$$f ( x ) = \begin{cases} \frac { 2 } { x } & \text { for } x < - 1 \\ x ^ { 2 } - 3 & \text { for } - 1 \leq x \leq 2 \\ 4 x - 3 & \text { for } x > 2 \end{cases}$$
Let $f$ be the function defined above. At what values of $x$, if any, is $f$ not differentiable?
(A) $x = - 1$ only
(B) $x = 2$ only
(C) $x = - 1$ and $x = - 2$
(D) $f$ is differentiable for all values of $x$.

Suppose that the function $f$ has a continuous second derivative for all $x$, and that $f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 3$, and $f ^ { \prime \prime } ( 0 ) = 0$. Let $g$ be a function whose derivative is given by $g ^ { \prime } ( x ) = e ^ { - 2 x } \left( 3 f ( x ) + 2 f ^ { \prime } ( x ) \right)$ for all $x$.
(a) Write an equation of the line tangent to the graph of $f$ at the point where $x = 0$.
(b) Is there sufficient information to determine whether or not the graph of $f$ has a point of inflection when $x = 0$ ? Explain your answer.
(c) Given that $g ( 0 ) = 4$, write an equation of the line tangent to the graph of $g$ at the point where $x = 0$.
(d) Show that $g ^ { \prime \prime } ( x ) = e ^ { - 2 x } \left( - 6 f ( x ) - f ^ { \prime } ( x ) + 2 f ^ { \prime \prime } ( x ) \right)$. Does $g$ have a local maximum at $x = 0$ ? Justify your answer.

The function $f$ is defined by $f ( x ) = \frac { x } { x + 2 }$. What points $( x , y )$ on the graph of $f$ have the property that the line tangent to $f$ at $( x , y )$ has slope $\frac { 1 } { 2 }$ ?
(A) $( 0,0 )$ only
(B) $\left( \frac { 1 } { 2 } , \frac { 1 } { 5 } \right)$ only
(C) $( 0,0 )$ and $( - 4,2 )$
(D) $( 0,0 )$ and $\left( 4 , \frac { 2 } { 3 } \right)$
(E) There are no such points.

The graph of $y = e ^ { \tan x } - 2$ crosses the $x$-axis at one point in the interval $[ 0,1 ]$. What is the slope of the graph at this point?
(A) 0.606
(B) 2
(C) 2.242
(D) 2.961
(E) 3.747

Let $f$ be the function whose graph goes through the point $(3, 6)$ and whose derivative is given by $f'(x) = \frac{1 + e^x}{x^2}$.
(a) Write an equation of the line tangent to the graph of $f$ at $x = 3$ and use it to approximate $f(3.1)$.
(b) Use Euler's method, starting at $x = 3$ with a step size of 0.05, to approximate $f(3.1)$. Use $f''$ to explain why this approximation is less than $f(3.1)$.
(c) Use $\int_{3}^{3.1} f'(x)\, dx$ to evaluate $f(3.1)$.

The function $f$ is defined by $f ( x ) = \frac { x } { x + 2 }$. What points $( x , y )$ on the graph of $f$ have the property that the line tangent to $f$ at $( x , y )$ has slope $\frac { 1 } { 2 }$ ?
(A) $( 0,0 )$ only
(B) $\left( \frac { 1 } { 2 } , \frac { 1 } { 5 } \right)$ only
(C) $( 0,0 )$ and $( - 4,2 )$
(D) $( 0,0 )$ and $\left( 4 , \frac { 2 } { 3 } \right)$
(E) There are no such points.

We consider the curve $\mathscr{C}$ with equation $y = \mathrm{e}^{x}$.
For every strictly positive real $m$, we denote by $\mathscr{D}_{m}$ the line with equation $y = mx$.

1. In this question, we choose $m = \mathrm{e}$.
   Prove that the line $\mathscr{D}_{\mathrm{e}}$, with equation $y = \mathrm{e}x$, is tangent to the curve $\mathscr{C}$ at its point with abscissa 1.
2. Conjecture, according to the values taken by the strictly positive real $m$, the number of intersection points of the curve $\mathscr{C}$ and the line $\mathscr{D}_{m}$.
3. Prove this conjecture.

