Consider the function $$f(x) = \frac{\sin x}{3 - 2\cos x} \quad (0 \leqq x \leqq \pi)$$
(1) The derivative of $f(x)$ is $$f'(x) = \frac{\mathbf{A}\cos x - \mathbf{B}}{(\mathbf{C} - \mathbf{D}\cos x)^2}.$$ Let $\alpha$ be the value of $x$ at which $f(x)$ has a local extremum. Then we have $$\cos\alpha = \frac{\mathbf{E}}{\mathbf{F}}.$$
(2) The portion of the plane bounded by the graph of the function $y = f(x)$ and the $x$-axis is divided into two parts by the straight line $x = \alpha$. Let $S_1$ be the area of the part located on the left side of the line. Then we have $$S_1 = \int_{\frac{\mathbf{G}}{\mathbf{H}}}^{\mathbf{I}} \frac{dt}{\mathbf{J} - \mathbf{K}t} = \frac{\mathbf{L}}{\mathbf{L}}\log\frac{\mathbf{L}}{\mathbf{L}}.$$ Let $S_2$ be the area of the part located on the right side. We have $$S_2 = \frac{\mathbf{P}}{2}\log\mathbf{Q}.$$

4. (a) Find $\frac { d y } { d x }$ for each of the functions
$$\begin{aligned} & y = \sin ( \ln x ) \\ & y = x \sin ( \ln x ) \\ & y = x \cos ( \ln x ) \end{aligned}$$
(b) Sketch the following curves using the axes provided on the next page:
(i) $y = \ln x$, for $1 \leqslant x \leqslant e ^ { \pi }$,
(ii) $y = \sin ( \ln x )$, for $1 \leqslant x \leqslant e ^ { \pi }$.
(c) Evaluate
$$\int _ { 1 } ^ { e ^ { \pi } } \sin ( \ln x ) d x$$
[Figure]

Given the function $f ( x ) = \left\{ \begin{array} { l l l } x e ^ { 2 x } & \text { if } & x < 0 \\ \frac { \ln ( x + 1 ) } { x + 1 } & \text { if } & x \geq 0 \end{array} \right.$, where $\ln$ means natural logarithm, it is requested:\ a) (1 point) Study the continuity and differentiability of $f ( x )$ at $x = 0$.\ b) (1 point) Calculate $\lim _ { x \rightarrow - \infty } f ( x )$ and $\lim _ { x \rightarrow + \infty } f ( x )$.\ c) (1 point) Calculate $\int _ { - 1 } ^ { 0 } f ( x ) d x$

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 $[3, 4]$

Find the derivative $\frac{\mathrm{d}y(x)}{\mathrm{d}x}$ of the following real function $y(x)$ defined for $0 < x < 1$: $$y(x) = (\arccos x)^{\log x}$$ where $0 < \arccos x < \pi/2$.

Consider the following function $f(x)$.
$$f(x) = \int_0^1 \frac{|t - x|}{1 + t^2}\,dt \qquad (0 \leq x \leq 1)$$
(1) Find the real number $\alpha$ satisfying $0 < \alpha < \dfrac{\pi}{4}$ such that $f'(\tan \alpha) = 0$.
(2) For the value of $\alpha$ found in (1), find the value of $\tan \alpha$.
(3) Find the maximum value and minimum value of the function $f(x)$ on the interval $0 \leq x \leq 1$. You may use the fact that $0.69 < \log 2 < 0.7$ if necessary.

$$f(x) = \ln\left(\sin^{2} x + e^{2x}\right)$$
Given this, what is $f'(0)$?
A) $e$
B) $1$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $2$

$$f ( x ) = e ^ { 2 x } - e ^ { - 2 x }$$
What is the value of the 15th order derivative of the function at the point $x = \ln 2$, that is $\mathbf { f } ^ { \mathbf { ( 1 5 ) } } ( \mathbf { \ln } \mathbf { 2 } )$?
A) $17 \cdot 2 ^ { 13 }$
B) $15 \cdot 2 ^ { 13 }$
C) $9 \cdot 2 ^ { 13 }$
D) $15 \cdot 2 ^ { 12 }$
E) $7 \cdot 2 ^ { 12 }$

What is the value of the 16th order derivative $f ^ { ( 16 ) } ( x )$ of the function $f ( x ) = e ^ { x } \cdot \cos x$ at the point $x = 0$?
A) 32 B) 64 C) 128 D) 256 E) 512

Let $a$ and $b$ be real numbers. A function $f$ is defined on the set of positive real numbers as
$$f ( x ) = a x ^ { a } + b x ^ { b }$$
$$\begin{aligned} & f ( 1 ) = 6 \\ & f ^ { \prime } ( 1 ) = 20 \end{aligned}$$
Given that, what is $f''(1)$?
A) 44
B) 46
C) 48
D) 50
E) 52