The derivative of a function $f$ is given by $f ^ { \prime } ( x ) = e ^ { \sin x } - \cos x - 1$ for $0 < x < 9$. On what intervals is $f$ decreasing?
(A) $0 < x < 0.633$ and $4.115 < x < 6.916$
(B) $0 < x < 1.947$ and $5.744 < x < 8.230$
(C) $0.633 < x < 4.115$ and $6.916 < x < 9$
(D) $1.947 < x < 5.744$ and $8.230 < x < 9$

Let $f$ be the function given by $f ( x ) = 300 x - x ^ { 3 }$. On which of the following intervals is the function $f$ increasing?
(A) $( - \infty , - 10 ]$ and $[ 10 , \infty )$
(B) $[ - 10,10 ]$
(C) $[ 0,10 ]$ only
(D) $[ 0,10 \sqrt { 3 } ]$ only
(E) $[ 0 , \infty )$

A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon $(t = 0)$ and 8 P.M. $(t = 8)$. The number of entries in the box $t$ hours after noon is modeled by a differentiable function $E$ for $0 \leq t \leq 8$. Values of $E(t)$, in hundreds of entries, at various times $t$ are shown in the table below.

\begin{tabular}{ c } $t$

(hours)

& 0 & 2 & 5 & 7 & 8 \hline

$E(t)$

(hundreds of

entries)

& 0 & 4 & 13 & 21 & 23 \hline \end{tabular}
(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time $t = 6$. Show the computations that lead to your answer.
(b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of $\frac{1}{8}\int_{0}^{8} E(t)\,dt$. Using correct units, explain the meaning of $\frac{1}{8}\int_{0}^{8} E(t)\,dt$ in terms of the number of entries.
(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function $P$, where $P(t) = t^3 - 30t^2 + 298t - 976$ hundreds of entries per hour for $8 \leq t \leq 12$. According to the model, how many entries had not yet been processed by midnight $(t = 12)$?
(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer.

From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by $A ( t ) = 450 \sqrt { \sin ( 0.62 t ) }$, where $t$ is the number of hours after 5 A.M. and $A ( t )$ is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line.
(a) Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. $( t = 1 )$ to 10 A.M. $( t = 5 )$.
(b) Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. $( t = 1 )$ to 10 A.M. $( t = 5 )$.
(c) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. ( $t = 1$ ) increasing or decreasing? Give a reason for your answer.
(d) A line forms whenever $A ( t ) \geq 400$. The number of vehicles in line at time $t$, for $a \leq t \leq 4$, is given by $N ( t ) = \int _ { a } ^ { t } ( A ( x ) - 400 ) d x$, where $a$ is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval $a \leq t \leq 4$. Justify your answer.

An invasive species of plant appears in a fruit grove at time $t = 0$ and begins to spread. The function $C$ defined by $C ( t ) = 7.6 \arctan ( 0.2 t )$ models the number of acres in the fruit grove affected by the species $t$ weeks after the species appears. It can be shown that $C ^ { \prime } ( t ) = \frac { 38 } { 25 + t ^ { 2 } }$.
(Note: Your calculator should be in radian mode.)
A. Find the average number of acres affected by the invasive species from time $t = 0$ to time $t = 4$ weeks. Show the setup for your calculations.
B. Find the time $t$ when the instantaneous rate of change of $C$ equals the average rate of change of $C$ over the time interval $0 \leq t \leq 4$. Show the setup for your calculations.
C. Assume that the invasive species continues to spread according to the given model for all times $t > 0$. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.
D. At time $t = 4$ weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function $A$, defined by $A ( t ) = C ( t ) - \int _ { 4 } ^ { t } 0.1 \cdot \ln ( x ) d x$, models the number of acres affected by the species over the time interval $4 \leq t \leq 36$. At what time $t$, for $4 \leq t \leq 36$, does $A$ attain its maximum value? Justify your answer.

A digital black and white image is composed of small squares (pixels) whose colour ranges from white to black through all shades of grey. Each shade is coded by a real number $x$ as follows:

• $x = 0$ for white;
• $x = 1$ for black;
• $x = 0.01; x = 0.02$ and so on up to $x = 0.99$ in steps of 0.01 for all intermediate shades (from light to dark).

A function $f$ defined on the interval $[0; 1]$ is called a ``retouching function'' if it has the following four properties:

• $f(0) = 0$;
• $f(1) = 1$;
• $f$ is continuous on the interval $[0; 1]$;
• $f$ is increasing on the interval $[0; 1]$.