Let $f$ and $g$ be the functions defined on the set $\mathbb { R }$ of real numbers by
$$f ( x ) = \mathrm { e } ^ { x } \quad \text { and } \quad g ( x ) = \mathrm { e } ^ { - x } .$$
We denote by $\mathscr { C } _ { f }$ the representative curve of function $f$ and $\mathscr { C } _ { g }$ that of function $g$ in an orthonormal coordinate system of the plane.
For every real number $a$, we denote by $M$ the point of $\mathscr { C } _ { f }$ with abscissa $a$ and $N$ the point of $\mathscr { C } _ { g }$ with abscissa $a$.
The tangent line to $\mathscr { C } _ { f }$ at $M$ intersects the $x$-axis at $P$, the tangent line to $\mathscr { C } _ { g }$ at $N$ intersects the $x$-axis at $Q$.
Questions 1 and 2 can be treated independently.

1. Prove that the tangent line to $\mathscr { C } _ { f }$ at $M$ is perpendicular to the tangent line to $\mathscr { C } _ { g }$ at $N$.
2. a. What can be conjectured about the length $PQ$? b. Prove this conjecture.

Question 1: Consider the function $g$ defined on $]0;+\infty[$ by $g(x) = x^2 + 2x - \frac{3}{x}$. An equation of the tangent line to the curve representing $g$ at the point with abscissa 1 is:

a. $y = 7(x-1)$

b. $y = x-1$

c. $y = 7x+7$

d. $y = x+1$

The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We can assert that: a. $f ^ { \prime } ( - 0.5 ) = 0$ b. if $x \in ] - \infty ; - 0.5 \left[ \right.$, then $f ^ { \prime } ( x ) < 0$ c. $f ^ { \prime } ( 0 ) = 15$ d. the derivative function $f ^ { \prime }$ does not change sign on $\mathbb { R }$.

We consider the function $g$ defined on the interval $] 0 ; + \infty [$, by
$$g ( x ) = \frac { \ln ( x ) } { 1 + x ^ { 2 } }$$
We admit that $g$ is differentiable on the interval $] 0 ; + \infty \left[ \right.$ and we denote $g ^ { \prime }$ its derivative function. We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in the plane with respect to a coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } )$.
We also consider the function $f$ defined on $]0;+\infty[$ by $f(x) = 1 + x^2 - 2x^2\ln(x)$, and $\alpha$ denotes the unique solution of $f(x)=0$ in $[1;+\infty[$. We admit that $g(\alpha) = \frac{1}{2\alpha^2}$.

1. Prove that for all real $x$ in the interval $] 0 ; + \infty \left[ , \quad g ^ { \prime } ( x ) = \frac { f ( x ) } { x \left( 1 + x ^ { 2 } \right) ^ { 2 } } \right.$.
2. Prove that the function $g$ admits a maximum at $x = \alpha$.
3. We denote $T _ { 1 }$ the tangent line to $\mathscr { C } _ { g }$ at the point with abscissa 1 and we denote $T _ { \alpha }$ the tangent line to $\mathscr { C } _ { g }$ at the point with abscissa $\alpha$. Determine, as a function of $\alpha$, the coordinates of the intersection point of the lines $T _ { 1 }$ and $T _ { \alpha }$.

Find the slope of a line L that satisfies both of the following properties: (i) L is tangent to the graph of $y = x ^ { 3 }$. (ii) L passes through the point $( 0, 200 )$.

Given a continuous function $f$, define the following subsets of the set $\mathbb{R}$ of real numbers.
$T =$ set of slopes of all possible tangents to the graph of $f$.
$S =$ set of slopes of all possible secants, i.e. lines joining two points on the graph of $f$.
For each statement below, state if it is true or false.
(i) If $f$ is differentiable, then $S \subset T$.
(ii) If $f$ is differentiable, then $T \subset S$.
(iii) If $T = S = \mathbb{R}$, then $f$ must be differentiable everywhere.
(iv) Suppose 0 and 1 are in $S$. Then every number between 0 and 1 must also be in $S$.

