The equation of the tangent line to the graph of the cubic function $f(x) = x^3 + ax^2 + 9x + 3$ at the point $(1, f(1))$ is $y = 2x + b$. What is the value of $a + b$? (Here, $a, b$ are constants.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a cubic function $f ( x )$, the tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = x f ( x )$ at the point $( 1,2 )$ coincide. What is the value of $f ^ { \prime } ( 2 )$? [4 points]
(1) $-18$
(2) $-17$
(3) $-16$
(4) $-15$
(5) $-14$

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. $y = x$

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. $y = x$

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. $a = \mathrm { e } ^ { - 1 } , b = - 1$

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. $a = \mathrm { e } ^ { - 1 } , b = - 1$

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. $a = \mathrm { e } ^ { - 1 } , b = - 1$

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 the tangent to the curve, $y = x ^ { 3 } + a x - b$ at the point $( 1 , - 5 )$ is perpendicular to the line, $- x + y + 4 = 0$, then which one of the following points lies on the curve?
(1) $( 2 , - 2 )$
(2) $( 2 , - 1 )$
(3) $( - 2,1 )$
(4) $( - 2,2 )$

If the lines $x + y = a$ and $x - y = b$ touch the curve $y = x^2 - 3x + 2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to ...

For the function $f(x) = 2x^{3} - ax^{2} + 3$, what should $a$ be so that the equation of the tangent line to the curve at some point is $y = 4$?
A) $-3$
B) $-1$
C) $0$
D) $1$
E) $3$

If the line tangent to the parabola $y = x^{2} + bx + c$ at the point $x = 2$ is $y = x$, what is the sum $b + c$?
A) $-2$
B) $-1$
C) $0$
D) $1$
E) $2$

Given that the function $f$ has derivative $f ^ { \prime } ( x ) = 3 x ^ { 2 }$ and the tangent line at the point $x = a ( a > 0 )$ is the line $y - 12 x + 14 = 0$, what is the value of $f ( 1 )$?
A) $- 2$
B) 0
C) 1
D) 3
E) 5

A third-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with leading coefficient 1 and its derivative $P ^ { \prime } ( x )$ satisfy
$$P ( 0 ) = P ( 1 ) = P ^ { \prime } ( 1 ) = 0$$
Accordingly, what is the value of $P ( - 1 )$?
A) 3
B) 1
C) 0
D) - 2
E) - 4

Let a and b be real numbers. In the rectangular coordinate plane, the parabola
$$y = a x ^ { 2 } + b x$$
passes through the point $( 1,2 )$, and the tangent line to the parabola at this point intersects the y-axis at the point $( 0,1 )$.
Accordingly, what is the product $a \cdot b$?
A) - 3
B) - 2
C) - 1
D) 2
E) 4

Let a and b be real numbers, and $$f ( x ) = a \cdot \ln x + b \cdot x ^ { 2 } + 3$$ The equation of the tangent line drawn to the graph of the function at the point $(1, f(1))$ is given as $y - 2x + 1 = 0$.
Accordingly, what is the product $\mathbf{a} \cdot \mathbf{b}$?\ A) $- 18$\ B) $- 16$\ C) $- 12$\ D) $- 8$\ E) $- 6$

Let a, b and c be real numbers. The equation of the tangent line to the curve
$$y = \frac { a } { x + a }$$
at point $P ( a , b )$ is given in the form
$$y = \frac { - x } { 8 } + c$$
Accordingly, what is the sum $a + b + c$?
A) $\frac { 7 } { 4 }$ B) $\frac { 11 } { 4 }$ C) $\frac { 13 } { 4 }$ D) 2 E) 3