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

13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$ .

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. $y = 2 x + 1$

A tangent line to the curve $y = \ln x + x + 1$ has slope 2. The equation of this tangent line is $\_\_\_\_$.

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

15. If the curve $y = ( x + a ) \mathrm { e } ^ { x }$ has two tangent lines passing through the origin, then the range of $a$ is $\_\_\_\_$ .

Given functions $f ( x ) = x ^ { 3 } - x , g ( x ) = x ^ { 2 } + a$. The tangent line to the curve $y = f ( x )$ at the point $\left( x _ { 1 } , f \left( x _ { 1 } \right) \right)$ is also tangent to the curve $y = g ( x )$ at some point.
(1) If $x _ { 1 } = - 1$ , find $a$ ;
(2) If $x_1 \neq 0$, prove that $a > \frac{1}{4}$ .

If the tangent line to the curve $y = \mathrm { e } ^ { x } + x$ at the point $( 0,1 )$ is also a tangent line to the curve $y = \ln ( x + 1 ) + a$ , then $a = $ $\_\_\_\_$ .

If the line $y = 2x + 5$ is tangent to the curve $y = e^x + x + a$, then $a = $ $\_\_\_\_$ .

Calculate for $a \neq 0$ the slope $m _ { a }$ of the line through the points $P$ and $Q _ { a }$ as a function of $a$. (for verification: $m _ { a } = \frac { a ^ { 3 } - 16 } { 8 a }$ )
)$} The tangent to the graph of $f$ at the point $Q _ { a }$ is denoted by $t _ { a }$. Determine computationally the value of $a \in \mathbb { R }$ for which $t _ { a }$ passes through $P$.
Given is the function $f : x \mapsto \frac { 6 x } { x ^ { 2 } - 4 }$ defined on $\mathbb { R } \backslash \{ - 2 ; 2 \}$. The graph of $f$ is denoted by $G _ { f }$ and is symmetric with respect to the origin. (1a) [3 marks] State the equations of all vertical asymptotes of $G _ { f }$. Justify that $G _ { f }$ has the x-axis as a horizontal asymptote.
(1b) [5 marks] Determine the monotonicity behavior of $f$ in each of the three subintervals $] - \infty ; - 2 [$, $] - 2 ; 2 [$ and $] 2 ; + \infty [$ of the domain. Also calculate the slope of the tangent to $G _ { f }$ at the point $( 0 \mid f ( 0 ) )$. (for verification: $\left. f ^ { \prime } ( x ) = - \frac { 6 \cdot \left( x ^ { 2 } + 4 \right) } { \left( x ^ { 2 } - 4 \right) ^ { 2 } } \right)$
The points $A ( 3 \mid 3,6 )$ and $B ( 8 \mid 0,8 )$ lie on $G _ { f }$; between these two points $G _ { f }$ runs below the line segment [AB].
(1c) [4 marks] Sketch $G _ { f }$ in the range $- 10 \leq x \leq 10$ using the information obtained so far in a coordinate system.
(1d) [5 marks] Calculate the area enclosed by $G _ { f }$ and the line segment $[ A B ]$.
Consider the family of functions $f _ { a , b , c } : x \mapsto \frac { a x + b } { x ^ { 2 } + c }$ with $a , b , c \in \mathbb { R }$ and maximal domain $D _ { a , b , c }$.
(2a) [1 marks] The function $f$ from Task 1 is a function of this family. State the corresponding values of $a , b$ and $c$.
(2b) [2 marks] Justify: If $a = 0$ and $b \neq 0$, then the graph of $f _ { a , b , c }$ is symmetric with respect to the y-axis and does not intersect the x-axis.
(2c) [3 marks] State all values for $a , b$ and $c$ such that both $D _ { a , b , c } = \mathbb { R }$ holds and the graph of $f _ { a , b , c }$ is symmetric with respect to the origin, but is not identical to the x-axis.
(2d) [4 marks] For the first derivative of $f _ { a , b , c }$ it holds: $f _ { a , b , c } ^ { \prime } ( x ) = - \frac { a x ^ { 2 } + 2 b x - a c } { \left( x ^ { 2 } + c \right) ^ { 2 } }$. Show: If $a \neq 0$ and $c > 0$, then the graph of $f _ { a , b , c }$ has exactly two extreme points.
Consider the function $p : x \mapsto \frac { 40 } { ( x - 12 ) ^ { 2 } + 4 }$ defined on $\mathbb { R }$; the figure shows the graph $G _ { p }$ of $p$. [Figure]
(3a) [4 marks] Describe how $G _ { p }$ is obtained step by step from the graph of the function $h : x \mapsto \frac { 5 } { x ^ { 2 } + 4 }$ defined on $\mathbb { R }$, and justify thereby that $G _ { p }$ is symmetric with respect to the line with equation $x = 12$.
A photovoltaic system installed on a house roof converts light energy into electrical energy. For $4 \leq x \leq 20$, the function $p$ describes the temporal development of the power output of the system on a particular day. Here $x$ is the time elapsed since midnight in hours and $p ( x )$ is the power in kW (kilowatts).
(3b) [4 marks] Determine computationally the time in the afternoon to the nearest minute from which the power output of the system is less than $40 \%$ of its daily maximum of 10 kW.
(3c) [2 marks] The function $p$ has an inflection point in the interval [4;12]. State the meaning of this inflection point in the context of the problem.
(3d) [3 marks] The electrical energy produced by the system is completely fed into the power grid. The homeowner receives a remuneration of 10 cents per kilowatt-hour (kWh) for the electrical energy fed in.
The function $x \mapsto E ( x )$ defined on [4;20] gives the electrical energy in kWh that the system feeds into the power grid on the day in question from 4:00 a.m. until $x$ hours after midnight.
It holds that $E ^ { \prime } ( x ) = p ( x )$ for $x \in [ 4 ; 20 ]$. Use the figure to determine an approximate value for the remuneration that the homeowner receives for the electrical energy fed into the power grid from 10:00 a.m. to 2:00 p.m.

