Consider $f : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined as follows: $$f(a,b) := \lim_{n \rightarrow \infty} \frac{1}{n} \log_{e}\left[e^{na} + e^{nb}\right]$$
For each statement, state if it is true or false.
(a) $f$ is not onto i.e. the range of $f$ is not all of $\mathbb{R}$.
(b) For every $a$ the function $x \mapsto f(a,x)$ is continuous everywhere.
(c) For every $b$ the function $x \mapsto f(x,b)$ is differentiable everywhere.
(d) We have $f(0,x) = x$ for all $x \geq 0$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$. For each statement, state if it is true or false.
(a) There is no continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx < \frac{1}{4}$.
(b) There is only one continuous function $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.
(c) There are infinitely many continuous functions $f$ for which $\int_{0}^{1} f(x)(1 - f(x))\,dx = \frac{1}{4}$.

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Show that the sequence $$\left[\int_0^1 |f(x)|^n\,\mathrm{d}x\right]^{\frac{1}{n}}$$ is convergent.

[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., $g\left(3^{\frac{1}{n}}\right) \rightarrow g(1)$.

Statements
(13) $\lim _ { x \rightarrow 0 } e ^ { \frac { 1 } { x } } = + \infty$. (14) The following inequality is true. $$\lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { 100 } } < \lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { \frac { 1 } { 100 } } }$$ (15) For any positive integer $n$, $$\int _ { - n } ^ { n } x ^ { 2023 } \cos ( n x ) \, dx < \frac { n } { 2023 }$$ (16) There is no polynomial $p ( x )$ for which there is a single line that is tangent to the graph of $p ( x )$ at exactly 100 points.

Let $f : \mathbb { R } _ { \geq 0 } \longrightarrow \mathbb { R }$ be the function
$$f ( x ) = \begin{cases} 1 , & x = 0 \\ x ^ { - x } , & x > 0 \end{cases}$$
Determine whether the following statement is true:
$$\int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x = \sum _ { i = 0 } ^ { \infty } n ^ { - n }$$

(A) (6 marks) Let $f , g : [ 0,1 ] \mapsto \mathbb { R }$ be monotonically increasing continuous functions. Show that
$$\left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right) \leq \int _ { 0 } ^ { 1 } f ( x ) g ( x ) d x$$
(Hint: try double integrals.)
(B) (4 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be an infinitely differentiable function such that $f ( 1 ) = f ( 0 ) = 0$. Also, suppose that for some $n > 0$, the first $n$ derivatives of $f$ vanish at zero. Then prove that for the $( n + 1 )$ th derivative of $f$, $f ^ { ( n + 1 ) } ( x ) = 0$ for some $x \in ( 0,1 )$.

(a) Find the domain of the function $g(x)$ defined by the following formula. $$g(x) = \int_{10}^{x} \log_{10}\left(\log_{10}\left(t^2 - 1000t + 10^{1000}\right)\right) dt$$ Calculate the quantities below. You may give an approximate answer where necessary, but clearly state which answers are exact and which are approximations.
(b) $g(1000)$.
(c) $x$ in $[10, 1000]$ where $g(x)$ has the maximum possible slope.
(d) $x$ in $[10, 1000]$ where $g(x)$ has the least possible slope.
(e) $\lim_{x \rightarrow \infty} \frac{\ln(x)}{g(x)}$ if it exists.

Let $f$ be a continuous real-valued function on $[ 0,1 ]$ such that
$$\int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } x f ( x ) d x = 0$$
Pick the correct statement(s) from below.
(A) $f$ must have a zero in $[ 0,1 ]$.
(B) $f$ has at least two zeros, counted with multiplicity, in $[ 0,1 ]$.
(C) If $f \not\equiv 0$, then $f$ has exactly two zeros in $[ 0,1 ]$.
(D) $f \equiv 0$.

The following shows the graph of a continuous function $y = f ( x )$ and two distinct points $\mathrm { P } ( a , f ( a ) ) , \mathrm { Q } ( b , f ( b ) )$ on this graph.
When the function $F ( x )$ satisfies $F ^ { \prime } ( x ) = f ( x )$, which of the following statements in $\langle$Remarks$\rangle$ are always correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. The function $F ( x )$ is increasing on the interval $[ a , b ]$. ㄴ. $\frac { F ( b ) - F ( a ) } { b - a }$ is equal to the slope of line PQ. ㄷ. $\int _ { a } ^ { b } \{ f ( x ) - f ( b ) \} d x \leqq \frac { ( b - a ) \{ f ( a ) - f ( b ) \} } { 2 }$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

