The function $f$ is differentiable on the closed interval $[-6, 5]$ and satisfies $f(-2) = 7$. The graph of $f'$, the derivative of $f$, consists of a semicircle and three line segments, as shown in the figure above.
(a) Find the values of $f(-6)$ and $f(5)$.
(b) On what intervals is $f$ increasing? Justify your answer.
(c) Find the absolute minimum value of $f$ on the closed interval $[-6, 5]$. Justify your answer.
(d) For each of $f''(-5)$ and $f''(3)$, find the value or explain why it does not exist.

The graph of the continuous function $g$, the derivative of the function $f$, is shown above. The function $g$ is piecewise linear for $- 5 \leq x < 3$, and $g ( x ) = 2 ( x - 4 ) ^ { 2 }$ for $3 \leq x \leq 6$.
(a) If $f ( 1 ) = 3$, what is the value of $f ( - 5 )$ ?
(b) Evaluate $\int _ { 1 } ^ { 6 } g ( x ) \, dx$.
(c) For $- 5 < x < 6$, on what open intervals, if any, is the graph of $f$ both increasing and concave up? Give a reason for your answer.
(d) Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason for your answer.

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.

The continuous function $f$ is defined on the closed interval $- 6 \leq x \leq 12$. The graph of $f$, consisting of two semicircles and one line segment, is shown in the figure.
Let $g$ be the function defined by $g ( x ) = \int _ { 6 } ^ { x } f ( t ) d t$.
A. Find $g ^ { \prime } ( 8 )$. Give a reason for your answer.
B. Find all values of $x$ in the open interval $- 6 < x < 12$ at which the graph of $g$ has a point of inflection. Give a reason for your answer.
C. Find $g ( 12 )$ and $g ( 0 )$. Label your answers.
D. Find the value of $x$ at which $g$ attains an absolute minimum on the closed interval $- 6 \leq x \leq 12$. 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?

4. Let $f$ be a function defined on the closed interval $- 3 \leq x \leq 4$ with $f ( 0 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of one line segment and a semicircle, as shown above.
(a) On what intervals, if any, is $f$ increasing? Justify your answer.
(b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $- 3 < x < 4$. Justify your answer.
(c) Find an equation for the line tangent to the graph of $f$ at the point ( 0,3 ).
(d) Find $f ( - 3 )$ and $f ( 4 )$. Show the work that leads to your answers. [Figure]

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 $x = 2$.

Let $f$ be a function defined on the closed interval $- 5 \leq x \leq 5$ with $f ( 1 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of two semicircles and two line segments, as shown above. (a) For $- 5 < x < 5$, find all values $x$ at which $f$ has a relative maximum. Justify your answer. (b) For $- 5 < x < 5$, find all values $x$ at which the graph of $f$ has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of $f$ is concave up and also has positive slope. Explain your reasoning. (d) Find the absolute minimum value of $f ( x )$ over the closed interval $- 5 \leq x \leq 5$. Explain your reasoning.

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 $f ( 3 )$.

Let $f$ be a twice-differentiable function defined on the interval $- 1.2 < x < 3.2$ with $f ( 1 ) = 2$. The graph of $f ^ { \prime }$, the derivative of $f$, is shown above. The graph of $f ^ { \prime }$ crosses the $x$-axis at $x = - 1$ and $x = 3$ and has a horizontal tangent at $x = 2$. Let $g$ be the function given by $g ( x ) = e ^ { f ( x ) }$.
(a) Write an equation for the line tangent to the graph of $g$ at $x = 1$.
(b) For $- 1.2 < x < 3.2$, find all values of $x$ at which $g$ has a local maximum. Justify your answer.
(c) The second derivative of $g$ is $g ^ { \prime \prime } ( x ) = e ^ { f ( x ) } \left[ \left( f ^ { \prime } ( x ) \right) ^ { 2 } + f ^ { \prime \prime } ( x ) \right]$. Is $g ^ { \prime \prime } ( - 1 )$ positive, negative, or zero? Justify your answer.
(d) Find the average rate of change of $g ^ { \prime }$, the derivative of $g$, over the interval $[ 1,3 ]$.

5. Let $f$ and $g$ be the functions defined by $f ( x ) = \frac { 1 } { x }$ and $g ( x ) = \frac { 4 x } { 1 + 4 x ^ { 2 } }$, for all $x > 0$.
(a) Find the absolute maximum value of $g$ on the open interval $( 0 , \infty )$ if the maximum exists. Find the absolute minimum value of $g$ on the open interval $( 0 , \infty )$ if the minimum exists. Justify your answers.
(b) Find the area of the unbounded region in the first quadrant to the right of the vertical line $x = 1$, below the graph of $f$, and above the graph of $g$.

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 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 $x = 3$.

Consider the function $f ( x ) = \frac { 1 } { x ^ { 2 } - k x }$, where $k$ is a nonzero constant. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { k - 2 x } { \left( x ^ { 2 } - k x \right) ^ { 2 } }$.
(a) Let $k = 3$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 3 x }$. Write an equation for the line tangent to the graph of $f$ at the point whose $x$-coordinate is 4.
(b) Let $k = 4$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 4 x }$. Determine whether $f$ has a relative minimum, a relative maximum, or neither at $x = 2$. Justify your answer.
(c) Find the value of $k$ for which $f$ has a critical point at $x = -5$.
(d) Let $k = 6$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 6 x }$. Find the partial fraction decomposition for the function $f$. Find $\int f ( x ) \, dx$.