Exercise III

Let $f$ be the function defined for every real number $x$ different from $1$ by $f ( x ) = \frac { 3 } { 1 - x }$ and $C _ { f }$ its representative curve in an orthonormal coordinate system. III-A- $\quad \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = - \infty$. III-B- An equation of the tangent line to the curve $C _ { f }$ at the point with abscissa $x = - 1$ is $y = \frac { 3 } { 4 } x + \frac { 3 } { 2 }$. III-C- $f$ is concave on $] 1 ; + \infty [$.
For each statement, indicate whether it is TRUE or FALSE.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$. Show that $f_h$ can be extended to a continuous function on all of $\mathbb{R}$ by setting $f_h(0) = \frac{\gamma_h}{2}$.

117- The function $f(x) = \begin{cases} ax^3 + bx & ; \ x < 1 \\ 2\sqrt{4x - 3} & ; \ x \geq 1 \end{cases}$ is differentiable on the set of real numbers. What is $b$?
(1) $\dfrac{1}{2}$ (2) $1$ (3) $\dfrac{3}{2}$ (4) $2$
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119- If $f(x) = xe^x$; $x > 0$, then the tangent line to the graph of $f^{-1}$ at points located along $e$, intersects the $y$-axis at which value?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{3}$ (3) $\dfrac{1}{2}$ (4) $\dfrac{1}{e}$

118. From the point $A(2,-1)$, two tangent lines to the curve $y = \dfrac{1}{2}x^2 - x$ are drawn. What is the angle between these two tangent lines?
(1) $\dfrac{\pi}{4}$ (2) $\dfrac{\pi}{3}$ (3) $\dfrac{\pi}{2}$ (4) $\tan^{-1}2$

117. If $\theta$ is the angle between the left and right tangents to the graph of the function $f(x) = \left[x + \frac{1}{2}\right]x + x^2$, at the point $x = \frac{1}{2}$, what is $\tan\theta$?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{2}$ (3) $\dfrac{2}{3}$ (4) $\dfrac{3}{4}$

118- The tangent line to the graph of $f(x) = (x+2)e^{1-x}$ at the point $x = 1$ meets the line connecting this point to the origin. What is $\tan\alpha$?
(1) $0.5$ (2) $1$ (3) $1.5$ (4) $2$

119- The line $y = 3x - 2$ at the point $x = 2$ is tangent to the curve $y = f(x)$. What is $\displaystyle\lim_{x \to 2} \dfrac{f^2(x) - 4f(x)}{x - 2}$?
(1) $2$ (2) $6$ (3) $12$ (4) $15$

117. If $\theta$ is the angle between the left and right tangents to the graph of $y = |\ln x|$ at the corner point, then $\tan\theta$ equals:
(1) $-1$ (2) $1$ (3) zero (4) $\infty$

119. The function $f(x) = x + \ln x$ is defined (given). The equation of the tangent line to the graph of $f^{-1}$ at the point where it meets the bisector of the first quadrant is:
(1) $y + 2x = 3$ (2) $2x - y = 1$ (3) $2x + y = 3$ (4) $2y - x = 1$

120. The $x$-intercept of the normal line to the curve $x^2 + y^2 = 3xy + 3$ at the point $(1, 2)$ is:
(1) $2$ (2) $3$ (3) $4$ (4) $5$

117- A line is tangent to the graph of the function $y = x^3 - 2x^2 + 3x$ at the point $x = 2$ and passes through it. The slope of this line is which of the following?
(1) $-\dfrac{2}{3}$ (2) $\dfrac{2}{3}$ (3) $\dfrac{4}{3}$ (4) $\dfrac{5}{3}$
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118- The line perpendicular to the graph of $f(x) = \dfrac{\cos 2x}{2 - \sin x}$, at the point of tangency with the $y$-axis, cuts the $x$-axis at which length?
(1) $0/1$ (2) $0/2$ (3) $0/3$ (4) $0/5$

116. The function with the formula $f(x) = \begin{cases} |x^2 - 2x| & ; \quad x < 2 \\ \dfrac{1}{2}x^2 + ax + b & ; \quad x \geq 2 \end{cases}$ is differentiable at $x = 2$. What is $a + b$?
(1) $2$ (2) $3$ (3) $4$ (4) $5$

121. The tangent line to the curve $f(x) = \dfrac{5x - 4}{\sqrt{x}}$ at the point $x = 4$. At which values does it intersect the $y$-axis?
(1) $-4$ (2) $-1$ (3) $2$ (4) $3$

122. If $\tan\alpha$ and $\tan\beta$ are the roots of the equation $2x^2 + 3x - 1 = 0$, what is $\tan(\alpha + \beta)$?
(1) $1$ (2) $\dfrac{3}{2}$ (3) $-3$ (4) $-1$