5. Let $R$ be the region in the first quadrant under the graph of $y = \frac { x } { x ^ { 2 } + 2 }$ for $0 \leqq x \leqq \sqrt { 6 }$.
(a) Find the area of $R$.
(b) If the line $\mathrm { x } = k$ divides $R$ into two regions of equal area, what is the value of $k$ ?
(c) What is the average value of $y = \frac { x } { x ^ { 2 } + 2 }$ on the interval $0 \leqq x \leqq \sqrt { 6 }$ ?
Solution Distribution of Points
(a) $A = \int _ { 0 } ^ { \sqrt { 6 } } \frac { x } { x ^ { 2 } + 2 } d x$
$$\begin{aligned} & = \left. \frac { 1 } { 2 } \ln \left( x ^ { 2 } + 2 \right) \right| _ { 0 } ^ { \sqrt { 6 } } \\ & = \frac { 1 } { 2 } \ln 8 - \frac { 1 } { 2 } \ln 2 = \ln 2 \end{aligned}$$
(b) $\frac { 1 } { 2 } \ln 2 = \int _ { 0 } ^ { k } \frac { x } { x ^ { 2 } + 2 } d x$
$$\begin{aligned} & = \left. \frac { 1 } { 2 } \ln \left( x ^ { 2 } + 2 \right) \right| _ { 0 } ^ { k } \\ & = \frac { 1 } { 2 } \ln \left( k ^ { 2 } + 2 \right) - \frac { 1 } { 2 } \ln 2 \end{aligned}$$
$$\begin{aligned} & \therefore \frac { 1 } { 2 } \ln \left( k ^ { 2 } + 2 \right) = \frac { 1 } { 2 } \ln 2 + \frac { 1 } { 2 } \ln 2 \\ & \quad \text { or } \ln \left( k ^ { 2 } + 2 \right) = \ln 4 \\ & \therefore k ^ { 2 } + 2 = 4 \text { and } k = \sqrt { 2 } \end{aligned}$$
(c) Average value $= \frac { 1 } { \sqrt { 6 } - 0 } \int _ { 0 } ^ { \sqrt { 6 } } \frac { \mathrm { x } } { \mathrm { x } ^ { 2 } + 2 } \mathrm { dx }$
$$= \frac { 1 } { \sqrt { 6 } } \ln 2$$
(a) 1: for correct integral $3 : \left\{ \begin{array} { l } 1 : \text { for antiderivative } \\ 1 : \text { for evaluation } \end{array} \right.$
(b) $\quad \begin{cases} 1 : & \text { for a correct equation } \\ & \text { involving integral(s) } \\ 1 : & \text { for antiderivative } \\ 1 : & \text { for evaluation of integral(s) } \end{cases}$
1: for finding the value of $k$
(c) $2 : \left\{ \begin{array} { l } 1 : \text { for correct integral } \\ 1 : \text { for evaluation } \end{array} \right.$

25. What is the area of the region between the graphs of $y = x ^ { 2 }$ and $y = - x$ from $x = 0$ to $x = 2$ ?
(A) $\frac { 2 } { 3 }$
(B) $\frac { 8 } { 3 }$
(C) 4
(D) $\frac { 14 } { 3 }$
(E) $\frac { 16 } { 3 }$

$x$	0	1	2
$f ( x )$	1	$k$	2

1. The function $f$ is continuous on the closed interval $[ 0,2 ]$ and has values that are given in the table above. The equation $f ( x ) = \frac { 1 } { 2 }$ must have at least two solutions in the interval $[ 0,2 ]$ if $k =$
   (A) 0
   (B) $\frac { 1 } { 2 }$
   (C) 1
   (D) 2
   (E) 3
2. What is the average value of $y = x ^ { 2 } \sqrt { x ^ { 3 } + 1 }$ on the interval $[ 0,2 ]$ ?
   (A) $\frac { 26 } { 9 }$
   (B) $\frac { 52 } { 9 }$
   (C) $\frac { 26 } { 3 }$
   (D) $\frac { 52 } { 3 }$
   (E) 24
3. If $f ( x ) = \tan ( 2 x )$, then $f ^ { \prime } \left( \frac { \pi } { 6 } \right) =$
   (A) $\sqrt { 3 }$
   (B) $2 \sqrt { 3 }$
   (C) 4
   (D) $4 \sqrt { 3 }$
   (E) 8

CALCULUS AB

SECTION I, Part B

Time - 50 minutes Number of questions - 17

A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED 76-92.

YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET.

