In the coordinate plane, a ``Bézier curve'' determined by four points $A$, $B$, $C$, $D$ refers to a polynomial function of degree at most 3 whose graph passes through points $A$ and $D$, and whose tangent line at point $A$ passes through point $B$, and whose tangent line at point $D$ passes through point $C$. Let $y = f(x)$ be the ``Bézier curve'' determined by the four points $A(0, 0)$, $B(1, 4)$, $C(3, 2)$, $D(4, 0)$. Answer the following questions.
(1) Let the equation of the tangent line to the graph of $y = f(x)$ at point $D$ be $y = ax + b$, where $a$ and $b$ are real numbers. Find the values of $a$ and $b$. (2 points)
(2) Prove that the polynomial $f(x)$ is divisible by $x^{2} - 4x$. (2 points)
(3) Find $f(x)$. (4 points)
(4) Find the value of the definite integral $\int_{2}^{6} |8f(x)|\, dx$. (4 points)

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $f(x) = \frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$, with domain $\{x \mid 1 < x < 8\}$. Using the linear approximation (first-order approximation) of $f(x)$, find the approximate value of $\cos\theta$ when $x = 4.96$. (Non-multiple choice question, 4 points)

On the coordinate plane, consider the graphs of two functions $f(x) = x^{5} - 5x^{3} + 5x^{2} + 5$ and $g(x) = \sin\left(\frac{\pi x}{3} + \frac{\pi}{2}\right)$ (where $\pi$ is the circumference ratio). Select the correct options.
(1) $f'(1) = 0$
(2) $y = f(x)$ is increasing on the closed interval $[0, 2]$
(3) $y = f(x)$ is concave up on the closed interval $[0, 2]$
(4) For any real number $x$, $g(x + 6\pi) = g(x)$
(5) Both $y = f(x)$ and $y = g(x)$ are increasing on the closed interval $[3, 4]$

Below is the graph of the derivative of a function f defined on the interval $[ - 5,5 ]$.
According to this graph, I. The function f is decreasing for $x > 0$. II. $f ( - 2 ) > f ( 0 ) > f ( 2 )$. III. The function f has local extrema at $x = - 2$ and $x = 2$. Which of the following statements are true?
A) Only I
B) Only II
C) I and II
D) I and III
E) I, II and III

$$\lim _ { x \rightarrow 1 ^ { + } } ( x - 1 ) \cdot \ln \left( x ^ { 2 } - 1 \right)$$
What is the value of this limit?
A) $\frac { -1 } { 2 }$
B) $-2$
C) 0
D) 1
E) 4

For a function f defined on the set of real numbers
$$\begin{aligned} & \lim _ { x \rightarrow 3 ^ { + } } f ( x ) = 1 \\ & \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = 2 \end{aligned}$$
Given this, what is the value of the limit $\lim _ { x \rightarrow 2 ^ { + } } \frac { f ( 2 x - 1 ) + f ( 5 - x ) } { f \left( x ^ { 2 } - 1 \right) }$?
A) $\frac { -1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) 1
D) 3
E) 4

$$f ( x ) = \begin{cases} 1 , & x \leq 1 \\ x ^ { 2 } + a x + b , & 1 < x < 3 \\ 5 , & x \geq 3 \end{cases}$$
If the function is continuous on the set of real numbers, what is the difference $a - b$?
A) $-4$
B) $-1$
C) 2
D) 3
E) 5

Below, the graph of the derivative of a function f that is defined and continuous on the set of real numbers is given.
Accordingly,
I. $f ( 2 ) - f ( 1 ) = -2$. II. The function f has a local maximum at the point $x = 0$. III. The second derivative function is defined at the point $x = 0$.
Which of the following statements are true?
A) Only I
B) Only III
C) I and II
D) II and III
E) I, II and III

$$f ( x ) = 2 x ( x - 1 ) ^ { 3 } + ( x - 1 ) ^ { 4 }$$
What is the value of the third derivative of the function at the point $x = 1$?
A) 10
B) 12
C) 14
D) 16
E) 18

In the open interval $(1, e)$, I. The function $\sin ( \ln ( x ) )$ is increasing. II. The function $\cos ( \ln ( x ) )$ is increasing. III. The function $\tan ( \ln ( \mathrm { x } ) )$ is increasing. Which of these statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

An internet company can serve at most 1000 customers and can reach this number when it sets the monthly internet fee at 40 TL. The company has observed that after each 5 TL increase in the monthly internet fee, the number of customers decreases by 50.
At what monthly internet fee should this company set its rate to maximize the total revenue from the monthly internet fee?
A) 55 B) 60 C) 65 D) 70 E) 75

A function f is continuous on the closed interval $[ 0,6 ]$ and differentiable on each of the open intervals $( 0,3 ) , ( 3,4 ) , ( 4,6 )$. The graph of its derivative $f ^ { \prime }$ is given in the rectangular coordinate plane below.
$$\begin{gathered} \text{Let } 0 < c < 2 \text{ and } \\ f ( 0 ) = 5 \end{gathered}$$
Accordingly, which of the following could be the value of f(6)?
A) 5,5
B) 7,3
C) 10,1
D) 12,7
E) 14,9

The amount of fuel consumed in 1 hour by a rocket moving at a constant speed of $V$ kilometers per hour is calculated by the function
$$f ( V ) = \frac { V ^ { 3 } } { 20 } - 7 \cdot V ^ { 2 } + 265 \cdot V$$
in units.
Accordingly, what is the minimum amount of fuel in units that this rocket must consume to travel 100 kilometers at a constant speed?
A) 1000
B) 2000
C) 3000
D) 4000
E) 5000

In the rectangular coordinate plane, the shaded region between the lines $y = 2 + 3x$ and $y = 12 - x$ and the positive $x$ and $y$-axes is given below.
Rectangles are constructed in this region with one side on the $x$-axis and one vertex each on the lines $y = 2 + 3x$ and $y = 12 - x$.
What is the length of the side on the $x$-axis of the rectangle with the largest area?
A) 6 B) $\dfrac{19}{3}$ C) $\dfrac{20}{3}$ D) 7 E) $\dfrac{22}{3}$