For $- 1.5 < x < 1.5$, let $f$ be a function with first derivative given by $f ^ { \prime } ( x ) = e ^ { \left( x ^ { 4 } - 2 x ^ { 2 } + 1 \right) } - 2$. Which of the following are all intervals on which the graph of $f$ is concave down?
(A) $(-0.418, 0.418)$ only
(B) $( - 1,1 )$
(C) $( - 1.354 , - 0.409 )$ and $( 0.409,1.354 )$
(D) $( - 1.5 , - 1 )$ and $( 0,1 )$
(E) $( - 1.5 , - 1.354 ) , ( - 0.409,0 )$, and $( 1.354,1.5 )$

The graph of $f ^ { \prime }$, the derivative of $f$, is shown in the figure above. The function $f$ has a local maximum at $x =$
(A) $-3$
(B) $-1$
(C) 1
(D) 3
(E) 4

The graph of $f ^ { \prime \prime }$, the second derivative of $f$, is shown above for $- 2 \leq x \leq 4$. What are all intervals on which the graph of the function $f$ is concave down?
(A) $-1 < x < 1$
(B) $0 < x < 2$
(C) $1 < x < 3$ only
(D) $-2 < x < -1$ only
(E) $-2 < x < -1$ and $1 < x < 3$

Let $f$ be a polynomial function with values of $f ^ { \prime } ( x )$ at selected values of $x$ given in the table above.

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

Which of the following must be true for $-2 < x < 6$ ?
(A) The graph of $f$ is concave up.
(B) The graph of $f$ has at least two points of inflection.
(C) $f$ is increasing.
(D) $f$ has no critical points.
(E) $f$ has at least two relative extrema.

The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $0 \leq x \leq 8$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$, $x = 3$, and $x = 5$. The areas of the regions between the graph of $f ^ { \prime }$ and the $x$-axis are labeled in the figure. The function $f$ is defined for all real numbers and satisfies $f ( 8 ) = 4$.
(a) Find all values of $x$ on the open interval $0 < x < 8$ for which the function $f$ has a local minimum. Justify your answer.
(b) Determine the absolute minimum value of $f$ on the closed interval $0 \leq x \leq 8$. Justify your answer.
(c) On what open intervals contained in $0 < x < 8$ is the graph of $f$ both concave down and increasing? Explain your reasoning.
(d) The function $g$ is defined by $g ( x ) = ( f ( x ) ) ^ { 3 }$. If $f ( 3 ) = - \frac { 5 } { 2 }$, find the slope of the line tangent to the graph of $g$ at $x = 3$.

Let $f$ be a differentiable function with $f(4) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f'$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above.
(a) Find $f(0)$ and $f(5)$.
(b) Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$. Justify your answer.
(c) Let $g$ be the function defined by $g(x) = f(x) - x$. On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$? Show the analysis that leads to your answer.
(d) For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$. Justify your answer.

The function $f$ is defined on the closed interval $[-2, 8]$ and satisfies $f(2) = 1$. The graph of $f'$, the derivative of $f$, consists of two line segments and a semicircle, as shown in the figure.
(a) Does $f$ have a relative minimum, a relative maximum, or neither at $x = 6$? Give a reason for your answer.
(b) On what open intervals, if any, is the graph of $f$ concave down? Give a reason for your answer.
(c) Find the value of $\lim_{x \to 2} \frac{6f(x) - 3x}{x^{2} - 5x + 6}$, or show that it does not exist. Justify your answer.
(d) Find the absolute minimum value of $f$ on the closed interval $[-2, 8]$. Justify your answer.

Let $f$ be the function given by $f(x) = 2xe^{2x}$.
(a) Find $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$.
(b) Find the absolute minimum value of $f$. Justify that your answer is an absolute minimum.
(c) What is the range of $f$?
(d) Consider the family of functions defined by $y = bxe^{bx}$, where $b$ is a nonzero constant. Show that the absolute minimum value of $bxe^{bx}$ is the same for all nonzero values of $b$.

Let $h$ be a function defined for all $x \neq 0$ such that $h(4) = -3$ and the derivative of $h$ is given by $h'(x) = \dfrac{x^2 - 2}{x}$ for all $x \neq 0$.
(a) Find all values of $x$ for which the graph of $h$ has a horizontal tangent, and determine whether $h$ has a local maximum, a local minimum, or neither at each of these values. Justify your answers.
(b) On what intervals, if any, is the graph of $h$ concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of $h$ at $x = 4$.
(d) Does the line tangent to the graph of $h$ at $x = 4$ lie above or below the graph of $h$ for $x > 4$? Why?