In this test:

(1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.
(2) Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number. [Figure] 76. The graph of a function $f$ is shown above. Which of the following statements about $f$ is false?
(A) $f$ is continuous at $x = a$.
(B) $f$ has a relative maximum at $x = a$.
(C) $x = a$ is in the domain of $f$.
(D) $\lim _ { x \rightarrow a ^ { + } } f ( x )$ is equal to $\lim _ { x \rightarrow a ^ { - } } f ( x )$.
(E) $\lim _ { x \rightarrow a } f ( x )$ exists. 77. Let $f$ be the function given by $f ( x ) = 3 e ^ { 2 x }$ and let $g$ be the function given by $g ( x ) = 6 x ^ { 3 }$. At what value of $x$ do the graphs of $f$ and $g$ have parallel tangent lines?
(A) - 0.701
(B) - 0.567
(C) - 0.391
(D) - 0.302
(E) - 0.258 78. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference $C$, what is the rate of change of the area of the circle, in square centimeters per second?
(A) $- ( 0.2 ) \pi C$
(B) $- ( 0.1 ) C$
(C) $- \frac { ( 0.1 ) C } { 2 \pi }$
(D) $( 0.1 ) ^ { 2 } \mathrm { C }$
(E) $( 0.1 ) ^ { 2 } \pi C$ [Figure] 79. The graphs of the derivatives of the functions $f , g$, and $h$ are shown above. Which of the functions $f , g$, or $h$ have a relative maximum on the open interval $a < x < b$ ?
(A) $f$ only
(B) $g$ only
(C) $h$ only
(D) $f$ and $g$ only
(E) $f , g$, and $h$ 80. The first derivative of the function $f$ is given by $f ^ { \prime } ( x ) = \frac { \cos ^ { 2 } x } { x } - \frac { 1 } { 5 }$. How many critical values does $f$ have on the open interval $( 0,10 )$ ?
(A) One
(B) Three
(C) Four
(D) Five
(E) Seven 81. Let $f$ be the function given by $f ( x ) = | x |$. Which of the following statements about $f$ are true? I. $f$ is continuous at $x = 0$. II. $f$ is differentiable at $x = 0$. III. $f$ has an absolute minimum at $x = 0$.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only 82. If $f$ is a continuous function and if $F ^ { \prime } ( x ) = f ( x )$ for all real numbers $x$, then $\int _ { 1 } ^ { 3 } f ( 2 x ) d x =$
(A) $2 F ( 3 ) - 2 F ( 1 )$
(B) $\frac { 1 } { 2 } F ( 3 ) - \frac { 1 } { 2 } F ( 1 )$
(C) $2 F ( 6 ) - 2 F ( 2 )$
(D) $F ( 6 ) - F ( 2 )$
(E) $\frac { 1 } { 2 } F ( 6 ) - \frac { 1 } { 2 } F ( 2 )$ 83. If $a \neq 0$, then $\lim _ { x \rightarrow a } \frac { x ^ { 2 } - a ^ { 2 } } { x ^ { 4 } - a ^ { 4 } }$ is
(A) $\frac { 1 } { a ^ { 2 } }$
(B) $\frac { 1 } { 2 a ^ { 2 } }$
(C) $\frac { 1 } { 6 a ^ { 2 } }$
(D) 0
(E) nonexistent 84. Population $y$ grows according to the equation $\frac { d y } { d t } = k y$, where $k$ is a constant and $t$ is measured in years. If the population doubles every 10 years, then the value of $k$ is
(A) 0.069
(B) 0.200
(C) 0.301
(D) 3.322
(E) 5.000

