The function $f(x) = e^{2x}$ has derivative:
(A) $e^{2x}$
(B) $2e^{x}$
(C) $2e^{2x}$
(D) $4e^{2x}$
(E) $e^{x}$

Let $f$ be a real valued continuous function defined on $\mathbb{R}$ satisfying $$f'\left(\tan^{2}\theta\right) = \cos 2\theta + \tan\theta \sin 2\theta, \text{ for all real numbers } \theta.$$ If $f'(0) = -\cos\frac{\pi}{12}$ then find $f(1)$.

(Calculus) For the function $f ( x ) = x + \sin x$, define the function $g ( x )$ as $$g ( x ) = ( f \circ f ) ( x )$$ Which of the following in are correct? [3 points]
ㄱ. The graph of function $f ( x )$ is concave down on the open interval $( 0 , \pi )$. ㄴ. The function $g ( x )$ is increasing on the open interval $( 0 , \pi )$. ㄷ. There exists a real number $x$ in the open interval $( 0 , \pi )$ such that $g ^ { \prime } ( x ) = 1$.
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ᄂ, ᄃ
(5) ᄀ, ᄂ, ᄃ

(Calculus) For the function $f(x) = 4\ln x + \ln(10 - x)$, which of the following statements in are correct? [3 points]
ㄱ. The maximum value of function $f(x)$ is $13\ln 2$. ㄴ. The equation $f(x) = 0$ has two distinct real roots. ㄷ. The graph of function $y = e^{f(x)}$ is concave downward on the interval $(4, 8)$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

[Calculus] The tangent line to the curve $y = e ^ { x }$ at the point $( 1 , e )$ is tangent to the curve $y = 2 \sqrt { x - k }$. What is the value of the real number $k$? [3 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { e ^ { 2 } }$
(3) $\frac { 1 } { e ^ { 4 } }$
(4) $\frac { 1 } { 1 + e }$
(5) $\frac { 1 } { 1 + e ^ { 2 } }$

For all functions $f ( x )$ that are differentiable on the set of all real numbers and satisfy the following conditions, what is the minimum value of $\int _ { 0 } ^ { 2 } f ( x ) d x$? [4 points] (가) $f ( 0 ) = 1 , f ^ { \prime } ( 0 ) = 1$ (나) If $0 < a < b < 2$, then $f ^ { \prime } ( a ) \leqq f ^ { \prime } ( b )$. (다) On the interval $( 0,1 )$, $f ^ { \prime \prime } ( x ) = e ^ { x }$.
(1) $\frac { 1 } { 2 } e - 1$
(2) $\frac { 3 } { 2 } e - 1$
(3) $\frac { 5 } { 2 } e - 1$
(4) $\frac { 7 } { 2 } e - 2$
(5) $\frac { 9 } { 2 } e - 2$

What is the value of $\lim _ { x \rightarrow 0 } \frac { e ^ { x } - 1 } { 5 x }$? [2 points]
(1) 5
(2) $e$
(3) 1
(4) $\frac { 1 } { e }$
(5) $\frac { 1 } { 5 }$

A cubic function $f ( x )$ satisfies $f ( 0 ) > 0$. Define the function $g ( x )$ as
$$g ( x ) = \left| \int _ { 0 } ^ { x } f ( t ) d t \right|$$
The graph of the function $y = g ( x )$ is as shown in the figure. Which of the following statements are correct? Choose all that apply. [4 points]

ㄱ. The equation $f ( x ) = 0$ has three distinct real roots. ㄴ. $f ^ { \prime } ( 0 ) < 0$ ㄷ. The number of natural numbers $m$ satisfying $\int _ { m } ^ { m + 2 } f ( x ) d x > 0$ is 3.
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the function $f ( x ) = x \ln x + 13 x$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = 5 e ^ { 3 x - 3 }$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = \cos x + 4 e ^ { 2 x }$, find the value of $f ^ { \prime } ( 0 )$. [3 points]

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) $\sqrt { 10 }$

For the function $f ( x ) = 4 \sin 7 x$, find the value of $f ^ { \prime } ( 2 \pi )$. [3 points]

For the function $f ( x ) = e ^ { - x } \int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$, which of the following statements are correct? [4 points]
ㄱ. $f ( \sqrt { \pi } ) > 0$ ㄴ. There exists at least one $a$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( a ) > 0$. ㄷ. There exists at least one $b$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( b ) = 0$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

