Let $g$ be a continuously differentiable function with $g ( 1 ) = 6$ and $g ^ { \prime } ( 1 ) = 3$. What is $\lim _ { x \rightarrow 1 } \frac { \int _ { 1 } ^ { x } g ( t ) d t } { g ( x ) - 6 }$ ?
(A) 0
(B) $\frac { 1 } { 2 }$
(C) 1
(D) 2
(E) The limit does not exist.

[14 points] Let $\mathbb{R}_+$ denote the set of positive real numbers. For a continuous function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, define $A_r =$ the area bounded by the graph of $f$, X-axis, $x = 1$ and $x = r$ $B_r =$ the area bounded by the graph of $f$, X-axis, $x = r$ and $x = r^2$. Find all continuous $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ for which $A_r = B_r$ for every positive number $r$. Hints (use these or your own method): Find an equation relating $f(x)$ and $f(x^2)$. Consider the function $xf(x)$. Suppose a sequence $x_n$ converges to $b$ where all $x_n$ and $b$ are in the domain of a continuous function $g$. Then $g(x_n)$ must converge to $g(b)$. E.g., $g\left(3^{\frac{1}{n}}\right) \rightarrow g(1)$.

For the function $F ( x ) = \int _ { 0 } ^ { x } \left( t ^ { 3 } - 1 \right) d t$, what is the value of $F ^ { \prime } ( 2 )$? [3 points]
(1) 11
(2) 9
(3) 7
(4) 5
(5) 3

For the function $f ( x ) = 3 ( x - 1 ) ^ { 2 } + 5$, define the function $F ( x )$ as $F ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$. A differentiable function $g ( x )$ satisfies the following for all real numbers $x$:
$$F ( g ( x ) ) = \frac { 1 } { 2 } F ( x )$$
When $g ^ { \prime } ( 2 ) = p$, find the value of $30 p$. [4 points]

The graph of a continuous function $y = f ( x )$ is symmetric about the origin, and for all real numbers $x$, $$f ( x ) = \frac { \pi } { 2 } \int _ { 1 } ^ { x + 1 } f ( t ) d t$$ When $f ( 1 ) = 1$, what is the value of $$\pi ^ { 2 } \int _ { 0 } ^ { 1 } x f ( x + 1 ) d x$$ ? [4 points]
(1) $2 ( \pi - 2 )$
(2) $2 \pi - 3$
(3) $2 ( \pi - 1 )$
(4) $2 \pi - 1$
(5) $2 \pi$

When the function $f ( x )$ is $$f ( x ) = \int _ { 0 } ^ { x } \frac { 1 } { 1 + e ^ { - t } } d t$$ what is the value of the real number $a$ that satisfies $( f \circ f ) ( a ) = \ln 5$? [4 points]
(1) $\ln 11$
(2) $\ln 13$
(3) $\ln 15$
(4) $\ln 17$
(5) $\ln 19$

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

A polynomial function $f ( x )$ satisfies the following conditions. (가) For all real numbers $x$, $$\int _ { 1 } ^ { x } f ( t ) d t = \frac { x - 1 } { 2 } \{ f ( x ) + f ( 1 ) \}$$ (나) $\int _ { 0 } ^ { 2 } f ( x ) d x = 5 \int _ { - 1 } ^ { 1 } x f ( x ) d x$ When $f ( 0 ) = 1$, find the value of $f ( 4 )$. [4 points]

A polynomial function $f(x)$ satisfies $$\int_{0}^{x} f(t)\, dt = 3x^{3} + 2x$$ for all real numbers $x$. What is the value of $f(1)$? [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Let $\alpha$ be a number strictly less than $\frac{M}{2}$ and $F$ be the antiderivative of $f$ vanishing at $\alpha$. Show that for all $x$ and $y$ in $]-\infty, \frac{M}{2}[$, with $y \neq \alpha$, we have: $$f(2x) = 2\frac{F(x+y) - F(x+\alpha) - \frac{1}{4}F(2y) + \frac{1}{4}F(2\alpha)}{y - \alpha}$$

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ By noting that, for all $t \in [0,1[$, $\frac{1}{1-t} = \sum_{n=0}^{\infty} t^{n}$, show that $$\forall x \in {]-1,+\infty[}, \quad f(x) = \int_{0}^{1} \frac{t^{x} - 1}{1 - t} \mathrm{~d}t$$

Let $f : [0,2] \rightarrow \mathbb{R}$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0) = 1$. Let
$$F(x) = \int_{0}^{x^2} f(\sqrt{t})\, dt$$
for $x \in [0,2]$. If $F'(x) = f'(x)$ for all $x \in (0,2)$, then $F(2)$ equals
(A) $e^2 - 1$
(B) $e^4 - 1$
(C) $e - 1$
(D) $e^4$

Let $F ( x ) = \int _ { x } ^ { x ^ { 2 } + \frac { \pi } { 6 } } 2 \cos ^ { 2 } t \, d t$ for all $x \in \mathbb { R }$ and $f : \left[ 0 , \frac { 1 } { 2 } \right] \rightarrow [ 0 , \infty )$ be a continuous function. For $a \in \left[ 0 , \frac { 1 } { 2 } \right]$, if $F ^ { \prime } ( a ) + 2$ is the area of the region bounded by $x = 0 , y = 0 , y = f ( x )$ and $x = a$, then $f ( 0 )$ is