$x$	2	5	7	8
$f ( x )$	10	30	40	20

1. The function $f$ is continuous on the closed interval $[ 2,8 ]$ and has values that are given in the table above. Using the subintervals [2,5], [5,7], and [7,8], what is the trapezoidal approximation of $\int _ { 2 } ^ { 8 } f ( x ) d x ?$
   (A) 110
   (B) 130
   (C) 160
   (D) 190
   (E) 210 [Figure]
2. The base of a solid is a region in the first quadrant bounded by the $x$-axis, the $y$-axis, and the line $x + 2 y = 8$, as shown in the figure above. If cross sections of the solid perpendicular to the $x$-axis are semicircles, what is the volume of the solid?
   (A) 12.566
   (B) 14.661
   (C) 16.755
   (D) 67.021
   (E) 134.041
3. Which of the following is an equation of the line tangent to the graph of $f ( x ) = x ^ { 4 } + 2 x ^ { 2 }$ at the point where $f ^ { \prime } ( x ) = 1$ ?
   (A) $y = 8 x - 5$
   (B) $y = x + 7$
   (C) $y = x + 0.763$
   (D) $y = x - 0.122$
   (E) $y = x - 2.146$
4. Let $F ( x )$ be an antiderivative of $\frac { ( \ln x ) ^ { 3 } } { x }$. If $F ( 1 ) = 0$, then $F ( 9 ) =$
   (A) 0.048
   (B) 0.144
   (C) 5.827
   (D) 23.308
   (E) $1,640.250$
5. If $g$ is a differentiable function such that $g ( x ) < 0$ for all real numbers $x$ and if $f ^ { \prime } ( x ) = \left( x ^ { 2 } - 4 \right) g ( x )$, which of the following is true?
   (A) $f$ has a relative maximum at $x = - 2$ and a relative minimum at $x = 2$.
   (B) $f$ has a relative minimum at $x = - 2$ and a relative maximum at $x = 2$.
   (C) $f$ has relative minima at $x = - 2$ and at $x = 2$.
   (D) $f$ has relative maxima at $x = - 2$ and at $x = 2$.
   (E) It cannot be determined if $f$ has any relative extrema.
6. If the base $b$ of a triangle is increasing at a rate of 3 inches per minute while its height $h$ is decreasing at a rate of 3 inches per minute, which of the following must be true about the area $A$ of the triangle?
   (A) $A$ is always increasing.
   (B) $A$ is always decreasing.
   (C) $A$ is decreasing only when $b < h$.
   (D) $A$ is decreasing only when $b > h$.
   (E) $A$ remains constant.
7. Let $f$ be a function that is differentiable on the open interval $( 1,10 )$. If $f ( 2 ) = - 5 , f ( 5 ) = 5$, and $f ( 9 ) = - 5$, which of the following must be true? I. $f$ has at least 2 zeros. II. The graph of $f$ has at least one horizontal tangent. III. For some $c , 2 < c < 5 , f ( c ) = 3$.
   (A) None
   (B) I only
   (C) I and II only
   (D) I and III only
   (E) I, II and III
8. If $0 \leq k < \frac { \pi } { 2 }$ and the area under the curve $y = \cos x$ from $x = k$ to $x = \frac { \pi } { 2 }$ is 0.1 , then $k =$
   (A) 1.471
   (B) 1.414
   (C) 1.277
   (D) 1.120
   (E) 0.436

END OF SECTION I

IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY. CHECK YOUR WORK ON PART B ONLY. DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO.
MAKE SURE YOU HAVE PLACED YOUR AP NUMBER LABEL ON YOUR ANSWER SHEET AND HAVE WRITTEN AND GRIDDED YOUR AP NUMBER IN THE APPROPRIATE SECTION OF YOUR ANSWER SHEET.
AFTER TIME HAS BEEN CALLED, ANSWER QUESTIONS 93-96. 93. Which graphing calculator did you use during the examination?
(A) Casio 6300, Casio 7000, Casio 7300, Casio 7400, or Casio 7700
(B) Texas Instruments TI-80 or TI-81
(C) Casio 9700, Casio 9800, Casio 9850, Sharp 9200, Sharp 9300, Texas Instruments TI-82, Texas Instruments TI-83, Texas Instruments TI-85, or Texas Instruments TI-86
(D) Hewlett-Packard HP-48 series or HP-38G
(E) Some other calculator 94. During your Calculus AB course, which of the following best describes your calculator use?
(A) I used my own graphing calculator.
(B) I used a graphing calculator furnished by my school, both in class and at home.
(C) I used a graphing calculator furnished by my school only in class.
(D) I used a graphing calculator furnished by my school mostly in class, but occasionally at home.
(E) I did not use a graphing calculator. 95. During your Calculus AB course, which of the following describes approximately how often a graphing calculator was used by you or your teacher in classroom learning activities?
(A) Almost every class
(B) About three-quarters of the classes
(C) About one-half of the classes
(D) About one-quarter of the classes
(E) Seldom or never 96. During your Calculus AB course, which of the following describes approximately how often you were allowed to use a graphing calculator on tests?
(A) Almost all of the time
(B) About three-quarters of the time
(C) About one-half of the time
(D) About one-quarter of the time
(E) Seldom or never

CALCULUS AB

SECTION II

Time -- 1 hour and 30 minutes

Number of problems - 6 Percent of total grade-50

GENERAL INSTRUCTIONS

You may wish to look over the problems before starting to work on them, since it is not expected that everyone will be able to complete all parts of all problems. All problems are given equal weight, but the parts of a particular problem are not necessarily given equal weight. The problems are printed in the booklet and in the green insert; it may be easier for you to first look over all problems in the insert. When you are told to begin, open your booklet, carefully tear out the green insert, and start to work.

A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION.

