10. What is the instantaneous rate of change at $x = 2$ of the function $f$ given by $f ( x ) = \frac { x ^ { 2 } - 2 } { x - 1 }$ ?
(A) - 2
(B) $\frac { 1 } { 6 }$
(C) $\frac { 1 } { 2 }$
(D) 2
(E) 6

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) $x ( \sin x - \cos x )$

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) $( - \infty , \infty )$

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

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

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

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 $\_\_\_\_$.

The function $f : x \mapsto 8 x ^ { 3 } + 3 x$ is defined on $\mathbb { R }$ with derivative function $f ^ { \prime }$. (1a) [2 marks] Calculate $f ^ { \prime } ( 1 )$.
(1b) [3 marks] Determine a term for the antiderivative $F$ of $f$ whose graph passes through the point ( $- 1 \mid 5$ ).
The figure shows the graph $G _ { g }$ of the function $g$ defined on $\mathbb { R }$ with $g ( x ) = 2 \cdot \sin \left( \frac { 1 } { 2 } x \right)$. [Figure]
(2a) [2 marks] Using the figure, assess whether the value of the integral $\int _ { - 2 } ^ { 8 } g ( x ) \mathrm { dx }$ is negative.
(2b) [3 marks] Prove by calculation that the following statement is true: The tangent to $G _ { g }$ at the origin is the line through the points $( - 1 \mid - 1 )$ and $( 1 \mid 1 )$.
Consider the family of functions $f _ { a }$ defined on $\mathbb { R }$ with $f _ { a } ( x ) = x \cdot e ^ { a x }$ and $a \in \mathbb { R } \backslash \{ 0 \}$. For each value of $a$, the function $f _ { a }$ has exactly one extremum.
(3a) [2 marks] Justify that the graph of $f _ { a }$ lies below the x-axis for $x < 0$. (3b) [3 marks] The displayed graphs I and II are graphs of the family; one of the two belongs to a positive value of $a$. Decide which graph this is and justify your decision. [Figure]
(4a) [2 marks] Give a term for a function $g$ defined on $\mathbb { R }$ that has range $[ - 2 ; 4 ]$.
(4b) [3 marks] Give a term for a function $h$ defined on $\mathbb { R }$ such that the term $\sqrt { h ( x ) }$ is defined exactly for $x \in [ - 2 ; 4 ]$. Explain the reasoning underlying your answer.
The figure shows a section of the graph $G$ of the function $f$ defined on $\mathbb { R } \backslash \{ - 3 \}$ with $f ( x ) = x - 3 + \frac { 5 } { x + 3 }$. $G$ has exactly one minimum point $T$. [Figure]

117. For the function with formula $f(x) = (x+2)\sqrt{4x+1}$, the average rate of change of the function on the interval $[0,2]$ is how much greater than the instantaneous rate of change at $x = \dfrac{3}{4}$?
(1) $0.1$ (2) $0.15$ (3) $0.20$ (4) $0.25$

Let for a differentiable function $f : ( 0 , \infty ) \rightarrow R , f ( x ) - f ( y ) \geq \log _ { e } \left( \frac { x } { y } \right) + x - y , \forall x , y \in ( 0 , \infty )$. Then $\sum _ { n = 1 } ^ { 20 } f ^ { \prime } \left( \frac { 1 } { n ^ { 2 } } \right)$ is equal to

Q71. Let $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + 41$ be such that $f ( 1 ) = 40 , f ^ { \prime } ( 1 ) = 2$ and $f ^ { \prime } ( 1 ) = 4$. Then $\mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } + \mathrm { c } ^ { 2 }$ is equal to:
(1) 73
(2) 62
(3) 51
(4) 54

Let $f(x) = x^3 + ax^2 + 2bx + c$, $x \in \mathbb{R}$. If $f'(1) = a$, $f''(2) = b$, and $f'''(3) = c$, then the value of $f'(5)$ is
(A) $\frac{117}{5}$
(B) $\frac{62}{5}$
(C) $\frac{675}{5}$
(D) $\frac{117}{5}$

On the coordinate plane, let $\Gamma$ be the graph of the cubic function $f(x) = x^{3} - 9x^{2} + 15x - 4$. Which of the following is the derivative of $f(x)$? (Single choice, 2 points)
(1) $x^{2} - 9x + 15$
(2) $3x^{3} - 18x^{2} + 15x - 4$
(3) $3x^{3} - 18x^{2} + 15x$
(4) $3x^{2} - 18x + 15$
(5) $x^{2} - 18x + 15$

The least possible value of the gradient of the curve $y = ( 2 x + a ) ( x - 2 a ) ^ { 2 }$ at the point where $x = 1$, as $a$ varies, is
A $- \frac { 49 } { 4 }$ B - 8 C $- \frac { 25 } { 4 }$ D $\frac { 7 } { 4 }$ E $\frac { 47 } { 16 }$

$$f ( x ) = \frac { \left( x ^ { 2 } + 5 \right) ( 2 x ) } { \sqrt [ 4 ] { x ^ { 3 } } } , \quad x > 0$$
Which one of the following is equal to $f ^ { \prime } ( x )$ ?

The function f is given, for $x > 0$, by
$$\mathrm { f } ( x ) = \frac { x ^ { 3 } - 4 x } { 2 \sqrt { x } }$$
Find the value of $f ^ { \prime } ( 4 )$.

Which of the following is an expression for the first derivative with respect to $x$ of
$$\frac { x ^ { 3 } - 5 x ^ { 2 } } { 2 x \sqrt { x } }$$
A $- \frac { \sqrt { x } } { 2 }$
B $\frac { \sqrt { x } } { 4 }$
C $\frac { 3 x - 5 } { 4 \sqrt { x } }$
D $\frac { 3 \sqrt { x } - 5 } { 4 \sqrt { x } }$
E $\frac { 3 \sqrt { x } - 10 } { 3 \sqrt { x } }$
F $\frac { 3 x ^ { 2 } - 10 x } { 3 \sqrt { x } }$