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) $\frac { 3 x ^ { 2 } } { 2 - 6 y }$

6. If $y ^ { 2 } - 2 x y = 16$, then $\frac { d y } { d x } =$
(A) $\frac { x } { y - x }$
(B) $\frac { y } { x - y }$
(C) $\frac { y } { y - x }$
(D) $\frac { y } { 2 y - x }$
(E) $\frac { 2 y } { x - y }$

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 $x y ^ { 2 } - x ^ { 3 } y = 6$.
(a) Show that $\frac { d y } { d x } = \frac { 3 x ^ { 2 } y - y ^ { 2 } } { 2 x y - x ^ { 3 } }$.
(b) Find all points on the curve whose $x$-coordinate is 1 , and write an equation for the tangent line at each of these points.
(c) Find the $x$-coordinate of each point on the curve where the tangent line is vertical.

Consider the curve given by $y ^ { 2 } = 2 + x y$.
(a) Show that $\frac { d y } { d x } = \frac { y } { 2 y - x }$.
(b) Find all points $( x , y )$ on the curve where the line tangent to the curve has slope $\frac { 1 } { 2 }$.
(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 $y ^ { 2 } = 2 + x y$. At time $t = 5$, the value of $y$ is 3 and $\frac { d y } { d t } = 6$. Find the value of $\frac { d x } { d t }$ at time $t = 5$.

$- 8 x ^ { 2 } + 5 x y + y ^ { 3 } = - 149$
a) $- 16 x + 5 y + 5 x a y + 3 y ^ { 2 } d y = 0$
$\operatorname { dy } \left( 5 x + 5 y ^ { 2 } \right) = 16 x - 5 y$
$d x$
$d u = 16 x - 54$
$d x 5 x + 3 y ^ { 2 }$
b) at (4,-1) $\frac { d y } { d x } = \frac { ( 6 \mid 4 ) - 5 ( - 1 ) } { 5 ( 4 ) + 3 ( 1 ) } = \frac { 64 + 5 } { 23 } = \frac { 69 } { 23 } = 3$
Equation of tangent:
$u - ( - 1 ) = 3 ( x - 4 )$
$4 + 1 = 3 x - 12$
$y = 3 x - 13$
C) $( 4,2 , k )$
$k = 3 ( 4.2 ) - 13$
$k = - 0.4$
d) $- 8 ( 4.2 ) ^ { 2 } + 5 ( 4.2 ) ( k ) + k ^ { 3 } = - 149$
$- 141.12 + 21 k + k 3 = - 149$
$k ^ { 3 } + 21 k + 7.88 = 0$
CJUsing G.D.C.
$k = - 0.373$ to 3.d.p

Consider the closed curve in the $x y$-plane given by $x ^ { 2 } + 2 x + y ^ { 4 } + 4 y = 5$. (a) Show that $\frac { d y } { d x } = \frac { - ( x + 1 ) } { 2 \left( y ^ { 3 } + 1 \right) }$. (b) Write an equation for the line tangent to the curve at the point $( - 2,1 )$. (c) Find the coordinates of the two points on the curve where the line tangent to the curve is vertical. (d) Is it possible for this curve to have a horizontal tangent at points where it intersects the $x$-axis? Explain your reasoning.

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 $G$ 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.

5. Consider the curve given by $x y ^ { 2 } - x ^ { 3 } y = 6$.
(a) Show that $\frac { d y } { d x } = \frac { 3 x ^ { 2 } y - y ^ { 2 } } { 2 x y - x ^ { 3 } }$.
(b) Find all points on the curve whose $x$-coordinate is 1 , and write an equation for the tangent line at each of these points.
(c) Find the $x$-coordinate of each point on the curve where the tangent line is vertical.

Consider the curve given by $x ^ { 2 } + 4 y ^ { 2 } = 7 + 3 x y$.
(a) Show that $\frac { d y } { d x } = \frac { 3 y - 2 x } { 8 y - 3 x }$.
(b) Show that there is a point $P$ with $x$-coordinate 3 at which the line tangent to the curve at $P$ is horizontal. Find the $y$-coordinate of $P$.
(c) Find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $P$ found in part (b). Does the curve have a local maximum, a local minimum, or neither at the point $P$ ? Justify your answer.