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous odd function, which vanishes exactly at one point and $f ( 1 ) = \frac { 1 } { 2 }$. Suppose that $F ( x ) = \int _ { - 1 } ^ { x } f ( t ) d t$ for all $x \in [ - 1,2 ]$ and $G ( x ) = \int _ { - 1 } ^ { x } t | f ( f ( t ) ) | d t$ for all $x \in [ - 1,2 ]$. If $\lim _ { x \rightarrow 1 } \frac { F ( x ) } { G ( x ) } = \frac { 1 } { 14 }$, then the value of $f \left( \frac { 1 } { 2 } \right)$ is

Let $f ( x ) = \int _ { 0 } ^ { x } g ( t ) \, dt$, where $g$ is a non-zero even function. If $f ( x + 5 ) = g ( x )$, then $\int _ { 0 } ^ { x } f ( t ) \, dt$ equals
(1) $\int _ { 5 } ^ { x + 5 } g ( t ) \, dt$
(2) $\int _ { x + 5 } ^ { 5 } g ( t ) \, dt$
(3) $5 \int _ { x + 5 } ^ { 5 } g ( t ) \, dt$
(4) $2 \int _ { 5 } ^ { x + 5 } g ( t ) \, dt$

If $f : R \rightarrow R$ is a differentiable function and $f ( 2 ) = 6$, then $\lim _ { x \rightarrow 2 } \int _ { 6 } ^ { f ( x ) } \frac { 2 t d t } { ( x - 2 ) }$ is:
(1) 0
(2) $2 f ^ { \prime } ( 2 )$
(3) $24 f ^ { \prime } ( 2 )$
(4) $12 f ^ { \prime } ( 2 )$

Let $f ( x ) = \int _ { 0 } ^ { x } e ^ { t } f ( t ) d t + e ^ { x }$ be a differentiable function for all $x \in R$. Then $f ( x )$ equals:

If $x \phi ( x ) = \int _ { 5 } ^ { x } \left( 3 t ^ { 2 } - 2 \phi ^ { \prime } ( t ) \right) d t , x > - 2 , \phi ( 0 ) = 4$, then $\phi ( 2 )$ is

Let f be a continuous function satisfying $\int _ { 0 } ^ { t ^ { 2 } } \left( f(x) + x^2 \right) dx = \frac { 4 } { 3 } t ^ { 3 } , \forall t > 0$. Then $f\left( \frac { \pi ^ { 2 } } { 4 } \right)$ is equal to
(1) $\pi ^ { 2 } \left( 1 - \frac { \pi ^ { 2 } } { 16 } \right)$
(2) $- \pi \left( 1 + \frac { \pi ^ { 3 } } { 16 } \right)$
(3) $\pi \left( 1 - \frac { \pi ^ { 3 } } { 16 } \right)$
(4) $- \pi ^ { 2 } \left( 1 + \frac { \pi ^ { 2 } } { 16 } \right)$

Let $f ( x ) = x + \frac { a } { \pi ^ { 2 } - 4 } \sin x + \frac { b } { \pi ^ { 2 } - 4 } \cos x , x \in \mathbb { R }$ be a function which satisfies $f ( x ) = x + \int _ { 0 } ^ { \pi / 2 } \sin ( x + y ) f ( y ) d y$. Then $( a + b )$ is equal to
(1) $- \pi ( \pi + 2 )$
(2) $- 2 \pi ( \pi + 2 )$
(3) $- 2 \pi ( \pi - 2 )$
(4) $- \pi ( \pi - 2 )$

Let $f : \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right] \rightarrow R$ be a differentiable function such that $f ( 0 ) = \frac { 1 } { 2 }$, If $\lim _ { x \rightarrow 0 } \frac { x \int _ { 0 } ^ { x } f ( t ) d t } { e ^ { x ^ { 2 } } - 1 } = \alpha$, then $8 \alpha ^ { 2 }$ is equal to :
(1) 16
(2) 2
(3) 1
(4) 4

Let for some function $\mathrm { y } = f ( x ) , \int _ { 0 } ^ { x } t f ( t ) d t = x ^ { 2 } f ( x ) , x > 0$ and $f ( 2 ) = 3$. Then $f ( 6 )$ is equal to
(1) 1
(2) 3
(3) 6
(4) 2

Let $f$ be a real valued continuous function defined on the positive real axis such that $g ( x ) = \int _ { 0 } ^ { x } \mathrm { t } f ( \mathrm { t } ) \mathrm { dt }$. If $\mathrm { g } \left( x ^ { 3 } \right) = x ^ { 6 } + x ^ { 7 }$, then value of $\sum _ { r = 1 } ^ { 15 } f \left( \mathrm { r } ^ { 3 } \right)$ is :
(1) 270
(2) 340
(3) 320
(4) 310

Let $f:(0,\infty) \rightarrow \mathbf{R}$ be a twice differentiable function. If for some $\mathrm{a} \neq 0$, $\int_0^1 f(\lambda x)\,\mathrm{d}\lambda = \mathrm{a}f(x)$, $f(1) = 1$ and $f(16) = \frac{1}{8}$, then $16 - f'\left(\frac{1}{16}\right)$ is equal to \_\_\_\_ .

Let $a$ and $b$ be real numbers. A function $f$ that is continuous on the set of real numbers is defined as
$$f ( x ) = \begin{cases} 6 - \frac { 3 x ^ { 2 } } { 2 } , & x < 2 \\ a x - b & x \geq 2 \end{cases}$$
$$\int _ { 0 } ^ { 4 } f ( x ) d x = \int _ { 2 } ^ { 6 } f ( x ) d x$$
Given that, what is the sum $a + b$?
A) 1
B) 2
C) 3
D) 4
E) 5