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 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.

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 $( 0, 200 )$.

For the quartic function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 6 x ^ { 2 } + 4$, the slope of the tangent line at the point $( a , b )$ on the graph is 4. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]

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) $\sqrt { 10 }$

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

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 positive number $a$, let the function $f ( x )$ be $$f ( x ) = x ^ { 3 } + 3 a x ^ { 2 } - 9 a ^ { 2 } x + 4$$ When the line $y = 5$ is tangent to the curve $y = f ( x )$, what is the value of $f ( 2 )$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

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

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.

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) $\tan^{-1}2$

124. Suppose $A$ and $B$ are the extreme points of $f(x) = 2x^3 - 3x^2 - 12x + 1$. How many points on the curve $f$ have a tangent line parallel to line $AB$?
(1) zero (2) $1$ (3) $2$ (4) $3$
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24. The point (s) on the curve $y ^ { 3 } + 3 x ^ { 2 } = 12 y$ where the tangent is vertical, is (are)
(A) $\quad ( \pm 4 / \sqrt { } 3 , - 2 )$
(B) $\quad ( \pm \sqrt { } 11 / 3 , - 0 )$
(C) $( 0,0 )$
(D) $\quad ( \pm 4 / \sqrt { } 3,2 )$

The intercepts on the $x$-axis made by tangents to the curve, $y = \int_0^x |t|\, dt, x \in R$, which are parallel to the line $y = 2x$, are equal to
(1) $\pm 3$
(2) $\pm 4$
(3) $\pm 1$
(4) $\pm 2$

The tangent to the curve $y = x ^ { 2 } - 5 x + 5$, parallel to the line $2 y = 4 x + 1$, also passes through the point :
(1) $\left( \frac { 1 } { 4 } , \frac { 7 } { 2 } \right)$
(2) $\left( \frac { 7 } { 2 } , \frac { 1 } { 4 } \right)$
(3) $\left( - \frac { 1 } { 8 } , 7 \right)$
(4) $\left( \frac { 1 } { 8 } , - 7 \right)$

If the tangent to the curve $y = x + \sin y$ at a point $(a, b)$ is parallel to the line joining $\left(0, \frac{3}{2}\right)$ and $\left(\frac{1}{2}, 2\right)$, then
(1) $b = a$
(2) $|b - a| = 1$
(3) $|a + b| = 1$
(4) $b = \frac{\pi}{2} + a$

If the tangent at the point $\left( x _ { 1 } , y _ { 1 } \right)$ on the curve $y = x ^ { 3 } + 3 x ^ { 2 } + 5$ passes through the origin, then $\left( x _ { 1 } , y _ { 1 } \right)$ does NOT lie on the curve

If the line $y = 4 + kx$, $k > 0$, is the tangent to the parabola $y = x - x^2$ at the point $P$ and $V$ is the vertex of the parabola, then the slope of the line through $P$ and $V$ is
(1) $\frac{3}{2}$
(2) $\frac{26}{9}$
(3) $\frac{5}{2}$
(4) $\frac{23}{6}$

Let $M$ and $N$ be the number of points on the curve $y ^ { 5 } - 9 x y + 2 x = 0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M + N$ equals $\_\_\_\_$ .

Consider the function $f(x) = x^3 - 4x + 4$. Let the straight line $\ell$ be the tangent to the graph of $y = f(x)$ at the point $\mathrm{A}(-1, 7)$, and the straight line $m$ be the tangent to the graph of $y = f(x)$ that passes through the point $\mathrm{B}(0, -12)$. Also, let C be the point of intersection of $\ell$ and $m$. Let us denote the angle formed by $\ell$ and $m$ at C by $\theta$ $\left(0 < \theta < \frac{\pi}{2}\right)$. We are to find $\tan\theta$.
(1) The derivative $f'(x)$ of $f(x)$ is $$f'(x) = \mathbf{A}x^{\mathbf{B}} - \mathbf{C}.$$ Hence, the slope of $\ell$ is $\mathbf{DE}$, and the equation of $\ell$ is $$y = \mathbf{DE}x + \mathbf{F}.$$
(2) Let us denote by $a$ the $x$-coordinate of the tangent point of the graph of $y = f(x)$ and line $m$. Then the equation of $m$ can be expressed in terms of $a$ as $$y = (\mathbf{G}a^{\mathbf{H}} - \mathbf{I})x - \mathbf{J}a^{\mathbf{K}} + \mathbf{K}.$$ Since line $m$ passes through point $\mathrm{B}(0, -12)$, we see that $a = \mathbf{M}$, and the equation of $m$ is $$y = \mathbf{N}x - \mathbf{OP}.$$ Hence, the coordinates of point C, the intersection of $\ell$ and $m$, are $(\mathbf{Q}, \mathbf{R})$.
(3) Let us denote by $\alpha$ the angle between the positive direction of the $x$-axis and line $\ell$, and by $\beta$ the angle between the positive direction of the $x$-axis and line $m$. Then we see that $$\tan\alpha = \mathbf{ST}, \quad \tan\beta = \mathbf{U},$$ and hence $$\tan\theta = \frac{\mathbf{V}}{\mathbf{W}}.$$

Given the functions $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 1$ and $g ( x ) = 6 x$, it is requested:\ a) (0.5 points) Justify, using the appropriate theorem, that there exists some point in the interval [1,10] where both functions take the same value.\ b) (1 point) Calculate the equation of the tangent line to the curve $y = f ( x )$ with minimum slope.\ c) (1 point) Calculate $\int _ { 1 } ^ { 2 } \frac { f ( x ) } { g ( x ) } d x$

The tangent line drawn from a point $A(x, y)$ on the parabola $y^{2} = 4x$ has slope 1.
Accordingly, what is $x + y$, the sum of the coordinates of point $A$?
A) 1
B) 2
C) 3
D) 4
E) 5

$$x ^ { 2 } - y ^ { 2 } = 1$$
What is the distance between the points where the lines tangent to the hyperbola curve and having slope 3 intersect the y-axis, in units?
A) $\sqrt { 2 }$
B) $2 \sqrt { 2 }$
C) $4 \sqrt { 2 }$
D) $\sqrt { 3 }$
E) $2 \sqrt { 3 }$