An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical shape. The volume of the sphere is decreasing at a constant rate of $2 \pi$ cubic meters per hour. At what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment when the radius is 5 meters? (Note: For a sphere of radius $r$, the surface area is $4 \pi r ^ { 2 }$ and the volume is $\frac { 4 } { 3 } \pi r ^ { 3 }$.)
(A) $\frac { 4 \pi } { 5 }$
(B) $40 \pi$
(C) $80 \pi ^ { 2 }$
(D) $100 \pi$

The radius of a right circular cylinder is increasing at a rate of 2 units per second. The height of the cylinder is decreasing at a rate of 5 units per second. Which of the following expressions gives the rate at which the volume of the cylinder is changing with respect to time in terms of the radius $r$ and height $h$ of the cylinder? (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r ^ { 2 } h$.)
(A) $- 20 \pi r$
(B) $- 2 \pi r h$
(C) $4 \pi r h - 5 \pi r ^ { 2 }$
(D) $4 \pi r h + 5 \pi r ^ { 2 }$

34. In the figure above, $P Q$ represents a 40-foot ladder with end $P$ against a vertical wall and end $Q$ on level ground. If the ladder is slipping down the wall, what is the distance $R Q$ at the instant when $Q$ is moving along the ground $\frac { 3 } { 4 }$ as fast as $P$ is moving down the wall?
(A) $\frac { 6 } { 5 } \sqrt { 10 }$
(B) $\frac { 8 } { 5 } \sqrt { 10 }$
(C) $\frac { 80 } { \sqrt { 7 } }$
(D) 24
(E) 32

(a) A point moves on the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ so that its $y$-coordinate is increasing at a constant rate of 4 units per second. How fast is the $x$-coordinate changing when $x = 4$ ? (b) For what values of $k$ will the line $2 x + 9 y + k = 0$ be normal to the hyperbola $3 x ^ { 2 } - y ^ { 2 } = 23$ ?

A container has the shape of an open right circular cone. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth $h$ is changing at the constant rate of $\frac { - 3 } { 10 } \text{ cm/hr}$. (Note: The volume of a cone of height $h$ and radius $r$ is given by $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.)
(a) Find the volume $V$ of water in the container when $h = 5 \text{ cm}$. Indicate units of measure.
(b) Find the rate of change of the volume of water in the container, with respect to time, when $h = 5 \text{ cm}$. Indicate units of measure.
(c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality?

Ship $A$ is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour ($\mathrm{km/hr}$). Ship $B$ is traveling due north away from Lighthouse Rock at a speed of $10\mathrm{~km/hr}$. Let $x$ be the distance between Ship $A$ and Lighthouse Rock at time $t$, and let $y$ be the distance between Ship $B$ and Lighthouse Rock at time $t$, as shown in the figure above.
(a) Find the distance, in kilometers, between Ship $A$ and Ship $B$ when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.
(b) Find the rate of change, in $\mathrm{km/hr}$, of the distance between the two ships when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.
(c) Let $\theta$ be the angle shown in the figure. Find the rate of change of $\theta$, in radians per hour, when $x = 4\mathrm{~km}$ and $y = 3\mathrm{~km}$.

Consider the curve given by $y^2 = 2 + xy$.
(a) Show that $\dfrac{dy}{dx} = \dfrac{y}{2y - x}$.
(b) Find all points $(x, y)$ on the curve where the line tangent to the curve has slope $\dfrac{1}{2}$.
(c) Show that there are no points $(x, y)$ on the curve where the line tangent to the curve is horizontal.
(d) Let $x$ and $y$ be functions of time $t$ that are related by the equation $y^2 = 2 + xy$. At time $t = 5$, the value of $y$ is 3 and $\dfrac{dy}{dt} = 6$. Find the value of $\dfrac{dx}{dt}$ at time $t = 5$.

Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius and height changing with time. (Note: The volume $V$ of a right circular cylinder with radius $r$ and height $h$ is given by $V = \pi r ^ { 2 } h$.)
(a) At the instant when the radius of the oil slick is 100 centimeters and the height is 0.5 centimeter, the radius is increasing at the rate of 2.5 centimeters per minute. At this instant, what is the rate of change of the height of the oil slick with respect to time, in centimeters per minute?
(b) A recovery device arrives on the scene and begins removing oil. The rate at which oil is removed is $R ( t ) = 400 \sqrt { t }$ cubic centimeters per minute, where $t$ is the time in minutes since the device began working. Oil continues to leak at the rate of 2000 cubic centimeters per minute. Find the time $t$ when the oil slick reaches its maximum volume. Justify your answer.
(c) By the time the recovery device began removing oil, 60,000 cubic centimeters of oil had already leaked. Write, but do not evaluate, an expression involving an integral that gives the volume of oil at the time found in part (b).

A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 feet per second. If the height of the streetlight is 15 feet, what is the rate at which the person's shadow is lengthening?
(A) $1.5 \text{ ft/sec}$
(B) $2.667 \text{ ft/sec}$
(C) $3.75 \text{ ft/sec}$
(D) $6 \text{ ft/sec}$
(E) $10 \text{ ft/sec}$

Consider the curve given by the equation $y^3 - xy = 2$. It can be shown that $\dfrac{dy}{dx} = \dfrac{y}{3y^2 - x}$.
(a) Write an equation for the line tangent to the curve at the point $(-1, 1)$.
(b) Find the coordinates of all points on the curve at which the line tangent to the curve at that point is vertical.
(c) Evaluate $\dfrac{d^2y}{dx^2}$ at the point on the curve where $x = -1$ and $y = 1$.

5. A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let $h$ be the depth of the coffee in the pot, measured in inches, where $h$ is a function of time $t$, measured in seconds. The volume $V$ of coffee in the pot is changing at the rate of $- 5 \pi \sqrt { h }$ cubic inches per second. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r ^ { 2 } h$.)
(a) Show that $\frac { d h } { d t } = - \frac { \sqrt { h } } { 5 }$.
(b) Given that $h = 17$ at time $t = 0$, solve the differential equation $\frac { d h } { d t } = - \frac { \sqrt { h } } { 5 }$ for $h$ as a function of $t$.
(c) At what time $t$ is the coffeepot empty?

The continuous function $f$ is defined on the interval $- 4 \leq x \leq 3$. The graph of $f$ consists of two quarter circles and one line segment, as shown in the figure above. Let $g ( x ) = 2 x + \int _ { 0 } ^ { x } f ( t ) d t$. (a) Find $g ( - 3 )$. Find $g ^ { \prime } ( x )$ and evaluate $g ^ { \prime } ( - 3 )$. (b) Determine the $x$-coordinate of the point at which $g$ has an absolute maximum on the interval $- 4 \leq x \leq 3$. Justify your answer. (c) Find all values of $x$ on the interval $- 4 < x < 3$ for which the graph of $g$ has a point of inflection. Give a reason for your answer. (d) Find the average rate of change of $f$ on the interval $- 4 \leq x \leq 3$. There is no point $c , - 4 < c < 3$, for which $f ^ { \prime } ( c )$ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.

The fuel consumption of a car, in miles per gallon (mpg), is modeled by $F ( s ) = 6 e ^ { \left( \frac { s } { 20 } - \frac { s ^ { 2 } } { 2400 } \right) }$, where $s$ is the speed of the car, in miles per hour. If the car is traveling at 50 miles per hour and its speed is changing at the rate of 20 miles/hour$^{2}$, what is the rate at which its fuel consumption is changing?
(A) 0.215 mpg per hour
(B) 4.299 mpg per hour
(C) 19.793 mpg per hour
(D) 25.793 mpg per hour
(E) 515.855 mpg per hour

(a) Compute $\dfrac{d}{dx}\left[\int_{0}^{e^{x}} \log(t)\cos^{4}(t)\,dt\right]$.
(b) For $x > 0$ define $F(x) = \int_{1}^{x} t\log(t)\,dt$.
i. Determine the open interval(s) (if any) where $F(x)$ is decreasing and the open interval(s) (if any) where $F(x)$ is increasing.
ii. Determine all the local minima of $F(x)$ (if any) and the local maxima of $F(x)$ (if any).

