$$f ( x ) = \begin{cases} \frac { 2 } { x } & \text { for } x < - 1 \\ x ^ { 2 } - 3 & \text { for } - 1 \leq x \leq 2 \\ 4 x - 3 & \text { for } x > 2 \end{cases}$$
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 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

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 $\mathbb { R }$.

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]

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]

18. (This question is worth 13 points) Given the function $f ( x ) = \ln \frac { 1 + x } { 1 - x }$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 0 , f ( 0 ) )$; (II) Prove: When $x \in ( 0,1 )$, $f ( x ) > 2 \left( x + \frac { x ^ { 3 } } { 3 } \right)$; (III) Let the real number $k$ be such that $f ( x ) > k \left( x + \frac { x ^ { 3 } } { 3 } \right)$ holds for all $x \in ( 0,1 )$. Find the maximum value of $k$.

(12 points)
Let $A$ and $B$ be two points on the curve $C: y = \frac{x^2}{4}$, and the sum of the $x$-coordinates of $A$ and $B$ is 4.
(1) Find the slope of line $AB$;
(2) Find the equation of line $AB$.

The equation of the tangent line to the curve $y = 2 \ln x$ at the point $( 1,0 )$ is \_\_\_\_.

The equation of the tangent line to the curve $y = 2 \ln ( x + 1 )$ at the point $( 0,0 )$ is $\_\_\_\_$.

10. The equation of the tangent line to the curve $y = 2 \sin x + \cos x$ at the point $( \pi , - 1 )$ is
A. $x - y - \pi - 1 = 0$
B. $2 x - y - 2 \pi - 1 = 0$
C. $2 x + y - 2 \pi + 1 = 0$
D. $x + y - \pi + 1 = 0$

The equation of the tangent line to the graph of $f ( x ) = x ^ { 4 } - 2 x ^ { 3 }$ at the point $( 1 , f ( 1 ) )$ is
A. $y = - 2 x - 1$
B. $y = - 2 x + 1$
C. $y = 2 x - 3$
D. $y = 2 x + 1$

13. The equation of the tangent line to the curve $y = \frac{2x - 1}{x + 2}$ at the point $(-1, -3)$ is $\_\_\_\_$.

The slope of the tangent to the curve $\left(y - x^5\right)^2 = x\left(1 + x^2\right)^2$ at the point $(1, 3)$ is

Let $a , b \in \mathbb { R }$ and $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = a \cos \left( \left| x ^ { 3 } - x \right| \right) + b | x | \sin \left( \left| x ^ { 3 } + x \right| \right)$. Then $f$ is
(A) differentiable at $x = 0$ if $a = 0$ and $b = 1$
(B) differentiable at $x = 1$ if $a = 1$ and $b = 0$
(C) NOT differentiable at $x = 0$ if $a = 1$ and $b = 0$
(D) NOT differentiable at $x = 1$ if $a = 1$ and $b = 1$

On the coordinate plane, let $\Gamma$ be the graph of the cubic function $f(x) = x^{3} - 9x^{2} + 15x - 4$. Show that $P(1, 3)$ is a point on $\Gamma$, and find the equation of the tangent line $L$ to $\Gamma$ at point $P$.

The tangent line drawn to the graph of the function $y = f ( x )$ at the point $( 2,4 )$ passes through the point $( - 1,3 )$.
Accordingly, what is the value of $f ^ { \prime } ( 2 )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 3 } { 5 }$