The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, on the closed interval $- 1 \leq x \leq 5$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$ and $x = 3$. The function $f$ is twice differentiable with $f ( 2 ) = 6$.
(a) Find the $x$-coordinate of each of the points of inflection of the graph of $f$. Give a reason for your answer.
(b) At what value of $x$ does $f$ attain its absolute minimum value on the closed interval $- 1 \leq x \leq 5$ ? At what value of $x$ does $f$ attain its absolute maximum value on the closed interval $- 1 \leq x \leq 5$ ? Show the analysis that leads to your answers.
(c) Let $g$ be the function defined by $g ( x ) = x f ( x )$. Find an equation for the line tangent to the graph of $g$ at $x = 2$.

The derivative of a function $f$ is given by $f ^ { \prime } ( x ) = ( x - 3 ) e ^ { x }$ for $x > 0$, and $f ( 1 ) = 7$.
(a) The function $f$ has a critical point at $x = 3$. At this point, does $f$ have a relative minimum, a relative maximum, or neither? Justify your answer.
(b) On what intervals, if any, is the graph of $f$ both decreasing and concave up? Explain your reasoning.
(c) Find the value of $f ( 3 )$.

The rate at which people enter an auditorium for a rock concert is modeled by the function $R$ given by $R(t) = 1380t^{2} - 675t^{3}$ for $0 \leq t \leq 2$ hours; $R(t)$ is measured in people per hour. No one is in the auditorium at time $t = 0$, when the doors open. The doors close and the concert begins at time $t = 2$.
(a) How many people are in the auditorium when the concert begins?
(b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer.
(c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function $w$ models the total wait time for all the people who enter the auditorium before time $t$. The derivative of $w$ is given by $w'(t) = (2 - t)R(t)$. Find $w(2) - w(1)$, the total wait time for those who enter the auditorium after time $t = 1$.
(d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c).

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.

The function $f$, whose graph is shown above, is defined on the interval $- 2 \leq x \leq 2$. Which of the following statements about $f$ is false?
(A) $f$ is continuous at $x = 0$.
(B) $f$ is differentiable at $x = 0$.
(C) $f$ has a critical point at $x = 0$.
(D) $f$ has an absolute minimum at $x = 0$.
(E) The concavity of the graph of $f$ changes at $x = 0$.

The graph of $f ^ { \prime }$, the derivative of the function $f$, is shown above. Which of the following statements must be true?
I. $f$ has a relative minimum at $x = - 3$.
II. The graph of $f$ has a point of inflection at $x = - 2$.
III. The graph of $f$ is concave down for $0 < x < 4$.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Let $f$ be a function that is twice differentiable on $- 2 < x < 2$ and satisfies the conditions in the table above. If $f ( x ) = f ( - x )$, what are the $x$-coordinates of the points of inflection of the graph of $f$ on $- 2 < x < 2$ ?

\cline{2-3} \multicolumn{1}{c|}{}	$0 < x < 1$	$1 < x < 2$
$f ( x )$	Positive	Negative
$f ^ { \prime } ( x )$	Negative	Negative
$f ^ { \prime \prime } ( x )$	Negative	Positive

(A) $x = 0$ only
(B) $x = 1$ only
(C) $x = 0$ and $x = 1$
(D) $x = - 1$ and $x = 1$
(E) There are no points of inflection on $- 2 < x < 2$.

For $- 1.5 < x < 1.5$, let $f$ be a function with first derivative given by $f ^ { \prime } ( x ) = e ^ { \left( x ^ { 4 } - 2 x ^ { 2 } + 1 \right) } - 2$. Which of the following are all intervals on which the graph of $f$ is concave down?
(A) (-0.418, 0.418) only
(B) $( - 1, 1 )$
(C) $( - 1.354 , - 0.409 )$ and $( 0.409, 1.354 )$
(D) $( - 1.5 , - 1 )$ and $( 0, 1 )$
(E) $( - 1.5 , - 1.354 ) , ( - 0.409, 0 )$, and $( 1.354, 1.5 )$

The derivative of a function $f$ is increasing for $x < 0$ and decreasing for $x > 0$. Which of the following could be the graph of $f$ ?
(A), (B), (C), (D), (E) [graphs shown in figures above]