10. A spherical ball of ice of radius $20 m$ is dropped in a vat of hot water. The ice melts in such a way that (i) the shape of the ball remains spherical, and (ii) the radius of the ball decreases at a constant rate of $0.5 m s ^ { - 1 }$. At what rate does the volume of the ice ball decrease, when the radius of the ball is $15 m$ ?
(a) $100 \pi m ^ { 3 } s ^ { - 1 }$
(b) $225 \pi m ^ { 3 } s ^ { - 1 }$
(c) $450 \pi m ^ { 3 } s ^ { - 1 }$
(d) $600 \pi m ^ { 3 } s ^ { - 1 }$

(Calculus) Point $\mathrm{A}$ is on circle $\mathrm{O}$ with radius 1. As shown in the figure, for a positive angle $\theta$, two points $\mathrm{B}$ and $\mathrm{C}$ on circle $\mathrm{O}$ are chosen such that $\angle \mathrm{BAC} = \theta$ and $\overline{\mathrm{AB}} = \overline{\mathrm{AC}}$. Let $r(\theta)$ denote the radius of the inscribed circle of triangle $\mathrm{ABC}$. If $\lim_{\theta \rightarrow \pi - 0} \frac{r(\theta)}{(\pi - \theta)^2} = \frac{q}{p}$, find the value of $p^2 + q^2$. (Given: $p$ and $q$ are coprime natural numbers.) [4 points]

For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. For the point $\mathrm { R } ( 0,1 )$, let $S _ { n }$ be the area of triangle PRQ and $l _ { n }$ be the length of line segment PQ. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } ^ { 2 } } { l _ { n } }$? [4 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 4 }$
(3) 1
(4) $\frac { 3 } { 4 }$
(5) $\frac { 1 } { 2 }$

As shown in the figure, there is a rhombus ABCD with side length 1. Let E be the foot of the perpendicular from point C to the extension of segment AB, let F be the foot of the perpendicular from point E to segment AC, and let G be the intersection of segment EF and segment BC. If $\angle \mathrm { DAB } = \theta$, let the area of triangle CFG be $S ( \theta )$.
What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { S ( \theta ) } { \theta ^ { 5 } }$? (Here, $0 < \theta < \frac { \pi } { 2 }$) [4 points]
(1) $\frac { 1 } { 24 }$
(2) $\frac { 1 } { 20 }$
(3) $\frac { 1 } { 16 }$
(4) $\frac { 1 } { 12 }$
(5) $\frac { 1 } { 8 }$

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is differentiable on $\mathbb { R } \backslash \{ - 1,1 \}$ and that $$\forall x \in \mathbb { R } \backslash \{ - 1,1 \} \quad f ^ { \prime } ( x ) = \int _ { 0 } ^ { \pi } \frac { 2 x - 2 \cos \theta } { x ^ { 2 } - 2 x \cos \theta + 1 } \mathrm { ~d} \theta$$

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Justify that $\varphi$ is of class $C^1$ on $\mathbb{R}$ and, for every real $t$, give $\varphi'(t)$.

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Deduce that in a neighbourhood of 0 $$g(ta) = g(0) + t\left(a_1 \mathrm{D}_1 g(0) + a_2 \mathrm{D}_2 g(0) + \cdots + a_n \mathrm{D}_n g(0)\right) + \mathrm{o}(t)$$

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 $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Show that $f$ belongs to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ if and only if, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$r^2 \frac{\partial^2 g}{\partial r^2}(r,\theta) + \frac{\partial^2 g}{\partial \theta^2}(r,\theta) + r\frac{\partial g}{\partial r}(r,\theta) = 0$$

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Determine the radial harmonic functions of $\mathbb{R}^2 \setminus \{(0,0)\}$, that is, the functions $f$ belonging to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ such that $(r,\theta) \mapsto f(r\cos(\theta), r\sin(\theta))$ is independent of $\theta$.