A shade coded $x$ is said to be darkened by the function $f$ if $f(x) > x$, and lightened if $f(x) < x$.

Part A

1. We consider the function $f_{1}$ defined on the interval $[0; 1]$ by: $$f_{1}(x) = 4x^{3} - 6x^{2} + 3x$$ a) Prove that the function $f_{1}$ is a retouching function. b) Solve graphically the inequality $f_{1}(x) \leq x$, using the graph given in the appendix, to be returned with your answer sheet, showing the necessary dotted lines. Interpret this result in terms of lightening or darkening.
2. We consider the function $f_{2}$ defined on the interval $[0; 1]$ by: $$f_{2}(x) = \ln[1 + (e - 1)x]$$ We admit that $f_{2}$ is a retouching function. We define on the interval $[0; 1]$ the function $g$ by: $g(x) = f_{2}(x) - x$. a) Establish that, for all $x$ in the interval $[0; 1]$: $g'(x) = \frac{(e - 2) - (e - 1)x}{1 + (e - 1)x}$; b) Determine the variations of the function $g$ on the interval $[0; 1]$. Prove that the function $g$ has a maximum at $\frac{e - 2}{e - 1}$, a maximum whose value rounded to the nearest hundredth is 0.12. c) Establish that the equation $g(x) = 0.05$ has two solutions $\alpha$ and $\beta$ on the interval $[0; 1]$, with $\alpha < \beta$. We will admit that: $0.08 < \alpha < 0.09$ and that: $0.85 < \beta < 0.86$.

Part B

We note that a modification of shade is visually perceptible only if the absolute value of the difference between the code of the initial shade and the code of the modified shade is greater than or equal to 0.05.

1. In the algorithm described below, $f$ denotes a retouching function. What is the role of this algorithm? \begin{verbatim} Variables : x (initial shade) y (retouched shade) E (difference) c (counter) k Initialization : c takes the value 0 Processing: For k ranging from 0 to 100, do x takes the value k/100 y takes the value f(x) E takes the value |y - x| If E >= 0.05, do c takes the value c + 1 End if End for Output: Display c \end{verbatim}
2. What value will this algorithm display if applied to the function $f_{2}$ defined in the second question of part $\mathbf{A}$?

Part C

In this part, we are interested in retouching functions $f$ whose effect is to lighten the image overall, that is, such that, for all real $x$ in the interval $[0; 1]$, $f(x) \leq x$. We decide to measure the overall lightening of the image by calculating the area $\mathscr{A}_{f}$ of the portion of the plane between the x-axis, the curve representing the function $f$, and the lines with equations $x = 0$ and $x = 1$ respectively. Between two functions, the one that has the effect of lightening the image the most is the one corresponding to the smallest area. We wish to compare the effect of the following two functions, which we admit are retouching functions:
$$f_{3}(x) = x\mathrm{e}^{(x^{2} - 1)} \quad f_{4}(x) = 4x - 15 + \frac{60}{x + 4}$$

1. a) Calculate $\mathscr{A}_{f_{3}}$. b) Calculate $\mathscr{A}_{f_{4}}$
2. Of these two functions, which one has the effect of lightening the image the most?

Exercise B (Main topics: Sequences, function study, Logarithm function)
Let the function $f$ defined on the interval $]1; +\infty[$ by $$f(x) = x - \ln(x-1).$$ Consider the sequence $(u_n)$ with initial term $u_0 = 10$ and such that $u_{n+1} = f(u_n)$ for every natural integer $n$.

Part I:

The spreadsheet below was used to obtain approximate values of the first terms of the sequence $(u_n)$.

	A	B
1	$n$	$u_n$
2	0	10
3	1	7.80277542
4	2	5.88544474
5	3	4.29918442
6	4	3.10550913
7	5	2.36095182
8	6	2.0527675
9	7	2.00134509
10	8	2.0000009

1. What formula was entered in cell B3 to allow the calculation of approximate values of $(u_n)$ by copying downward?
2. Using these values, conjecture the direction of variation and the limit of the sequence $(u_n)$.

Part II:

We recall that the function $f$ is defined on the interval $]1; +\infty[$ by $$f(x) = x - \ln(x-1).$$

1. Calculate $\lim_{x \rightarrow 1} f(x)$. We will admit that $\lim_{x \rightarrow +\infty} f(x) = +\infty$.
2. a. Let $f^{\prime}$ be the derivative function of $f$. Show that for all $x \in ]1; +\infty[$, $f^{\prime}(x) = \frac{x-2}{x-1}$. b. Deduce the table of variations of $f$ on the interval $]1; +\infty[$, completed by the limits. c. Justify that for all $x \geqslant 2$, $f(x) \geqslant 2$.

