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 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 $\_\_\_\_$.

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 $$\tan\theta = \frac{\mathbf{V}}{\mathbf{W}}.$$

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) $2 \sqrt { 3 }$