$\lim _ { x \rightarrow \pi } \frac { \cos x + \sin ( 2 x ) + 1 } { x ^ { 2 } - \pi ^ { 2 } }$ is
(A) $\frac { 1 } { 2 \pi }$
(B) $\frac { 1 } { \pi }$
(C) 1
(D) nonexistent

If $f ( x ) = \int _ { 1 } ^ { x ^ { 3 } } \frac { 1 } { 1 + \ln t } d t$ for $x \geq 1$, then $f ^ { \prime } ( 2 ) =$
(A) $\frac { 1 } { 1 + \ln 2 }$
(B) $\frac { 12 } { 1 + \ln 2 }$
(C) $\frac { 1 } { 1 + \ln 8 }$
(D) $\frac { 12 } { 1 + \ln 8 }$

The functions $f$ and $g$ are differentiable for all real numbers, and $g$ is strictly increasing. The table below gives values of the functions and their first derivatives at selected values of $x$.

$x$	$f(x)$	$f^{\prime}(x)$	$g(x)$	$g^{\prime}(x)$
1	6	4	2	5
2	9	2	3	1
3	10	-4	4	2
4	-1	3	6	7

The function $h$ is given by $h(x) = f(g(x)) - 6$.
(a) Explain why there must be a value $r$ for $1 < r < 3$ such that $h(r) = -5$.
(b) Explain why there must be a value $c$ for $1 < c < 3$ such that $h^{\prime}(c) = -5$.
(c) Let $w$ be the function given by $w(x) = \int_{1}^{g(x)} f(t)\, dt$. Find the value of $w^{\prime}(3)$.
(d) If $g^{-1}$ is the inverse function of $g$, write an equation for the line tangent to the graph of $y = g^{-1}(x)$ at $x = 2$.

If $y = x \sin x$, then $\frac { d y } { d x } =$
(A) $\sin x + \cos x$
(B) $\sin x + x \cos x$
(C) $\sin x - x \cos x$
(D) $x ( \sin x + \cos x )$
(E) $x ( \sin x - \cos x )$

If $y = \left( x ^ { 3 } - \cos x \right) ^ { 5 }$, then $y ^ { \prime } =$
(A) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 }$
(B) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 }$
(C) $5 \left( 3 x ^ { 2 } + \sin x \right)$
(D) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 } \cdot ( 6 x + \cos x )$
(E) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 } \cdot \left( 3 x ^ { 2 } + \sin x \right)$

If $f ( x ) = \sqrt { x ^ { 2 } - 4 }$ and $g ( x ) = 3 x - 2$, then the derivative of $f ( g ( x ) )$ at $x = 3$ is
(A) $\frac { 7 } { \sqrt { 5 } }$
(B) $\frac { 14 } { \sqrt { 5 } }$
(C) $\frac { 18 } { \sqrt { 5 } }$
(D) $\frac { 15 } { \sqrt { 21 } }$
(E) $\frac { 30 } { \sqrt { 21 } }$

If $y = \sin ^ { 3 } x$, then $\frac { d y } { d x } =$
(A) $\cos ^ { 3 } x$
(B) $3 \cos ^ { 2 } x$
(C) $3 \sin ^ { 2 } x$
(D) $- 3 \sin ^ { 2 } x \cos x$
(E) $3 \sin ^ { 2 } x \cos x$

Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = x ^ { 4 }$. On what intervals is the rate of change of $f ( x )$ greater than the rate of change of $g ( x )$ ?
(A) $( 0.831, 7.384 )$ only
(B) $( - \infty , 0.831 )$ and $( 7.384 , \infty )$
(C) $( - \infty , - 0.816 )$ and $( 1.430, 8.613 )$
(D) $( - 0.816, 1.430 )$ and $( 8.613 , \infty )$
(E) $( - \infty , \infty )$

Let $f_0(x) = x$. For $x > 0$, define functions inductively by $f_{n+1}(x) = x^{f_n(x)}$. So $f_1(x) = x^x$, $f_2(x) = x^{x^x}$, etc. Note that $f_0(1) = f_0'(1) = 1$.
Statements
(25) As $x \rightarrow 0^+$, $f_1(x) \rightarrow 1$. (26) As $x \rightarrow 0^+$, $f_2(x) \rightarrow 1$. (27) $\int_0^1 f_3(x)\, dx = 1$. (28) The derivative of $f_{123}$ at $x = 1$ is 1.