• You should write all work for each part of each problem in the space provided for that part in the booklet. Be sure to write clearly and legibly. If you make an error, you may save time by crossing it out rather than trying to erase it. Erased or crossed-out work will not be graded.
• Show all your work. You will be graded on the correctness and completeness of your methods as well as the accuracy of your final answers. Correct answers without supporting work may not receive credit.
• Justifications require that you give mathematical (noncalculator) reasons and that you clearly identify functions, graphs, tables, or other objects you use.
• You are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results.
• Your work must be expressed in standard mathematical notation rather than calculator syntax. For example, $\int _ { 1 } ^ { 5 } x ^ { 2 } d x$ may not be written as $f n \operatorname { Int } \left( X ^ { 2 } , X , 1,5 \right)$.
• Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation. it should be correct to three places after the decimal point.
• Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number.

CALCULUS AB SECTION II Time - 1 hour and 30 minutes Number of problems - 6 Percent of total grade-50

A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION.

REMEMBER TO SHOW YOUR SETUPS AS DESCRIBED IN THE GENERAL INSTRUCTIONS.

General instructions for this section are printed on the back cover of this booklet.

1. Let $R$ be the region bounded by the $x$-axis, the graph of $y = \sqrt { x }$, and the line $x = 4$.
   (a) Find the area of the region $R$.
   (b) Find the value of $h$ such that the vertical line $x = h$ divides the region $R$ into two regions of equal area.
   (c) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
   (d) The vertical line $x = k$ divides the region $R$ into two regions such that when these two regions are revolved about the $x$-axis, they generate solids with equal volumes. Find the value of $k$.
2. Let $f$ be the function given by $f ( x ) = 2 x e ^ { 2 x }$.
   (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 $y = b x e ^ { b x }$, where $b$ is a nonzero constant. Show that the absolute minimum value of $b x e ^ { b x }$ is the same for all nonzero values of $b$. [Figure]

\begin{tabular}{ c } $t$

(seconds)

&

$v ( t )$

(feet per second)

\hline 0 & 0
5 & 12 10 & 20 15 & 30 20 & 55 25 & 70 30 & 78 35 & 81 40 & 75 45 & 60 50 & 72 \hline \end{tabular}

1. The graph of the velocity $v ( t )$, in $\mathrm { ft } / \mathrm { sec }$, of a car traveling on a straight road, for $0 \leq t \leq 50$, is shown above. A table of values for $v ( t )$, at 5 second intervals of time $t$, is shown to the right of the graph.
   (a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer.
   (b) Find the average acceleration of the car, in $\mathrm { ft } / \mathrm { sec } ^ { 2 }$, over the interval $0 \leq t \leq 50$.
   (c) Find one approximation for the acceleration of the car, in $\mathrm { ft } / \mathrm { sec } ^ { 2 }$, at $t = 40$. Show the computations you used to arrive at your answer.
   (d) Approximate $\int _ { 0 } ^ { 50 } v ( t ) d t$ with a Riemann sum, using the midpoints of five subintervals of equal length. Using correct units, explain the meaning of this integral.
2. Let $f$ be a function with $f ( 1 ) = 4$ such that for all points $( x , y )$ on the graph of $f$ the slope is given by $\frac { 3 x ^ { 2 } + 1 } { 2 y }$.
   (a) Find the slope of the graph of $f$ at the point where $x = 1$.
   (b) Write an equation for the line tangent to the graph of $f$ at $x = 1$ and use it to approximate $f ( 1.2 )$.
   (c) Find $f ( x )$ by solving the separable differential equation $\frac { d y } { d x } = \frac { 3 x ^ { 2 } + 1 } { 2 y }$ with the initial condition $f ( 1 ) = 4$.
   (d) Use your solution from part (c) to find $f ( 1.2 )$.
3. The temperature outside a house during a 24 -hour period is given by

$$F ( t ) = 80 - 10 \cos \left( \frac { \pi t } { 12 } \right) , 0 \leq t \leq 24 ,$$
where $F ( t )$ is measured in degrees Fahrenheit and $t$ is measured in hours.
(a) Sketch the graph of $F$ on the grid below. [Figure]
(b) Find the average temperature, to the nearest degree Fahrenheit, between $t = 6$ and $t = 14$.
(c) An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit. For what values of $t$ was the air conditioner cooling the house?
(d) The cost of cooling the house accumulates at the rate of $\$ 0.05$ per hour for each degree the outside temperature exceeds 78 degrees Fahrenheit. What was the total cost, to the nearest cent, to cool the house for this 24 -hour period?

3. Let $f$ be the function given by $f ( x ) = \sqrt { x - 3 } . \wedge B$
(a) On the axes provided below, sketch the graph of $f$ and shade the region $R$ enclosed by the graph of $f$, the $x$-axis, and the vertical line $x = 6$.
Note: The axes for this graph are provided in the pink test booklet only.
(b) Find the area of the region $R$ described in part (a).
(c) Rather than using the line $x = 6$ as in part (a), consider the line $x = w$, where $w$ can be any number greater than 3 . Let $A ( w )$ be the area of the region enclosed by the graph of $f$, the $x$-axis, and the vertical line $x = w$. Write an integral expression for $A ( w )$.
(d) Let $A ( w )$ be as described in part (c). Find the rate of change of $A$ with respect to $w$ when $w = 6$.