Part III:

1. Using the results of Part II, prove by induction that $u_n \geqslant 2$ for every natural integer $n$.
2. Show that the sequence $(u_n)$ is decreasing.
3. Deduce that the sequence $(u_n)$ is convergent. We denote its limit by $\ell$.
4. We admit that $\ell$ satisfies $f(\ell) = \ell$. Give the value of $\ell$.

We are given a function $f$, assumed to be differentiable on $\mathbb{R}$, and we denote $f^{\prime}$ its derivative function.
Below is the variation table of $f$:

$x$	$-\infty$	$-1$	$+\infty$
$f(x)$
	$-\infty$	0

According to this variation table: a. $f^{\prime}$ is positive on $\mathbb{R}$. b. $f^{\prime}$ is positive on $\left.]-\infty;-1\right]$ c. $f^{\prime}$ is negative on $\mathbb{R}$ d. $f^{\prime}$ is positive on $[-1;+\infty[$

Part I

We consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = x - \mathrm{e}^{-2x}$$
We call $\Gamma$ the representative curve of the function $f$ in an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath})$.

1. Determine the limits of the function $f$ at $-\infty$ and at $+\infty$.
2. Study the monotonicity of the function $f$ on $\mathbb{R}$ and draw up its variation table.
3. Show that the equation $f(x) = 0$ has a unique solution $\alpha$ on $\mathbb{R}$, and give an approximate value to $10^{-2}$ precision.
4. Deduce from the previous questions the sign of $f(x)$ according to the values of $x$.

Part II

In the orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath})$, we call $\mathscr{C}$ the representative curve of the function $g$ defined on $\mathbb{R}$ by:
$$g(x) = \mathrm{e}^{-x}$$
The curves $\mathscr{C}$ and the curve $\Gamma$ (which represents the function $f$ from Part I) are drawn on the graph provided in the appendix which is to be completed and returned with your paper. The purpose of this part is to determine the point on the curve $\mathscr{C}$ closest to the origin O of the coordinate system and to study the tangent to $\mathscr{C}$ at this point.

1. For any real number $t$, we denote by $M$ the point with coordinates $(t; \mathrm{e}^{-t})$ on the curve $\mathscr{C}$.
   We consider the function $h$ which, to the real number $t$, associates the distance $OM$. We therefore have: $h(t) = OM$, that is:
   $$h(t) = \sqrt{t^2 + \mathrm{e}^{-2t}}$$
   a. Show that, for any real number $t$,
   $$h'(t) = \frac{f(t)}{\sqrt{t^2 + \mathrm{e}^{-2t}}}$$
   where $f$ denotes the function studied in Part I. b. Prove that the point A with coordinates $(\alpha; \mathrm{e}^{-\alpha})$ is the point on the curve $\mathscr{C}$ for which the length $OM$ is minimal. Place this point on the graph provided in the appendix, to be returned with your paper.
   
2. We call $T$ the tangent to the curve $\mathscr{C}$ at A. a. Express in terms of $\alpha$ the slope of the tangent $T$.
   We recall that the slope of the line (OA) is equal to $\frac{\mathrm{e}^{-\alpha}}{\alpha}$. We also recall the following result which may be used without proof: In an orthonormal coordinate system of the plane, two lines $D$ and $D'$ with slopes $m$ and $m'$ respectively are perpendicular if and only if the product $mm'$ is equal to $-1$. b. Prove that the line (OA) and the tangent $T$ are perpendicular.
   Draw these lines on the graph provided in the appendix, to be returned with your paper.

(a) Compute $\dfrac{d}{dx}\left[\int_{0}^{e^{x}} \log(t)\cos^{4}(t)\,dt\right]$.
(b) For $x > 0$ define $F(x) = \int_{1}^{x} t\log(t)\,dt$.
i. Determine the open interval(s) (if any) where $F(x)$ is decreasing and the open interval(s) (if any) where $F(x)$ is increasing.
ii. Determine all the local minima of $F(x)$ (if any) and the local maxima of $F(x)$ (if any).

Find the maximum value of the real number $a$ such that the function $f ( x ) = x ^ { 3 } + a x ^ { 2 } - \left( a ^ { 2 } - 8 a \right) x + 3$ is increasing on the entire set of real numbers. [3 points]

Let the function $f ( x ) = \frac { x ^ { 2 } } { 2 } - k \ln x$, $k > 0$\n(I) Find the monotonic intervals and extreme values of $f ( x )$;\n(II) Prove that if $f ( x )$ has a zero point, then $f ( x )$ has exactly one zero point on the interval $( 1 , \sqrt { e } )$.