The figure above shows the graph of $f ^ { \prime }$, the derivative of a twice-differentiable function $f$, on the closed interval $0 \leq x \leq 8$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = 1$, $x = 3$, and $x = 5$. The areas of the regions between the graph of $f ^ { \prime }$ and the $x$-axis are labeled in the figure. The function $f$ is defined for all real numbers and satisfies $f ( 8 ) = 4$.
(a) Find all values of $x$ on the open interval $0 < x < 8$ for which the function $f$ has a local minimum. Justify your answer.
(b) Determine the absolute minimum value of $f$ on the closed interval $0 \leq x \leq 8$. Justify your answer.
(c) On what open intervals contained in $0 < x < 8$ is the graph of $f$ both concave down and increasing? Explain your reasoning.
(d) The function $g$ is defined by $g ( x ) = ( f ( x ) ) ^ { 3 }$. If $f ( 3 ) = - \frac { 5 } { 2 }$, find the slope of the line tangent to the graph of $g$ at $x = 3$.

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.

The function $f$ is defined on the closed interval $[-2, 8]$ and satisfies $f(2) = 1$. The graph of $f'$, the derivative of $f$, consists of two line segments and a semicircle, as shown in the figure.
(a) Does $f$ have a relative minimum, a relative maximum, or neither at $x = 6$? Give a reason for your answer.
(b) On what open intervals, if any, is the graph of $f$ concave down? Give a reason for your answer.
(c) Find the value of $\lim_{x \rightarrow 2} \frac{6f(x) - 3x}{x^{2} - 5x + 6}$, or show that it does not exist. Justify your answer.
(d) Find the absolute minimum value of $f$ on the closed interval $[-2, 8]$. 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.

Let $f$ be the differentiable function defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \mathrm { e } ^ { x } + \frac { 1 } { x }$$
1. Study of an auxiliary function
a. Let the function $g$ be differentiable, defined on $[ 0 ; + \infty [$ by
$$g ( x ) = x ^ { 2 } \mathrm { e } ^ { x } - 1 .$$
Study the direction of variation of the function $g$.
b. Prove that there exists a unique real number $a$ belonging to $[ 0 ; + \infty [$ such that $g ( a ) = 0$.
Prove that $a$ belongs to the interval $[ 0{,}703 ; 0{,}704 [$.
c. Determine the sign of $g ( x )$ on $[ 0 ; + \infty [$.
2. Study of the function $f$
a. Determine the limits of the function $f$ at 0 and at $+ \infty$.
b. Let $f ^ { \prime }$ denote the derivative function of $f$ on the interval $] 0 ; + \infty [$.
Prove that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { g ( x ) } { x ^ { 2 } }$.
c. Deduce the direction of variation of the function $f$ and draw its variation table on the interval $] 0 ; + \infty [$.
d. Prove that the function $f$ has as its minimum the real number
$$m = \frac { 1 } { a ^ { 2 } } + \frac { 1 } { a } .$$
e. Justify that $3{,}43 < m < 3{,}45$.

Consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = ( x + 2 ) \mathrm { e } ^ { - x }.$$
We denote by $\mathscr { C }$ the representative curve of the function $f$ in an orthogonal coordinate system.

1. Study of the function $f$. a. Determine the coordinates of the intersection points of the curve $\mathscr { C }$ with the axes of the coordinate system. b. Study the limits of the function $f$ at $- \infty$ and at $+ \infty$. Deduce any possible asymptotes of the curve $\mathscr { C }$. c. Study the variations of $f$ on $\mathbb { R }$.
2. Calculation of an approximate value of the area under a curve.

We denote by $\mathscr { D }$ the region between the $x$-axis, the curve $\mathscr { C }$ and the lines with equations $x = 0$ and $x = 1$. We approximate the area of the region $\mathscr { D }$ by calculating a sum of areas of rectangles. a. In this question, we divide the interval $[ 0 ; 1 ]$ into four intervals of equal length:

• On the interval $\left[ 0 ; \frac { 1 } { 4 } \right]$, we construct a rectangle of height $f ( 0 )$
• On the interval $\left[ \frac { 1 } { 4 } ; \frac { 1 } { 2 } \right]$, we construct a rectangle of height $f \left( \frac { 1 } { 4 } \right)$
• On the interval $\left[ \frac { 1 } { 2 } ; \frac { 3 } { 4 } \right]$, we construct a rectangle of height $f \left( \frac { 1 } { 2 } \right)$
• On the interval $\left[ \frac { 3 } { 4 } ; 1 \right]$, we construct a rectangle of height $f \left( \frac { 3 } { 4 } \right)$