The following is the graph of a continuous function $y = f ( x )$.
When the inverse function $g ( x )$ of function $f ( x )$ exists and is continuous on the interval $[ 0,1 ]$, the limit value $$\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left\{ g \left( \frac { k } { n } \right) - g \left( \frac { k - 1 } { n } \right) \right\} \frac { k } { n }$$ has the same value as which of the following? [4 points]
(1) $\int _ { 0 } ^ { 1 } g ( x ) d x$
(2) $\int _ { 0 } ^ { 1 } x g ( x ) d x$
(3) $\int _ { 0 } ^ { 1 } f ( x ) d x$
(4) $\int _ { 0 } ^ { 1 } x f ( x ) d x$
(5) $\int _ { 0 } ^ { 1 } \{ f ( x ) - g ( x ) \} d x$

The graph of the function $f ( x ) = x ^ { 3 }$ is translated $a$ units in the $x$-direction and $b$ units in the $y$-direction to obtain the graph of the function $y = g ( x )$.
$$g ( 0 ) = 0 \text { and } \int _ { a } ^ { 3 a } g ( x ) dx - \int _ { 0 } ^ { 2 a } f ( x ) dx = 32$$
Find the value of $a ^ { 4 }$. [3 points]

The following is a graph showing the velocity $v ( t )$ at time $t$ ( $0 \leqq t \leqq d$ ) of a point P moving on a number line starting from the origin.
When $\int _ { 0 } ^ { a } | v ( t ) | d t = \int _ { a } ^ { d } | v ( t ) | d t$, which of the following statements in are correct? (Here, $0 < a < b < c < d$.) [3 points]
Remarks ㄱ. Point P passes through the origin again after starting. ㄴ. $\int _ { 0 } ^ { c } v ( t ) d t = \int _ { c } ^ { d } v ( t ) d t$ ㄷ. $\int _ { 0 } ^ { b } v ( t ) d t = \int _ { b } ^ { d } | v ( t ) | d t$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

(Calculus) Find the length of the curve $y = \frac { 1 } { 3 } \left( x ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } }$ from $x = 0$ to $x = 6$. [4 points]

When the function $f ( x ) = 6 x ^ { 2 } + 2 a x$ satisfies $\int _ { 0 } ^ { 1 } f ( x ) d x = f ( 1 )$, what is the value of the constant $a$? [2 points]
(1) $- 4$
(2) $- 2$
(3) 0
(4) 2
(5) 4

(Calculus) A continuous function $f(x)$ defined on the closed interval $[0, 1]$ satisfies $f(0) = 0$, $f(1) = 1$, has a second derivative on the open interval $(0, 1)$, and $f'(x) > 0$, $f''(x) > 0$. Which of the following is equal to $\int_0^1 \{f^{-1}(x) - f(x)\} dx$? [3 points]
(1) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{2n}$
(2) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{2}{n}$
(3) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
(4) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{2n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
(5) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{2k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$

There is a function $f ( x ) = x ^ { 2 } + a x + b \quad ( a \geqq 0 , b > 0 )$. For a natural number $n \geq 2$, divide the closed interval $[ 0,1 ]$ into $n$ equal parts, and let the division points (including both endpoints) be $$0 = x _ { 0 } , x _ { 1 } , x _ { 2 } , \cdots , x _ { n - 1 } , x _ { n } = 1$$ respectively. Let $A _ { k }$ be the area of the rectangle with base $\left[ x _ { k - 1 } , x _ { k } \right]$ and height $f \left( x _ { k } \right)$. $( k = 1,2 , \cdots , n )$
Given that the sum of the areas of the two rectangles at the ends is $$A _ { 1 } + A _ { n } = \frac { 7 n ^ { 2 } + 1 } { n ^ { 3 } }$$ find the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 8 k } { n } A _ { k }$. [4 points]

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) $\frac { 9 } { 2 } e - 2$

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

A quadratic function $f ( x )$ satisfies $f ( 0 ) = - 1$ and
$$\int _ { - 1 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { - 1 } ^ { 0 } f ( x ) d x$$
What is the value of $f ( 2 )$? [4 points]
(1) 11
(2) 10
(3) 9
(4) 8
(5) 7

Find the value of $\int _ { 0 } ^ { 5 } ( 4 x - 3 ) d x$. [3 points]

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]

For the function $f(x) = x + 1$, $$\int _ { -1 } ^ { 1 } \{ f(x) \} ^ { 2 } dx = k \left( \int _ { -1 } ^ { 1 } f(x) dx \right) ^ { 2 }$$ what is the value of the constant $k$? [3 points]
(1) $\frac{1}{6}$
(2) $\frac{1}{3}$
(3) $\frac{1}{2}$
(4) $\frac{2}{3}$
(5) $\frac{5}{6}$

For the cubic function $f(x) = x^3 - 3x + a$, the function $$F(x) = \int_{0}^{x} f(t)\, dt$$ has exactly one extremum. What is the minimum value of the positive number $a$? [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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) $2 \pi$