If $x ^ { 3 } - 2 x y + 3 y ^ { 2 } = 7$, then $\frac { d y } { d x } =$
(A) $\frac { 3 x ^ { 2 } + 4 y } { 2 x }$
(B) $\frac { 3 x ^ { 2 } - 2 y } { 2 x - 6 y }$
(C) $\frac { 3 x ^ { 2 } } { 2 x - 6 y }$
(D) $\frac { 3 x ^ { 2 } } { 2 - 6 y }$

Consider the curve defined by $2y^3 + 6x^2y - 12x^2 + 6y = 1$.
(a) Show that $\dfrac{dy}{dx} = \dfrac{4x - 2xy}{x^2 + y^2 + 1}$.
(b) Write an equation of each horizontal tangent line to the curve.
(c) The line through the origin with slope $-1$ is tangent to the curve at point $P$. Find the $x$- and $y$-coordinates of point $P$.

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 $t = 5$.

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 $y = 1$.

Consider the function $y = f(x)$ whose curve is given by the equation $2y^{2} - 6 = y\sin x$ for $y > 0$.
(a) Show that $\frac{dy}{dx} = \frac{y\cos x}{4y - \sin x}$.
(b) Write an equation for the line tangent to the curve at the point $(0, \sqrt{3})$.
(c) For $0 \leq x \leq \pi$ and $y > 0$, find the coordinates of the point where the line tangent to the curve is horizontal.
(d) Determine whether $f$ has a relative minimum, a relative maximum, or neither at the point found in part (c). Justify your answer.

Consider the curve given by the equation $6xy = 2 + y^{3}$.
(a) Show that $\frac{dy}{dx} = \frac{2y}{y^{2} - 2x}$.
(b) Find the coordinates of a point on the curve at which the line tangent to the curve is horizontal, or explain why no such point exists.
(c) Find the coordinates of a point on the curve at which the line tangent to the curve is vertical, or explain why no such point exists.
(d) A particle is moving along the curve. At the instant when the particle is at the point $\left(\frac{1}{2}, -2\right)$, its horizontal position is increasing at a rate of $\frac{dx}{dt} = \frac{2}{3}$ unit per second. What is the value of $\frac{dy}{dt}$, the rate of change of the particle's vertical position, at that instant?

Consider the curve defined by the equation $x^2 + 3y + 2y^2 = 48$. It can be shown that $\frac{dy}{dx} = \frac{-2x}{3 + 4y}$.
(a) There is a point on the curve near $(2, 4)$ with $x$-coordinate 3. Use the line tangent to the curve at $(2, 4)$ to approximate the $y$-coordinate of this point.
(b) Is the horizontal line $y = 1$ tangent to the curve? Give a reason for your answer.
(c) The curve intersects the positive $x$-axis at the point $(\sqrt{48}, 0)$. Is the line tangent to the curve at this point vertical? Give a reason for your answer.
(d) For time $t \geq 0$, a particle is moving along another curve defined by the equation $y^3 + 2xy = 24$. At the instant the particle is at the point $(4, 2)$, the $y$-coordinate of the particle's position is decreasing at a rate of 2 units per second. At that instant, what is the rate of change of the $x$-coordinate of the particle's position with respect to time?

Consider the curve $G$ defined by the equation $y ^ { 3 } - y ^ { 2 } - y + \frac { 1 } { 4 } x ^ { 2 } = 0$.
A. Show that $\frac { d y } { d x } = \frac { - x } { 2 \left( 3 y ^ { 2 } - 2 y - 1 \right) }$.
B. There is a point $P$ on the curve $G$ near $( 2 , - 1 )$ with $x$-coordinate 1.6. Use the line tangent to the curve at $( 2 , - 1 )$ to approximate the $y$-coordinate of point $P$.
C. For $x > 0$ and $y > 0$, there is a point $S$ on the curve $G$ at which the line tangent to the curve at that point is vertical. Find the $y$-coordinate of point $S$. Show the work that leads to your answer.
D. A particle moves along the curve $H$ defined by the equation $2 x y + \ln y = 8$. At the instant when the particle is at the point $( 4,1 ) , \frac { d x } { d t } = 3$. Find $\frac { d y } { d t }$ at that instant. Show the work that leads to your answer.

Let $A$, $B$ and $C$ be unknown constants. Consider the function $f(x)$ defined by
$$\begin{aligned} f(x) &= Ax^2 + Bx + C \text{ when } x \leq 0 \\ &= \ln(5x + 1) \text{ when } x > 0 \end{aligned}$$
Write the values of the constants $A$, $B$ and $C$ such that $f''(x)$, i.e., the double derivative of $f$, exists for all real $x$. If this is not possible, write ``not possible''. If some of the constants cannot be uniquely determined, write ``not unique'' for each such constant.

Let $$f(x,y) = \begin{cases} \frac{x^3 y^3}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$$ Choose the correct statement(s) from below:
(A) $f$ is continuous on $\mathbb{R}^2$;
(B) $f$ is continuous at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;
(C) $f$ is differentiable at every point of $\mathbb{R}^2 \backslash \{(0,0)\}$;
(D) $f$ is not differentiable at $(0,0)$.