The algorithm below allows us to obtain an approximate value of the area of the region $\mathscr { D }$ by adding the areas of the four preceding rectangles:

Variables :	$k$ is an integer
	$S$ is a real number
Initialization :	Assign to $S$ the value 0
Processing:	For $k$ varying from 0 to 3
	$\mid$ Assign to $S$ the value $S + \frac { 1 } { 4 } f \left( \frac { k } { 4 } \right)$
	End For
Output :	Display $S$

Give an approximate value to $10 ^ { - 3 }$ of the result displayed by this algorithm. b. In this question, $N$ is an integer strictly greater than 1. We divide the interval $[ 0 ; 1 ]$ into $N$ intervals of equal length. On each of these intervals, we construct a rectangle by proceeding in the same manner as in question 2.a. Modify the preceding algorithm so that it displays as output the sum of the areas of the $N$ rectangles thus constructed.
3. Calculation of the exact value of the area under a curve.
Let $g$ be the function defined on $\mathbb { R }$ by
$$g ( x ) = ( - x - 3 ) \mathrm { e } ^ { - x }$$
We admit that $g$ is an antiderivative of the function $f$ on $\mathbb { R }$. a. Calculate the area $\mathscr { A }$ of the region $\mathscr { D }$, expressed in square units. b. Give an approximate value to $10 ^ { - 3 }$ of the error made by replacing $\mathscr { A }$ by the approximate value found using the algorithm of question 2.a, that is the difference between these two values.

In the plane with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath }$), the representative curve $\mathscr { C }$ of a function $f$ defined and differentiable on the interval $] 0 ; + \infty [$ is given.
We have the following information:

• the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ have coordinates $(1,0)$, $(1,2)$, $(0,2)$ respectively;
• the curve $\mathscr { C }$ passes through point B and the line (BC) is tangent to $\mathscr { C }$ at B;
• there exist two positive real numbers $a$ and $b$ such that for every strictly positive real number $x$, $$f ( x ) = \frac { a + b \ln x } { x } .$$

1. 1. [a.] Using the graph, give the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
   2. [b.] Verify that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { ( b - a ) - b \ln x } { x ^ { 2 } }$.
   3. [c.] Deduce the real numbers $a$ and $b$.
2. 1. [a.] Justify that for every real number $x$ in the interval $] 0 , + \infty [$, $f ^ { \prime } ( x )$ has the same sign as $- \ln x$.
   2. [b.] Determine the limits of $f$ at 0 and at $+ \infty$. We may note that for every strictly positive real number $x$, $f ( x ) = \frac { 2 } { x } + 2 \frac { \ln x } { x }$.
   3. [c.] Deduce the table of variations of the function $f$.
3. 1. [a.] Prove that the equation $f ( x ) = 1$ has a unique solution $\alpha$ on the interval $] 0,1 ]$.
   2. [b.] By similar reasoning, we prove that there exists a unique real number $\beta$ in the interval $] 1 , + \infty [$ such that $f ( \beta ) = 1$. Determine the integer $n$ such that $n < \beta < n + 1$.
4. The following algorithm is given.
   

   	\begin{tabular}{l} Variables:
   $a , b$ and $m$ are real numbers.
   Initialization:
   Assign to $a$ the value 0.
   Assign to $b$ the value 1.
   Processing:
   While $b - a > 0.1$
   Assign to $m$ the value $\frac { 1 } { 2 } ( a + b )$.
   If $f ( m ) < 1$ then Assign to $a$ the value $m$. Otherwise Assign to $b$ the value $m$.
   End If.
   End While.
   Output:
   Display $a$.
   Display $b$.

   \hline \end{tabular}
   
   1. [a.] Run this algorithm by completing the table below, which you will copy onto your answer sheet.
      

      	step 1	step 2	step 3	step 4	step 5
      $a$	0
      $b$	1
      $b - a$
      $m$

      
      
   2. [b.] What do the values displayed by this algorithm represent?
   3. [c.] Modify the algorithm above so that it displays the two bounds of an interval containing $\beta$ with amplitude $10 ^ { - 1 }$.
5. The purpose of this question is to prove that the curve $\mathscr { C }$ divides the rectangle OABC into two regions of equal area.
   1. [a.] Justify that this amounts to proving that $\int _ { \frac { 1 } { \mathrm { e } } } ^ { 1 } f ( x ) \mathrm { d } x = 1$.
   2. [b.] By noting that the expression of $f ( x )$ can be written as $\frac { 2 } { x } + 2 \times \frac { 1 } { x } \times \ln x$, complete the proof.