$x$	$f ( x )$	$f ^ { \prime } ( x )$	$g ( x )$	$g ^ { \prime } ( x )$
1	2	- 4	- 5	3
2	- 3	1	8	4

The table above gives values of the differentiable functions $f$ and $g$ and their derivatives at selected values of $x$. If $h$ is the function defined by $h ( x ) = f ( x ) g ( x ) + 2 g ( x )$, then $h ^ { \prime } ( 1 ) =$
(A) 32
(B) 30
(C) - 6
(D) - 16

The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions: $$f(0) = 2,\quad f'(0) = -4,\quad \text{and}\quad f''(0) = 3.$$
(a) The function $g$ is given by $g(x) = e^{ax} + f(x)$ for all real numbers, where $a$ is a constant. Find $g'(0)$ and $g''(0)$ in terms of $a$. Show the work that leads to your answers.
(b) The function $h$ is given by $h(x) = \cos(kx)f(x)$ for all real numbers, where $k$ is a constant. Find $h'(x)$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.

The twice-differentiable functions $f$ and $g$ are defined for all real numbers $x$. Values of $f$, $f ^ { \prime }$, $g$, and $g ^ { \prime }$ for various values of $x$ are given in the table below.

$x$	-2	$- 2 < x < - 1$	-1	$- 1 < x < 1$	1	$1 < x < 3$	3
$f ( x )$	12	Positive	8	Positive	2	Positive	7
$f ^ { \prime } ( x )$	-5	Negative	0	Negative	0	Positive	$\frac { 1 } { 2 }$
$g ( x )$	-1	Negative	0	Positive	3	Positive	1
$g ^ { \prime } ( x )$	2	Positive	$\frac { 3 } { 2 }$	Positive	0	Negative	-2

(a) Find the $x$-coordinate of each relative minimum of $f$ on the interval $[ - 2, 3 ]$. Justify your answers.
(b) Explain why there must be a value $c$, for $- 1 < c < 1$, such that $f ^ { \prime \prime } ( c ) = 0$.
(c) The function $h$ is defined by $h ( x ) = \ln ( f ( x ) )$. Find $h ^ { \prime } ( 3 )$. Show the computations that lead to your answer.
(d) Evaluate $\displaystyle\int _ { - 2 } ^ { 3 } f ^ { \prime } ( g ( x ) ) g ^ { \prime } ( x ) \, dx$.

The functions $f$ and $g$ have continuous second derivatives. The table below gives values of the functions and their derivatives at selected values of $x$.

$x$	$f ( x )$	$f ^ { \prime } ( x )$	$g ( x )$	$g ^ { \prime } ( x )$
1	$-6$	3	2	8
2	2	$-2$	$-3$	0
3	8	7	6	2
6	4	5	3	$-1$

(a) Let $k ( x ) = f ( g ( x ) )$. Write an equation for the line tangent to the graph of $k$ at $x = 3$.
(b) Let $h ( x ) = \frac { g ( x ) } { f ( x ) }$. Find $h ^ { \prime } ( 1 )$.
(c) Evaluate $\int _ { 1 } ^ { 3 } f ^ { \prime \prime } ( 2 x ) \, d x$.

Let $f$ be the function defined by $f(x) = \cos(2x) + e^{\sin x}$.
Let $g$ be a differentiable function. The table below gives values of $g$ and its derivative $g'$ at selected values of $x$.

\multicolumn{1}{|c|}{$x$}	$g(x)$	$g'(x)$
-5	10	-3
-4	5	-1
-3	2	4
-2	3	1
-1	1	-2
0	0	-3

Let $h$ be the function whose graph, consisting of five line segments, is shown in the figure above.
(a) Find the slope of the line tangent to the graph of $f$ at $x = \pi$.
(b) Let $k$ be the function defined by $k(x) = h(f(x))$. Find $k'(\pi)$.
(c) Let $m$ be the function defined by $m(x) = g(-2x) \cdot h(x)$. Find $m'(2)$.
(d) Is there a number $c$ in the closed interval $[-5, -3]$ such that $g'(c) = -4$? Justify your answer.

Let $f$ be the function defined by $f ( x ) = e ^ { x } \cos x$.
(a) Find the average rate of change of $f$ on the interval $0 \leq x \leq \pi$.
(b) What is the slope of the line tangent to the graph of $f$ at $x = \frac { 3 \pi } { 2 }$ ?
(c) Find the absolute minimum value of $f$ on the interval $0 \leq x \leq 2 \pi$. Justify your answer.
(d) Let $g$ be a differentiable function such that $g \left( \frac { \pi } { 2 } \right) = 0$. The graph of $g ^ { \prime }$, the derivative of $g$, is shown below. Find the value of $\lim _ { x \rightarrow \pi / 2 } \frac { f ( x ) } { g ( x ) }$ or state that it does not exist. Justify your answer.