Consider the part of the graph of $y ^ { 2 } + x ^ { 3 } = 15 x y$ that is strictly to the right of the $Y$-axis, i.e., take only the points on the graph with $x > 0$.
Questions
(33) Write the least possible value of $y$ among considered points. If there is no such real number, write NONE. (34) Write the largest possible value of $y$ among considered points. If there is no such real number, write NONE.

On the coordinate plane, what is the slope of the tangent line to the curve $y ^ { 3 } = \ln \left( 5 - x ^ { 2 } \right) + x y + 4$ at the point $( 2,2 )$? [3 points]
(1) $- \frac { 3 } { 5 }$
(2) $- \frac { 1 } { 2 }$
(3) $- \frac { 2 } { 5 }$
(4) $- \frac { 3 } { 10 }$
(5) $- \frac { 1 } { 5 }$

The function $$f(x) = \begin{cases} x^3 + ax & (x < 1) \\ bx^2 + x + 1 & (x \geq 1) \end{cases}$$ is differentiable at $x = 1$. What is the value of $a + b$? (Here, $a, b$ are constants.) [4 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

Find the slope of the tangent line to the curve $2 x + x ^ { 2 } y - y ^ { 3 } = 2$ at the point $( 1,1 )$. [3 points]

What is the slope of the tangent line to the curve $e ^ { x } - x e ^ { y } = y$ at the point $( 0,1 )$? [3 points]
(1) $3 - e$
(2) $2 - e$
(3) $1 - e$
(4) $- e$
(5) $- 1 - e$

A function $f ( x )$ that is differentiable on the entire set of real numbers satisfies the following conditions. What is the value of $f ( - 1 )$? [4 points] (가) For all real numbers $x$, $$2 \{ f ( x ) \} ^ { 2 } f ^ { \prime } ( x ) = \{ f ( 2 x + 1 ) \} ^ { 2 } f ^ { \prime } ( 2 x + 1 ).$$ (나) $f \left( - \frac { 1 } { 8 } \right) = 1 , f ( 6 ) = 2$
(1) $\frac { \sqrt [ 3 ] { 3 } } { 6 }$
(2) $\frac { \sqrt [ 3 ] { 3 } } { 3 }$
(3) $\frac { \sqrt [ 3 ] { 3 } } { 2 }$
(4) $\frac { 2 \sqrt [ 3 ] { 3 } } { 3 }$
(5) $\frac { 5 \sqrt [ 3 ] { 3 } } { 6 }$

What is the slope of the tangent line to the curve $x ^ { 2 } - 3 x y + y ^ { 2 } = x$ at the point $( 1,0 )$? [3 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $\mathrm{D}_{i,j} f_k(x)$ or $\frac{\partial^2 f_k}{\partial x_i \partial x_j}(x)$, or also $f_{i,j,k}(x)$.
Justify that, for all $x$ in $\mathbb{R}^n$ and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, we have $f_{i,j,k}(x) = f_{j,i,k}(x)$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Show that for all $x$ in $\mathbb{R}^n$, and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $f_{i,j,k}(x) = -f_{i,k,j}(x)$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Deduce that, for all $x$ in $\mathbb{R}^n$ and all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, we have $f_{i,j,k}(x) = 0$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Show that there exist a real square matrix $A$ of size $n$ and an element $b$ of $\mathbb{R}^n$ such that for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$. Justify that $A$ is antisymmetric.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. What is the necessary and sufficient condition on $f$ for the Jacobian matrix $J_f(x)$ to be antisymmetric for all $x$ in $\mathbb{R}^n$?

Now $f$ is a function of class $C^1$ from $\mathbb{R}^n$ to itself.
Show that the Jacobian matrix $J_f(x)$ is symmetric for all $x$ in $\mathbb{R}^n$ if and only if there exists $g$ of class $C^2$ on $\mathbb{R}^n$ with values in $\mathbb{R}$ such that $$\forall x \in \mathbb{R}^n, \forall i \in \llbracket 1, n \rrbracket, \quad f_i(x) = \mathrm{D}_i g(x)$$
One may consider the map $g$ defined by $g(x) = \sum_{i=1}^n x_i \int_0^1 f_i(tx)\, \mathrm{d}t$ and express $\mathrm{D}_i g(x)$ as a single integral.

Let $\lambda \in \mathbb{R}^*$ and $(x_0, y_0) \in \mathbb{R}^2$ be fixed. We define: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ Let $g : \Omega_{x_0, y_0, \lambda} \rightarrow \mathbb{R}$ be a harmonic application.
Show that the application $(x,y) \mapsto g\left(\lambda(x,y) + (x_0, y_0)\right)$ is harmonic on $\Omega$.

Show that the applications $$h_1 : \left|\begin{array}{rll} \mathbb{R}^2 \backslash \{(0,0)\} & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \ln(x^2 + y^2) \end{array}\right. \quad \text{and} \quad h_2 : \left|\begin{array}{rll} \mathbb{R}^2 \backslash \{(0,0)\} & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \dfrac{1}{x^2 + y^2} \end{array}\right.$$ are harmonic.