < 20$, determine whether the temperature of the coffee is changing at a decreasing rate or at an increasing rate. Give a reason for your answer.

[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.)

The distance between point O and point E is 40 m. As shown in the figure on the right, person A departs from point O and runs along the half-line OS perpendicular to segment OE at a constant speed of 3 m/s, and person B departs from point E 10 seconds after person A starts and runs along the half-line EN perpendicular to segment OE at a constant speed of 4 m/s. The angle formed by the intersection of the segment connecting the positions of persons A and B with segment OE is $\theta$ (in radians). What is the rate of change of $\theta$ at the moment 20 seconds after person A departs? [4 points]
(1) $\frac { 21 } { 290 }$ radians/second
(2) $\frac { 13 } { 290 }$ radians/second
(3) $\frac { 7 } { 290 }$ radians/second
(4) $\frac { 3 } { 290 }$ radians/second
(5) $\frac { 1 } { 290 }$ radians/second

As shown in the figure, there is a sector OAB with radius 1 and central angle $\frac { \pi } { 2 }$. Let H be the foot of the perpendicular from point P on arc AB to line segment OA, and let Q be the intersection of line segment PH and line segment AB. When $\angle \mathrm { POH } = \theta$, let $S ( \theta )$ be the area of triangle AQH. What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { S ( \theta ) } { \theta ^ { 4 } }$? (Here, $0 < \theta < \frac { \pi } { 2 }$) [4 points]
(1) $\frac { 1 } { 8 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 5 } { 8 }$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Justify that $g$ is of class $\mathcal{C}^2$ on $\mathbb{R}^{*+} \times \mathbb{R}$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Justify that $g$ is of class $\mathcal{C}^2$ on $\mathbb{R}^{*+} \times \mathbb{R}$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ For all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, express $\frac{\partial g}{\partial r}(r,\theta)$ and $\frac{\partial g}{\partial \theta}(r,\theta)$ in terms of $$\frac{\partial f}{\partial x}(r\cos(\theta), r\sin(\theta)) \quad \text{and} \quad \frac{\partial f}{\partial y}(r\cos(\theta), r\sin(\theta))$$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ For all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, express $\frac{\partial g}{\partial r}(r,\theta)$ and $\frac{\partial g}{\partial \theta}(r,\theta)$ in terms of $$\frac{\partial f}{\partial x}(r\cos(\theta), r\sin(\theta)) \quad \text{and} \quad \frac{\partial f}{\partial y}(r\cos(\theta), r\sin(\theta))$$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Also express $\frac{\partial^2 g}{\partial r^2}(r,\theta)$ and $\frac{\partial^2 g}{\partial \theta^2}(r,\theta)$ in terms of the first and second partial derivatives of $f$ at $(r\cos(\theta), r\sin(\theta))$.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Also express $\frac{\partial^2 g}{\partial r^2}(r,\theta)$ and $\frac{\partial^2 g}{\partial \theta^2}(r,\theta)$ in terms of the first and second partial derivatives of $f$ at $(r\cos(\theta), r\sin(\theta))$.

Let $t \in \mathbb{R}_{+}^{*}$ and $x \in\ ]0,1[$. Give the limit, as $\theta$ tends to zero, of $\frac{f(t+\theta, x) - f(t, x)}{\theta}$.

Let $t \in \mathbb{R}_{+}^{*}$ and $x \in\ ]0,1[$. Show that $\lim_{h \rightarrow 0} \frac{f(t, x+h) - 2f(t, x) + f(t, x-h)}{h^{2}} = \frac{\partial^{2} f}{\partial x^{2}}(x, t)$.

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change (in ft./sec.) in the length of his shadow is
(A) 2.4
(B) 3
(C) 3.6
(D) 12

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change in the length of his shadow is
(a) $3.6 \mathrm { ft } . / \mathrm { sec }$.
(b) $2.4 \mathrm { ft } . / \mathrm { sec }$.
(c) $3 \mathrm { ft } . / \mathrm { sec }$.
(d) $12 \mathrm { ft } . / \mathrm { sec }$.

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change in the length of his shadow is
(a) $3.6 \mathrm { ft } . / \mathrm { sec }$.
(b) $2.4 \mathrm { ft } . / \mathrm { sec }$.
(c) $3 \mathrm { ft } . / \mathrm { sec }$.
(d) $12 \mathrm { ft } . / \mathrm { sec }$.

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change (in ft./sec.) in the length of his shadow is
(A) 2.4
(B) 3
(C) 3.6
(D) 12

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change (in ft./sec.) in the length of his shadow is
(A) 2.4
(B) 3
(C) 3.6
(D) 12

A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base. The depth of the pond is 6 m. The square at the bottom has side length 2 m and the top square has side length 8 m. Water is filled in at a rate of $\frac { 19 } { 3 }$ cubic meters per hour. At what rate is the water level rising exactly 1 hour after the water started to fill the pond?

Consider a container of the shape obtained by revolving a segment of the parabola $x = 1 + y ^ { 2 }$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1 \mathrm {~cm} ^ { 3 } / \mathrm { s }$ into the container. Let $h ( t )$ be the height of water inside the container at time $t$. Find the time $t$ when the rate of change of $h ( t )$ is maximum.

If $g ( x ) = \int _ { \sin x } ^ { \sin ( 2 x ) } \sin ^ { - 1 } ( t ) d t$, then
[A] $g ^ { \prime } \left( \frac { \pi } { 2 } \right) = - 2 \pi$
[B] $g ^ { \prime } \left( - \frac { \pi } { 2 } \right) = 2 \pi$
[C] $g ^ { \prime } \left( \frac { \pi } { 2 } \right) = 2 \pi$
[D] $g ^ { \prime } \left( - \frac { \pi } { 2 } \right) = - 2 \pi$

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be the function defined as $f ( x ) = \sqrt { n }$ if $x \in \left[ \frac { 1 } { n + 1 } , \frac { 1 } { n } \right)$ where $n \in \mathbb { N }$. Let $g : ( 0,1 ) \rightarrow \mathbb { R }$ be a function such that $\int _ { x ^ { 2 } } ^ { x } \sqrt { \frac { 1 - t } { t } } d t < g ( x ) < 2 \sqrt { x }$ for all $x \in ( 0,1 )$. Then $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$
(A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3