Let $R$ be the region in the first quadrant under the graph of $y = \frac { x } { x ^ { 2 } + 2 }$ for $0 \leqq x \leqq$ (a) Find the area of $R \cdot \left( - x - \alpha , \frac { 1 } { 1 } \right)$ (b) If the line $x = k$ divides $R$ into two regions of equal area, what is the value of $k$ ? (c) What is the average value of $y = \frac { x } { x ^ { 2 } + 2 }$ on the interval $0 \leqq x \leqq \sqrt { 6 }$ ?

Let $R$ be the region in the first quadrant bounded by the $x$-axis and the graphs of $y = \ln x$ and $y = 5 - x$, as shown in the figure above.
(a) Find the area of $R$.
(b) Region $R$ is the base of a solid. For the solid, each cross section perpendicular to the $x$-axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.
(c) The horizontal line $y = k$ divides $R$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $k$.

What is the area of the region in the first quadrant bounded by the graph of $y = e ^ { x / 2 }$ and the line $x = 2$ ?
(A) $2 e - 2$
(B) $2 e$
(C) $\frac { e } { 2 } - 1$
(D) $\frac { e - 1 } { 2 }$
(E) $e - 1$

The graph above gives the velocity, $v$, in ft/sec, of a car for $0 \leq t \leq 8$, where $t$ is the time in seconds. Of the following, which is the best estimate of the distance traveled by the car from $t = 0$ until the car comes to a complete stop?
(A) 21 ft
(B) 26 ft
(C) 180 ft
(D) 210 ft
(E) 260 ft

The density of a bacteria population in a circular petri dish at a distance $r$ centimeters from the center of the dish is given by an increasing, differentiable function $f$, where $f(r)$ is measured in milligrams per square centimeter. Values of $f(r)$ for selected values of $r$ are given in the table below.

\begin{tabular}{ c } $r$

(centimeters)

& 0 & 1 & 2 & 2.5 & 4 \hline

$f ( r )$

(milligrams per square centimeter)

& 1 & 2 & 6 & 10 & 18 \hline \end{tabular}
(a) Use the data in the table to estimate $f^{\prime}(2.25)$. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) The total mass, in milligrams, of bacteria in the petri dish is given by the integral expression $2\pi \int_{0}^{4} r f(r)\, dr$. Approximate the value of $2\pi \int_{0}^{4} r f(r)\, dr$ using a right Riemann sum with the four subintervals indicated by the data in the table.
(c) Is the approximation found in part (b) an overestimate or underestimate of the total mass of bacteria in the petri dish? Explain your reasoning.
(d) The density of bacteria in the petri dish, for $1 \leq r \leq 4$, is modeled by the function $g$ defined by $g(r) = 2 - 16(\cos(1.57\sqrt{r}))^{3}$. For what value of $k$, $1 < k < 4$, is $g(k)$ equal to the average value of $g(r)$ on the interval $1 \leq r \leq 4$?

The graph of the differentiable function $f$, shown for $-6 \leq x \leq 7$, has a horizontal tangent at $x = -2$ and is linear for $0 \leq x \leq 7$. Let $R$ be the region in the second quadrant bounded by the graph of $f$, the vertical line $x = -6$, and the $x$- and $y$-axes. Region $R$ has area 12.
(a) The function $g$ is defined by $g(x) = \int_{0}^{x} f(t)\, dt$. Find the values of $g(-6)$, $g(4)$, and $g(6)$.
(b) For the function $g$ defined in part (a), find all values of $x$ in the interval $0 \leq x \leq 6$ at which the graph of $g$ has a critical point. Give a reason for your answer.
(c) The function $h$ is defined by $h(x) = \int_{-6}^{x} f'(t)\, dt$. Find the values of $h(6)$, $h'(6)$, and $h''(6)$. Show the work that leads to your answers.

Let $\ell$ be the line tangent to the graph of $y = x ^ { n }$ at the point $( 1,1 )$, where $n > 1$, as shown above.
(a) Find $\int _ { 0 } ^ { 1 } x ^ { n } d x$ in terms of $n$.
(b) Let $T$ be the triangular region bounded by $\ell$, the $x$-axis, and the line $x = 1$. Show that the area of $T$ is $\frac { 1 } { 2 n }$.
(c) Let $S$ be the region bounded by the graph of $y = x ^ { n }$, the line $\ell$, and the $x$-axis. Express the area of $S$ in terms of $n$ and determine the value of $n$ that maximizes the area of $S$.