Two real numbers $a$ and $b$ satisfy $\lim _ { x \rightarrow 2 } \frac { \sqrt { x ^ { 2 } + a } - b } { x - 2 } = \frac { 2 } { 5 }$. Find the value of $a + b$. [3 points]

For two points $\mathrm { P } ( n , f ( n ) )$ and $\mathrm { Q } ( n + 1 , f ( n + 1 ) )$ on the graph of the quadratic function $f ( x ) = 3 x ^ { 2 }$, let $a _ { n }$ be the distance between them. Find the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$. (Here, $n$ is a natural number.) [4 points]
(1) 9
(2) 8
(3) 7
(4) 6
(5) 5

For the function $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } + 1$, find the value of $$\lim _ { h \rightarrow 0 } \frac { f ( 1 + 2 h ) - f ( 1 ) } { h }$$. [3 points]

(Calculus) Let the function $f(x)$ be defined as $$f(x) = \int_a^x \{2 + \sin(t^2)\} dt$$ If $f''(a) = \sqrt{3}a$, find the value of $(f^{-1})'(0)$. (Given: $a$ is a constant satisfying $0 < a < \sqrt{\frac{\pi}{2}}$) [4 points]
(1) $\frac{1}{10}$
(2) $\frac{1}{5}$
(3) $\frac{3}{10}$
(4) $\frac{2}{5}$
(5) $\frac{1}{2}$

For two constants $a , b$, if $\lim _ { x \rightarrow 3 } \frac { \sqrt { x + a } - b } { x - 3 } = \frac { 1 } { 4 }$, what is the value of $a + b$? [2 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11

[Calculus] As shown in the figure, let Q be the point where the tangent line to the circle $x ^ { 2 } + y ^ { 2 } = 1$ at point P on the circle meets the $x$-axis. For point $\mathrm { A } ( - 1,0 )$ and the origin O, let $\angle \mathrm { PAO } = \theta$. Find the value of $\lim _ { \theta \rightarrow \frac { \pi } { 4 } - 0 } \frac { \overline { \mathrm { PQ } } - \overline { \mathrm { OQ } } } { \theta - \frac { \pi } { 4 } }$. (where point P is in the first quadrant) [3 points]
(1) 2
(2) $\sqrt { 3 }$
(3) $\frac { 3 } { 2 }$
(4) 1
(5) $\frac { \sqrt { 2 } } { 2 }$

Find the value of $\lim _ { x \rightarrow 1 } \frac { ( x - 1 ) \left( x ^ { 2 } + 3 x + 7 \right) } { x - 1 }$. [3 points]

For the function $f ( x ) = x ^ { 3 } + 7 x + 3$, what is the value of $f ^ { \prime } ( 1 )$? [3 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

For a function $f ( x )$ differentiable on the set of all real numbers, let the function $g ( x )$ be defined as $$g ( x ) = \frac { f ( x ) } { e ^ { x - 2 } }$$ If $\lim _ { x \rightarrow 2 } \frac { f ( x ) - 3 } { x - 2 } = 5$, what is the value of $g ^ { \prime } ( 2 )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

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

For the function $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + x - 1$, what is the value of $f ^ { \prime } ( 1 )$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

11. Given the function $f ( x ) = a x \ln x , x \in ( 0 , + \infty )$, where $a$ is a real number, and $f ^ { \prime } ( x )$ is the derivative of $f ( x )$. If $f ^ { \prime } ( 1 ) = 3$, then the value of $a$ is $\_\_\_\_$.

13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$.

13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$ .

Let $x , y \in C ^ { 1 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$. Prove that $$\forall t \in \left[ 0 , + \infty \left[ , \frac { d } { d t } ( \langle A x ; y \rangle ) ( t ) = \left\langle A x ^ { \prime } ( t ) ; y ( t ) \right\rangle + \left\langle A x ( t ) ; y ^ { \prime } ( t ) \right\rangle \right. \right.$$

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Justify that $\varphi$ is of class $C^1$ on $\mathbb{R}$ and, for every real $t$, give $\varphi'(t)$.