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

For the function $f ( x ) = x ^ { 2 } - 4 x - 3$, let $l$ be the tangent line to the curve $y = f ( x )$ at the point $( 1 , - 6 )$, and for the function $g ( x ) = \left( x ^ { 3 } - 2 x \right) f ( x )$, let $m$ be the tangent line to the curve $y = g ( x )$ at the point $( 1,6 )$. What is the area of the figure enclosed by the two lines $l , m$ and the $y$-axis? [4 points]
(1) 21
(2) 28
(3) 35
(4) 42
(5) 49

15. The tangent line to the curve $y = e ^ { x }$ at the point $( 0,1 )$ is perpendicular to the tangent line to the curve $y = \frac { 1 } { x } ( x > 0 )$ at point P. The coordinates of P are $\_\_\_\_$

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 $AB$.

Let $f ( x ) = x ^ { 3 } + ( a - 1 ) x ^ { 2 } + a x$. If $f ( x )$ is an odd function, then the equation of the tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ is
A. $y = - 2 x$
B. $y = - x$
C. $y = 2 x$
D. $y = x$

Let $f ( x ) = x ^ { 3 } + ( a - 1 ) x ^ { 2 } + a x$. If $f ( x )$ is an odd function, then the equation of the tangent line to $y = f ( x )$ at the point $( 0,0 )$ is
A. $y = - 2 x$
B. $y = - x$
C. $y = 2 x$
D. $y = x$

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 $\_\_\_\_$.

The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

6. The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a \mathrm { e } )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , \quad b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

7. The tangent line to the curve $y = a \mathrm { e } ^ { x } + x \ln x$ at the point $( 1 , a e )$ has equation $y = 2 x + b$ . Then
A. $a = \mathrm { e } , b = - 1$
B. $a = \mathrm { e } , b = 1$
C. $a = \mathrm { e } ^ { - 1 } , b = 1$
D. $a = \mathrm { e } ^ { - 1 } , b = - 1$

10. The system of inequalities $\left\{ \begin{array} { l } x - 1 \geq 0 , \\ k x - y \leq 0 , \\ x + \sqrt { 3 } y - 3 \sqrt { 3 } \leq 0 \end{array} \right.$ represents a planar region that is an equilateral triangle. The minimum value of $z = x + 3 y$ is
A. $2 + 3 \sqrt { 3 }$ B. $1 + 3 \sqrt { 3 }$ C. $2 + \sqrt { 3 }$ D. $1 + \sqrt { 3 }$

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. $y = 2 x + 1$

If line $l$ is tangent to both the curve $y = \sqrt { x }$ and the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 5 }$ , then the equation of $l$ is
A. $y = 2 x + 1$
B. $y = 2 x + \frac { 1 } { 2 }$
C. $y = \frac { 1 } { 2 } x + 1$
D. $y = \frac { 1 } { 2 } x + \frac { 1 } { 2 }$

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 $\_\_\_\_$ .

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 $y = \mathrm{e}^x + x + a$, then $a = $ $\_\_\_\_$ .

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

If $x^{2/3} + y^{2/3} = a^{1/3}$, find the equation of the tangent to the curve at a point, and show that the length of the tangent intercepted between the axes is constant.

The tangent to the curve $y = e^x$ drawn at the point $(c, e^c)$ intersects the line joining the points $(c-1, e^{c-1})$ and $(c+1, e^{c+1})$
(A) on the left of $x = c$
(B) on the right of $x = c$
(C) at no point
(D) at all points

If $st = 1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is
(A) $\frac{(t^2+1)^2}{2t^3}$
(B) $\frac{a(t^2+1)^2}{2t^3}$
(C) $\frac{a(t^2+1)^2}{t^3}$
(D) $\frac{a(t^2+2)^2}{t^3}$