Functions $f$, $g$, and $h$ are twice-differentiable functions with $g(2) = h(2) = 4$. The line $y = 4 + \dfrac{2}{3}(x - 2)$ is tangent to both the graph of $g$ at $x = 2$ and the graph of $h$ at $x = 2$.
(a) Find $h'(2)$.
(b) Let $a$ be the function given by $a(x) = 3x^3 h(x)$. Write an expression for $a'(x)$. Find $a'(2)$.
(c) The function $h$ satisfies $h(x) = \dfrac{x^2 - 4}{1 - (f(x))^3}$ for $x \neq 2$. It is known that $\lim_{x \to 2} h(x)$ can be evaluated using L'H\^{o}pital's Rule. Use $\lim_{x \to 2} h(x)$ to find $f(2)$ and $f'(2)$. Show the work that leads to your answers.
(d) It is known that $g(x) \leq h(x)$ for $1 < x < 3$. Let $k$ be a function satisfying $g(x) \leq k(x) \leq h(x)$ for $1 < x < 3$. Is $k$ continuous at $x = 2$? Justify your answer.

The functions $f$ and $g$ are twice differentiable. The table shown gives values of the functions and their first derivatives at selected values of $x$.

$x$	0	2	4	7
$f(x)$	10	7	4	5
$f'(x)$	$\frac{3}{2}$	$-8$	3	6
$g(x)$	1	2	$-3$	0
$g'(x)$	5	4	2	8

(a) Let $h$ be the function defined by $h(x) = f(g(x))$. Find $h'(7)$. Show the work that leads to your answer.
(b) Let $k$ be a differentiable function such that $k'(x) = (f(x))^{2} \cdot g(x)$. Is the graph of $k$ concave up or concave down at the point where $x = 4$? Give a reason for your answer.
(c) Let $m$ be the function defined by $m(x) = 5x^{3} + \int_{0}^{x} f'(t)\, dt$. Find $m(2)$. Show the work that leads to your answer.
(d) Is the function $m$ defined in part (c) increasing, decreasing, or neither at $x = 2$? Justify your answer.

The function $f$ is defined on the interval $[ 0 ; 1 ]$ by: $$f ( x ) = \frac { 0,96 x } { 0,93 x + 0,03 }$$

1. Prove that, for all $x$ belonging to the interval $[ 0 ; 1 ]$, $$f ^ { \prime } ( x ) = \frac { 0,0288 } { ( 0,93 x + 0,03 ) ^ { 2 } }$$
2. Determine the direction of variation of the function $f$ on the interval $[ 0 ; 1 ]$.

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

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

For the function $f ( x ) = \frac { x ^ { 2 } - 2 x - 6 } { x - 1 }$, find the value of $f ^ { \prime } ( 0 )$.

For a polynomial function $f ( x )$, define the function $g ( x )$ as $$g ( x ) = x ^ { 2 } f ( x )$$ If $f ( 2 ) = 1$ and $f ^ { \prime } ( 2 ) = 3$, what is the value of $g ^ { \prime } ( 2 )$? [3 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

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

For the function $f(x) = \left(x^{2} + 1\right)\left(3x^{2} - x\right)$, what is the value of $f'(1)$? [3 points]
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16

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

Express the derivatives $f^{\prime}, f^{\prime\prime}$ and $f^{(3)}$ using usual functions, where $f$ is defined on $I = ]-\pi/2, \pi/2[$ by $$\forall x \in I, \quad f(x) = \frac{\sin x + 1}{\cos x}.$$

Let $f$ and $g$ be real valued functions defined on interval $( - 1,1 )$ such that $g ^ { \prime \prime } ( x )$ is continuous, $g ( 0 ) \neq 0 , g ^ { \prime } ( 0 ) = 0 , g ^ { \prime \prime } ( 0 ) \neq 0$, and $f ( x ) = g ( x ) \sin x$.
STATEMENT-1 : $\lim _ { x \rightarrow 0 } [ g ( x ) \cot x - g ( 0 ) \operatorname { cosec } x ] = f ^ { \prime \prime } ( 0 )$. and STATEMENT-2 : $\quad f ^ { \prime } ( 0 ) = g ( 0 )$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True