A baker is creating a birthday cake. The base of the cake is the region $R$ in the first quadrant under the graph of $y = f ( x )$ for $0 \leq x \leq 30$, where $f ( x ) = 20 \sin \left( \frac { \pi x } { 30 } \right)$. Both $x$ and $y$ are measured in centimeters. The region $R$ is shown in the figure above. The derivative of $f$ is $f ^ { \prime } ( x ) = \frac { 2 \pi } { 3 } \cos \left( \frac { \pi x } { 30 } \right)$.
(a) The region $R$ is cut out of a 30 -centimeter-by- 20 -centimeter rectangular sheet of cardboard, and the remaining cardboard is discarded. Find the area of the discarded cardboard.
(b) The cake is a solid with base $R$. Cross sections of the cake perpendicular to the $x$-axis are semicircles. If the baker uses 0.05 gram of unsweetened chocolate for each cubic centimeter of cake, how many grams of unsweetened chocolate will be in the cake?
(c) Find the perimeter of the base of the cake.

Let $R$ be the region in the first quadrant bounded by the graph of $y = 2\sqrt{x}$, the horizontal line $y = 6$, and the $y$-axis.
(a) Find the area of $R$.
(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = 7$.
(c) Region $R$ is the base of a solid. For each $y$, where $0 \leq y \leq 6$, the cross section of the solid taken perpendicular to the $y$-axis is a rectangle whose height is 3 times the length of its base in region $R$. Write, but do not evaluate, an integral expression that gives the volume of the solid.

The graph of the differentiable function $y = f ( x )$ with domain $0 \leq x \leq 10$ is shown in the figure above. The area of the region enclosed between the graph of $f$ and the $x$-axis for $0 \leq x \leq 5$ is 10 , and the area of the region enclosed between the graph of $f$ and the $x$-axis for $5 \leq x \leq 10$ is 27 . The arc length for the portion of the graph of $f$ between $x = 0$ and $x = 5$ is 11, and the arc length for the portion of the graph of $f$ between $x = 5$ and $x = 10$ is 18 . The function $f$ has exactly two critical points that are located at $x = 3$ and $x = 8$.
(a) Find the average value of $f$ on the interval $0 \leq x \leq 5$.
(b) Evaluate $\int _ { 0 } ^ { 10 } ( 3 f ( x ) + 2 ) d x$. Show the computations that lead to your answer.
(c) Let $g ( x ) = \int _ { 5 } ^ { x } f ( t ) d t$. On what intervals, if any, is the graph of $g$ both concave up and decreasing? Explain your reasoning.
(d) The function $h$ is defined by $h ( x ) = 2 f \left( \frac { x } { 2 } \right)$. The derivative of $h$ is $h ^ { \prime } ( x ) = f ^ { \prime } \left( \frac { x } { 2 } \right)$. Find the arc length of the graph of $y = h ( x )$ from $x = 0$ to $x = 20$.

The graph of $f ^ { \prime }$, the derivative of a function $f$, consists of two line segments and a semicircle, as shown in the figure above. If $f ( 2 ) = 1$, then $f ( - 5 ) =$
(A) $2 \pi - 2$
(B) $2 \pi - 3$
(C) $2 \pi - 5$
(D) $6 - 2 \pi$
(E) $4 - 2 \pi$

What is the average value of $y = \sqrt { \cos x }$ on the interval $0 \leq x \leq \frac { \pi } { 2 }$ ?
(A) - 0.637
(B) 0.500
(C) 0.763
(D) 1.198
(E) 1.882

The figure above shows the graph of the piecewise-linear function $f$. For $- 4 \leq x \leq 12$, the function $g$ is defined by $g ( x ) = \int _ { 2 } ^ { x } f ( t ) \, d t$.
(a) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 10$ ? Justify your answer.
(b) Does the graph of $g$ have a point of inflection at $x = 4$ ? Justify your answer.
(c) Find the absolute minimum value and the absolute maximum value of $g$ on the interval $- 4 \leq x \leq 12$. Justify your answers.
(d) For $- 4 \leq x \leq 12$, find all intervals for which $g ( x ) \leq 0$.

We consider the function $g$ defined for all real $x$ in the interval $[0;1]$ by: $$g(x) = 1 + \mathrm{e}^{-x}$$ We admit that, for all real $x$ in the interval $[0;1], g(x) > 0$.
We denote $\mathscr{C}$ the representative curve of function $g$ in an orthogonal coordinate system, and $\mathscr{D}$ the plane region bounded on one hand between the $x$-axis and curve $\mathscr{C}$, on the other hand between the lines with equations $x = 0$ and $x = 1$.
The purpose of this exercise is to divide region $\mathscr{D}$ into two regions of equal area, first by a line parallel to the $y$-axis (part A), then by a line parallel to the $x$-axis (part B).