In the plane equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, we denote by $\mathscr{C}_{u}$ the curve representing the function $u$ defined on the interval $]0; +\infty[$ by: $$u(x) = a + \frac{b}{x} + \frac{c}{x^2}$$ where $a, b$ and $c$ are fixed real numbers.
The curve $\mathscr{C}_{u}$ passes through the points $\mathrm{A}(1; 0)$ and $\mathrm{B}(4; 0)$ and the $y$-axis and the line $\mathscr{D}$ with equation $y = 1$ are asymptotes to the curve $\mathscr{C}_{u}$.

1. Give the values of $u(1)$ and $u(4)$.
2. Give $\lim_{x \rightarrow +\infty} u(x)$. Deduce the value of $a$.
3. Deduce that, for all strictly positive real $x$, $u(x) = \frac{x^2 - 5x + 4}{x^2}$.

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) $- 1 - e$

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) $\frac { 5 } { 12 }$

$\mathbf{15}$ ▷ Let $M \in \mathcal{M}_n(\mathbf{R})$, which can also be considered as a complex matrix, and let the application $\delta_M : \mathbf{R} \rightarrow \mathbf{R},\ t \mapsto \delta_M(t) = \operatorname{det}\left(I_n + tM\right)$. Using a Taylor expansion to order 1, show that $\delta_M$ is differentiable at 0 and compute $\delta_M'(0)$.

$\mathbf{16}$ ▷ Show that the differential at the point $I_n$ of the application $\det: \mathcal{M}_n(\mathbf{R}) \rightarrow \mathbf{R}$ is the linear form ``trace''.

119- From the relation $y^2 + xy^2 + x = 7$, the value of $\dfrac{d^2y}{dx^2}$ at the point $(1,2)$ is which of the following?
(1) $\dfrac{3}{4}$ (2) $\dfrac{4}{5}$ (3) $\dfrac{6}{5}$ (4) $\dfrac{3}{2}$

120- The function $f : \mathbb{R} \to \mathbb{R}$ is twice differentiable. For every real number $x$, the function $g(x) = f(4 - x^2)$ is defined. If $f^{-1}(1) = -5$ and $f^{-1}(1) = -1$, and $f''(1) = -1$, what is the value of $g''(\sqrt{3})$?
(1) $-3$ (2) $-2$ (3) $2$ (4) $3$

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
If $f ( - 10 \sqrt { 2 } ) = 2 \sqrt { 2 }$, then $f ^ { \prime \prime } ( - 10 \sqrt { 2 } ) =$
(A) $\frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 ^ { 2 } }$
(B) $- \frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 ^ { 2 } }$
(C) $\frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 }$
(D) $- \frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 }$

The normal to the curve $x^2 + 2xy - 3y^2 = 0$ at $(1, 1)$:
(1) does not meet the curve again
(2) meets the curve again in the second quadrant
(3) meets the curve again in the third quadrant
(4) meets the curve again in the fourth quadrant

The normal to the curve $x ^ { 2 } + 2 x y - 3 y ^ { 2 } = 0$, at $( 1,1 )$
(1) Meets the curve again in the fourth quadrant
(2) Does not meet the curve again
(3) Meets the curve again in the second quadrant
(4) Meets the curve again in the third quadrant

If $x ^ { 2 } + y ^ { 2 } + \sin y = 4$, then the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( - 2,0 )$ is :
(1) $-34$
(2) 4
(3) $-2$
(4) $-32$

For the curve $C : \left( x ^ { 2 } + y ^ { 2 } - 3 \right) + \left( x ^ { 2 } - y ^ { 2 } - 1 \right) ^ { 5 } = 0$, the value of $3 y ^ { \prime } - y ^ { 3 } y ^ { \prime \prime }$, at the point $( \alpha , \alpha ) , \alpha > 0$, on $C$, is equal to $\_\_\_\_$ .

Let $\mathrm { f } ( x ) = \left\{ \begin{array} { l l } 3 x , & x < 0 \\ \min \{ 1 + x + [ x ] , x + 2 [ x ] \} , & 0 \leq x \leq 2 \\ 5 , & x > 2 , \end{array} \right.$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where f is not continuous and is not differentiable, respectively, then $\alpha + \beta$ equals