[Calculus] The tangent line to the curve $y = e ^ { x }$ at the point $( 1 , e )$ is tangent to the curve $y = 2 \sqrt { x - k }$. What is the value of the real number $k$? [3 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { e ^ { 2 } }$
(3) $\frac { 1 } { e ^ { 4 } }$
(4) $\frac { 1 } { 1 + e }$
(5) $\frac { 1 } { 1 + e ^ { 2 } }$

The equation of the tangent line to the curve $y = - x ^ { 3 } + 4 x$ at the point $( 1,3 )$ is $y = a x + b$. Find the value of $10 a + b$. (where $a , b$ are constants) [4 points]

The equation of the tangent line to the graph of the cubic function $f(x) = x^3 + ax^2 + 9x + 3$ at the point $(1, f(1))$ is $y = 2x + b$. What is the value of $a + b$? (Here, $a, b$ are constants.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$.
Let A be the point where the tangent line to the curve $y = 3 ^ { x }$ at point P meets the $x$-axis, and let B be the point where the tangent line to the curve $y = a ^ { x - 1 }$ at point P meets the $x$-axis. For point $\mathrm { H } ( k , 0 )$, when $\overline { \mathrm { AH } } = 2 \overline { \mathrm { BH } }$, what is the value of $a$? [4 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For the curve $y = 3 e ^ { x - 1 }$, when the tangent line at point A passes through the origin O, what is the length of segment OA? [3 points]
(1) $\sqrt { 6 }$
(2) $\sqrt { 7 }$
(3) $2 \sqrt { 2 }$
(4) 3
(5) $\sqrt { 10 }$

Two polynomial functions $f ( x ) , g ( x )$ satisfy the following conditions. (가) $g ( x ) = x ^ { 3 } f ( x ) - 7$ (나) $\lim _ { x \rightarrow 2 } \frac { f ( x ) - g ( x ) } { x - 2 } = 2$
When the equation of the tangent line to the curve $y = g ( x )$ at the point $( 2 , g ( 2 ) )$ is $y = a x + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (where $a , b$ are constants.) [4 points]

For the curve $y = x ^ { 3 } - a x + b$, the slope of the line perpendicular to the tangent line at the point $( 1,1 )$ is $- \frac { 1 } { 2 }$. For the two constants $a$ and $b$, find the value of $a + b$. [4 points]

From the point $\left( - \frac { \pi } { 2 } , 0 \right)$, tangent lines are drawn to the curve $y = \sin x ( x > 0 )$, and when the $x$-coordinates of the points of tangency are listed in increasing order, the $n$-th number is denoted as $a _ { n }$. For all natural numbers $n$, which of the following statements in the given options are correct? [4 points]
Options ㄱ. $\tan a _ { n } = a _ { n } + \frac { \pi } { 2 }$ ㄴ. $\tan a _ { n + 2 } - \tan a _ { n } > 2 \pi$ ㄷ. $a _ { n + 1 } + a _ { n + 2 } > a _ { n } + a _ { n + 3 }$
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

The tangent line to the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 2 x + 2$ at point $\mathrm { A } ( 0,2 )$ is perpendicular to a line passing through point A. What is the $x$-intercept of this line? [3 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

For a cubic function $f ( x )$, the tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = x f ( x )$ at the point $( 1,2 )$ coincide. What is the value of $f ^ { \prime } ( 2 )$? [4 points]
(1) $-18$
(2) $-17$
(3) $-16$
(4) $-15$
(5) $-14$

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

For a real number $a > \sqrt{2}$, define the function $f(x)$ as $$f(x) = -x^3 + ax^2 + 2x$$ The tangent line to the curve $y = f(x)$ at the point $\mathrm{O}(0,0)$ intersects the curve $y = f(x)$ at another point A. The tangent line to the curve $y = f(x)$ at point A intersects the $x$-axis at point B. If point A lies on the circle with diameter OB, find the value of $\overline{\mathrm{OA}} \times \overline{\mathrm{AB}}$. [4 points]