21. (This question is worth 14 points) Given the function $f ( x ) = - 2 \ln x + x ^ { 2 } - 2 a x + a ^ { 2 }$, where $a > 0$. (1) Let $g ( x )$ be the derivative of $f ( x )$. Discuss the monotonicity of $g ( x )$; (2) Prove: there exists $a \in ( 0,1 )$ such that $f ( x ) \geq 0$ holds for all $x$

9. If the function $f ( x ) = \frac { 1 } { 2 } ( m - 2 ) x ^ { 2 } + ( n - 8 ) x + 1$ $(m \geq 0, n \geq 0)$ is monotonically decreasing on the interval $\left[ \frac { 1 } { 2 }, 2 \right]$, then the maximum value of $m n$ is
(A) $16$
(B) $18$
(C) $25$
(D) $\frac { 81 } { 2 }$

21. Given the function $f ( x ) = - 2 ( x + a ) \ln x + x ^ { 2 } - 2 a x - 2 a ^ { 2 } + a$, where $a > 0$.
(1) Let $g ( x )$ be the derivative of $f ( x )$. Discuss the monotonicity of $g ( x )$;
(2) Prove: there exists $a \in ( 0, 1 )$ such that $f ( x ) \geq 0$ holds on the interval $(1, + \infty)$, and $f ( x ) = 0$ has a unique solution in $(1, + \infty)$.

Given the function $f ( x ) = \frac { 1 } { 3 } x ^ { 3 } - a \left( x ^ { 2 } + x + 1 \right)$.
(1) When $a = 3$, find the monotonic intervals of $f ( x )$;
(2) Prove: $f ( x )$ has exactly one zero.

20. Let $f ( x ) = a ^ { 2 } x ^ { 2 } + a x - 3 \ln x + 1$, where $a > 0$.
(1) Discuss the monotonicity of $f ( x )$;
(2) If the graph of $y = f ( x )$ has no common points with the $x$-axis, find the range of values for $a$.

Given $f(x) = ax - \frac{\sin x}{\cos^{2} x} , \quad x \in \left(0 , \frac{\pi}{2}\right)$ ,
(1) When $a = 8$ , discuss the monotonicity of $f(x)$ ;
(2) If $f(x) < \sin 2x$ , find the range of values for $a$ .

Given function $f ( x ) = \left\{ \begin{array} { l l } - x ^ { 2 } - 2 a x - a , & x < 0 , \\ \mathrm { e } ^ { x } + \ln ( x + 1 ) , & x \geqslant 0 \end{array} \right.$ is monotonically increasing on $\mathbb { R }$ , then the range of $a$ is
A. $( - \infty , 0 ]$
B. $[ - 1,0 ]$
C. $[ - 1,1 ]$
D. $[ 0 , + \infty )$

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

If $f(x) = \cos(x) - 1 + \frac{x^2}{2}$, then
(A) $f(x)$ is an increasing function on the real line
(B) $f(x)$ is a decreasing function on the real line
(C) $f(x)$ is increasing on $-\infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f(x)$ is decreasing on $-\infty < x \leq 0$ and increasing on $0 \leq x < \infty$

Let $f(x) = ax^3 + bx^2 + cx + d$ be a strictly increasing function with $a > 0$. Define $g(x) = f'(x) - 6ax - 2b + 6a$. Then $g(x)$ is
(A) always negative (B) always positive (C) sometimes positive sometimes negative (D) always zero

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(a) $f ( x )$ is an increasing function on the real line
(b) $f ( x )$ is a decreasing function on the real line
(c) $f ( x )$ is increasing on the interval $- \infty < x \leq 0$ and decreasing on the interval $0 \leq x < \infty$
(d) $f ( x )$ is decreasing on the interval $- \infty < x \leq 0$ and increasing on the interval $0 \leq x < \infty$.

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(a) $\alpha \geq 2$
(b) $\alpha < 2$
(c) $\alpha < - 1$
(d) $\alpha > 2$.

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(a) $f ( x )$ is an increasing function on the real line
(b) $f ( x )$ is a decreasing function on the real line
(c) $f ( x )$ is increasing on the interval $- \infty < x \leq 0$ and decreasing on the interval $0 \leq x < \infty$
(d) $f ( x )$ is decreasing on the interval $- \infty < x \leq 0$ and increasing on the interval $0 \leq x < \infty$.