A cubic function $f ( x )$ with leading coefficient 1 and $f ( 1 ) = 0$ satisfies $$\lim _ { x \rightarrow 2 } \frac { f ( x ) } { ( x - 2 ) \left\{ f ^ { \prime } ( x ) \right\} ^ { 2 } } = \frac { 1 } { 4 }$$ Find the value of $f ( 3 )$. [4 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

For the function $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

A polynomial function $f ( x )$ satisfies for all real numbers $x$: $$\int _ { 1 } ^ { x } \left\{ \frac { d } { d t } f ( t ) \right\} d t = x ^ { 3 } + a x ^ { 2 } - 2$$ What is the value of $f ^ { \prime } ( a )$? (Here, $a$ is a constant.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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

A function $f ( x )$ that is differentiable on the entire set of real numbers satisfies the following conditions. What is the value of $f ( - 1 )$? [4 points] (가) For all real numbers $x$, $$2 \{ f ( x ) \} ^ { 2 } f ^ { \prime } ( x ) = \{ f ( 2 x + 1 ) \} ^ { 2 } f ^ { \prime } ( 2 x + 1 ).$$ (나) $f \left( - \frac { 1 } { 8 } \right) = 1 , f ( 6 ) = 2$
(1) $\frac { \sqrt [ 3 ] { 3 } } { 6 }$
(2) $\frac { \sqrt [ 3 ] { 3 } } { 3 }$
(3) $\frac { \sqrt [ 3 ] { 3 } } { 2 }$
(4) $\frac { 2 \sqrt [ 3 ] { 3 } } { 3 }$
(5) $\frac { 5 \sqrt [ 3 ] { 3 } } { 6 }$

For a real number $t$, let the equation of the tangent line to the curve $y = e ^ { x }$ at the point $\left( t , e ^ { t } \right)$ be $y = f ( x )$. Let $g ( t )$ be the minimum value of the real number $k$ such that the function $y = | f ( x ) + k - \ln x |$ is differentiable on the entire set of positive real numbers. For two real numbers $a , b ( a < b )$, let $\int _ { a } ^ { b } g ( t ) d t = m$. Which of the following statements in the given options are correct? [4 points]
$\langle$Options$\rangle$
ㄱ. There exist two real numbers $a , b ( a < b )$ such that $m < 0$. ㄴ. If $g ( c ) = 0$ for a real number $c$, then $g ( - c ) = 0$. ㄷ. If $m$ is minimized when $a = \alpha , b = \beta ( \alpha < \beta )$, then
$$\frac { 1 + g ^ { \prime } ( \beta ) } { 1 + g ^ { \prime } ( \alpha ) } < - e ^ { 2 }$$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the function $f ( x ) = x ^ { 3 } \ln x$, find the value of $\frac { f ^ { \prime } ( e ) } { e ^ { 2 } }$. [3 points]

For a positive real number $t$, let $f ( t )$ be the value of the real number $a$ such that the curve $y = t ^ { 3 } \ln ( x - t )$ meets the curve $y = 2 e ^ { x - a }$ at exactly one point. Find the value of $\left\{ f ^ { \prime } \left( \frac { 1 } { 3 } \right) \right\} ^ { 2 }$. [4 points]

For the function $f ( x ) = \left( x ^ { 2 } - 2 x - 7 \right) e ^ { x }$, let the local maximum value and local minimum value be $a$ and $b$ respectively. What is the value of $a \times b$? [3 points]
(1) - 32
(2) - 30
(3) - 28
(4) - 26
(5) - 24

For a function $f ( x )$ that is differentiable on $x > 0$, $$f ^ { \prime } ( x ) = 2 - \frac { 3 } { x ^ { 2 } } , \quad f ( 1 ) = 5$$ For a function $g ( x )$ that is differentiable on $x < 0$ and satisfies the following conditions, what is the value of $g ( - 3 )$? [4 points] (가) For all real numbers $x < 0$, $g ^ { \prime } ( x ) = f ^ { \prime } ( - x )$. (나) $f ( 2 ) + g ( - 2 ) = 9$
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a function $f ( x )$, if $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 4 x + 5$ and $f ( 0 ) = 4$, find the value of $f ( 1 )$. [3 points]