Part A

Let $a$ be a real number such that $0 \leqslant a \leqslant 1$. We denote $\mathscr{A}_1$ the area of the region between curve $\mathscr{C}$, the $x$-axis, the lines with equations $x = 0$ and $x = a$, and $\mathscr{A}_2$ that of the region between curve $\mathscr{C}$, the $x$-axis and the lines with equations $x = a$ and $x = 1$. $\mathscr{A}_1$ and $\mathscr{A}_2$ are expressed in square units.

1. a. Prove that $\mathscr{A}_1 = a - \mathrm{e}^{-a} + 1$. b. Express $\mathscr{A}_2$ as a function of $a$.
2. Let $f$ be the function defined for all real $x$ in the interval $[0;1]$ by: $$f(x) = 2x - 2\mathrm{e}^{-x} + \frac{1}{\mathrm{e}}$$ a. Draw the variation table of function $f$ on the interval $[0;1]$. The exact values of $f(0)$ and $f(1)$ will be specified. b. Prove that function $f$ vanishes once and only once on the interval $[0;1]$, at a real number $\alpha$. Give the value of $\alpha$ rounded to the nearest hundredth.
3. Using the previous questions, determine an approximate value of the real $a$ for which the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal.

Part B

Let $b$ be a positive real number. In this part, we propose to divide region $\mathscr{D}$ into two regions of equal area by the line with equation $y = b$. We admit that there exists a unique positive real $b$ that is a solution.

1. Justify the inequality $b < 1 + \frac{1}{\mathrm{e}}$. You may use a graphical argument.
2. Determine the exact value of the real $b$.

1. Determine the area $\mathscr{A}$, expressed in square units, of the shaded region in the graph of Part A (the region bounded by the curve $\mathscr{C}_u$ where $u(x) = \frac{x^2 - 5x + 4}{x^2}$ between its zeros $x=1$ and $x=4$).
2. For all real $\lambda$ greater than or equal to 4, we denote by $\mathscr{A}_{\lambda}$ the area, expressed in square units, of the region formed by the points $M$ with coordinates $(x; y)$ such that $$4 \leqslant x \leqslant \lambda \quad \text{and} \quad 0 \leqslant y \leqslant u(x).$$ Does there exist a value of $\lambda$ for which $\mathscr{A}_{\lambda} = \mathscr{A}$?

A factory produces mineral water in bottles. The shape of the bottle labels is bounded by the x-axis and the curve $\mathscr { C }$ with equation $y = a \cos x$ with $x \in \left[ - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \right]$ and $a$ a strictly positive real number.
A disk located inside is intended to receive information given to buyers. We consider the disk with centre at point A with coordinates $\left( 0 ; \frac { a } { 2 } \right)$ and radius $\frac { a } { 2 }$. It is admitted that this disk is entirely below the curve $\mathscr { C }$ for values of $a$ less than 1.4.

1. Justify that the area of the region between the x-axis, the lines with equations $x = - \frac { \pi } { 2 }$ and $x = \frac { \pi } { 2 }$, and the curve $\mathscr { C }$ equals $2 a$ square units.
2. For aesthetic reasons, it is desired that the area of the disk equals the area of the shaded surface. What value should be given to the real number $a$ to satisfy this constraint?

Let $f$ be a function defined on the interval $[0;1]$, continuous and positive on this interval, and $a$ a real number such that $0 < a < 1$.
We denote:

• $\mathscr{C}$ the representative curve of the function $f$ in an orthogonal coordinate system;
• $\mathscr{A}_1$ the area of the plane region bounded by the $x$-axis and the curve $\mathscr{C}$ on one hand, the lines with equations $x = 0$ and $x = a$ on the other hand.
• $\mathscr{A}_2$ the area of the plane region bounded by the $x$-axis and the curve $\mathscr{C}$ on one hand, the lines with equations $x = a$ and $x = 1$ on the other hand.

The purpose of this exercise is to determine, for different functions $f$, a value of the real number $a$ satisfying condition (E): ``the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal''. We admit the existence of such a real number $a$ for each of the functions considered.
Part A: Study of some examples

1. Verify that in the following cases, condition (E) is satisfied for a unique real number $a$ and determine its value. a. $f$ is a strictly positive constant function. b. $f$ is defined on $[0;1]$ by $f(x) = x$.
   
2. a. Using integrals, express, in units of area, the areas $\mathscr{A}_1$ and $\mathscr{A}_2$. b. Let $F$ be a primitive of the function $f$ on the interval $[0;1]$. Prove that if the real number $a$ satisfies condition (E), then $F(a) = \dfrac{F(0) + F(1)}{2}$. Is the converse true?
   