The function $f$ is differentiable on the closed interval $[-6, 5]$ and satisfies $f(-2) = 7$. The graph of $f'$, the derivative of $f$, consists of a semicircle and three line segments, as shown in the figure.
(a) Find the values of $f(-6)$ and $f(5)$.
(b) On what intervals is $f$ increasing? Justify your answer.
(c) Find the absolute minimum value of $f$ on the closed interval $[-6, 5]$. Justify your answer.
(d) For each of $f''(-5)$ and $f''(3)$, find the value or explain why it does not exist.

Let $f$ be a differentiable function with $f ( 4 ) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f ^ { \prime }$, 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 $\lim_{x \rightarrow 2} \frac{6f(x) - 3x}{x^{2} - 5x + 6}$, 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.

We consider the function $f$ defined on the interval $[0;+\infty[$ by $$f(x) = 5 - \frac{4}{x+2}$$ It will be admitted that $f$ is differentiable on the interval $[0;+\infty[$. The curve $\mathscr{C}$ representing $f$ and the line $\mathscr{D}$ with equation $y = x$ have been drawn in an orthonormal coordinate system in Appendix 1.

1. Prove that $f$ is increasing on the interval $[0;+\infty[$.
2. Solve the equation $f(x) = x$ on the interval $[0;+\infty[$. We denote the solution by $\alpha$. The exact value of $\alpha$ will be given, then an approximate value to $10^{-2}$ will be given.
3. We consider the sequence $(u_{n})$ defined by $u_{0} = 1$ and, for every natural integer $n$, $u_{n+1} = f(u_{n})$.
   On the figure in Appendix 1, using the curve $\mathscr{C}$ and the line $\mathscr{D}$, place the points $M_{0}$, $M_{1}$ and $M_{2}$ with zero ordinate and abscissae $u_{0}$, $u_{1}$ and $u_{2}$ respectively. What conjectures can be made about the direction of variation and the convergence of the sequence $(u_{n})$?
4. a. Prove, by induction, that for every natural integer $n$, $$0 \leqslant u_{n} \leqslant u_{n+1} \leqslant \alpha$$ where $\alpha$ is the real number defined in question 2. b. Can we affirm that the sequence $(u_{n})$ is convergent? The answer will be justified.
5. For every natural integer $n$, we define the sequence $(S_{n})$ by $$S_{n} = \sum_{k=0}^{n} u_{k} = u_{0} + u_{1} + \cdots + u_{n}$$ a. Calculate $S_{0}$, $S_{1}$ and $S_{2}$. Give an approximate value of the results to $10^{-2}$ near. b. Complete the algorithm given in Appendix 2 so that it displays the sum $S_{n}$ for the value of the integer $n$ requested from the user. c. Show that the sequence $(S_{n})$ diverges to $+\infty$.

For each real number $a$, we consider the function $f _ { a }$ defined on the set of real numbers $\mathbb { R }$ by
$$f _ { a } ( x ) = \mathrm { e } ^ { x - a } - 2 x + \mathrm { e } ^ { a } .$$

1. Show that for every real number $a$, the function $f _ { a }$ has a minimum.
2. Does there exist a value of $a$ for which this minimum is as small as possible?

The profile of a slide is modelled by the curve $\mathcal { C }$ representing the function $f$ defined on the interval [1;8] by
$$f ( x ) = ( a x + b ) \mathrm { e } ^ { - x } \text { where } a \text { and } b \text { are two natural integers. }$$
The curve $\mathcal { C }$ is drawn in an orthonormal coordinate system with unit of one metre.

Part A Modelling

1. We want the tangent to the curve $\mathcal { C }$ at its point with abscissa 1 to be horizontal. Determine the value of the integer $b$.
2. We want the top of the slide to be located between 3.5 and 4 metres high. Determine the value of the integer $a$.

Part B An amenity for visitors

We assume in the following that the function $f$ introduced in Part A is defined for all real $x \in [ 1 ; 8 ]$ by
$$f ( x ) = 10 x \mathrm { e } ^ { - x }$$
The retaining wall of the slide will be painted by an artist on a single face. In the quote he proposes, he asks for a flat fee of 300 euros plus 50 euros per square metre painted.

1. Let $g$ be the function defined on [ $1 ; 8$ ] by
   $$g ( x ) = 10 ( - x - 1 ) \mathrm { e } ^ { - x }$$
   Determine the derivative of the function $g$.
2. What is the amount of the artist's quote?

Part C A constraint to verify

Safety considerations require limiting the maximum slope of the slide. Consider a point $M$ on the curve $\mathcal { C }$, with abscissa different from 1. We call $\alpha$ the acute angle formed by the tangent to $\mathcal { C }$ at $M$ and the horizontal axis. The constraints require that the angle $\alpha$ be less than 55 degrees.

1. We denote $f ^ { \prime }$ the derivative of the function $f$ on the interval $[ 1 ; 8 ]$. We admit that, for all $x$ in the interval $[ 1 ; 8 ] , f ^ { \prime } ( x ) = 10 ( 1 - x ) \mathrm { e } ^ { - x }$. Study the variations of the function $f ^ { \prime }$ on the interval [ $1 ; 8$ ].
2. Let $x$ be a real number in the interval ] 1; 8] and let $M$ be the point with abscissa $x$ on the curve $\mathcal { C }$. Justify that $\tan \alpha = \left| f ^ { \prime } ( x ) \right|$.
3. Is the slide compliant with the imposed constraints?