3. In this question, we consider two other particular functions. a. The function $f$ is defined for all real $x$ in $[0;1]$ by $f(x) = \mathrm{e}^x$. Verify that condition (E) is satisfied for a unique real number $a$ and give its value. b. The function $f$ defined for all real $x$ in $[0;1]$ by $f(x) = \dfrac{1}{(x+2)^2}$. Verify that the value $a = \dfrac{2}{5}$ works.

Part B: Using a sequence to determine an approximate value of $a$
In this part, we consider the function $f$ defined for all real $x$ in $[0;1]$ by $f(x) = 4 - 3x^2$.

1. Prove that if $a$ is a real number satisfying condition (E), then $a$ is a solution of the equation: $$x = \frac{x^3}{4} + \frac{3}{8}$$ In the rest of the exercise, we will admit that this equation has a unique solution in the interval $[0;1]$. We denote this solution by $a$.
   
2. We consider the function $g$ defined for all real $x$ in $[0;1]$ by $g(x) = \dfrac{x^3}{4} + \dfrac{3}{8}$ and the sequence $(u_n)$ defined by: $u_0 = 0$ and, for all natural number $n$, $u_{n+1} = g(u_n)$. a. Calculate $u_1$. b. Prove that the function $g$ is increasing on the interval $[0;1]$. c. Prove by induction that, for all natural number $n$, we have $0 \leqslant u_n \leqslant u_{n+1} \leqslant 1$. d. Prove that the sequence $(u_n)$ is convergent. Using operations on limits, prove that the limit is $a$. e. We admit that the real number $a$ satisfies the inequality $0 < a - u_{10} < 10^{-9}$. Calculate $u_{10}$ to $10^{-8}$ precision.

A homeowner wants to have a water tank built. This water tank must comply with the following specifications:

• it must be located two metres from his house;
• the maximum depth must be two metres;
• it must measure five metres long;
• it must follow the natural slope of the land.

The curved part is modelled by the curve $\mathscr{C}_f$ of the function $f$ on the interval $[2; 2e]$ defined by: $$f(x) = x \ln\left(\frac{x}{2}\right) - x + 2$$
The curve $\mathscr{C}_f$ is represented in an orthonormal coordinate system with unit $1\mathrm{m}$ and constitutes a profile view of the tank. We consider the points $\mathrm{A}(2; 2)$, $\mathrm{I}(2; 0)$ and $\mathrm{B}(2\mathrm{e}; 2)$.
Part A
The objective of this part is to evaluate the volume of the tank.

1. Justify that the points B and I belong to the curve $\mathscr{C}_f$ and that the x-axis is tangent to the curve $\mathscr{C}_f$ at point I.
2. We denote by $\mathscr{T}$ the tangent to the curve $\mathscr{C}_f$ at point B, and D the point of intersection of the line $\mathscr{T}$ with the x-axis. a. Determine an equation of the line $\mathscr{T}$ and deduce the coordinates of D. b. We call $S$ the area of the region bounded by the curve $\mathscr{C}_f$, the lines with equations $y = 2$, $x = 2$ and $x = 2\mathrm{e}$. $S$ can be bounded by the area of triangle ABI and that of trapezoid AIDB. What bounds on the volume of the tank can we deduce?
3. a. Show that, on the interval $[2; 2\mathrm{e}]$, the function $G$ defined by $$G(x) = \frac{x^2}{2} \ln\left(\frac{x}{2}\right) - \frac{x^2}{4}$$ is an antiderivative of the function $g$ defined by $g(x) = x \ln\left(\frac{x}{2}\right)$. b. Deduce an antiderivative $F$ of the function $f$ on the interval $[2; 2\mathrm{e}]$. c. Determine the exact value of the area $S$ and deduce an approximate value of the volume $V$ of the tank to the nearest $\mathrm{m}^3$.

Part B
For any real number $x$ between 2 and $2\mathrm{e}$, we denote by $v(x)$ the volume of water, expressed in $\mathrm{m}^3$, in the tank when the water level in the tank is equal to $f(x)$. We admit that, for any real number $x$ in the interval $[2; 2\mathrm{e}]$, $$v(x) = 5\left[\frac{x^2}{2}\ln\left(\frac{x}{2}\right) - 2x\ln\left(\frac{x}{2}\right) - \frac{x^2}{4} + 2x - 3\right]$$

1. What volume of water, to the nearest $\mathrm{m}^3$, is in the tank when the water level in the tank is one metre?
2. We recall that $V$ is the total volume of the tank, $f$ is the function defined at the beginning of the exercise and $v$ the function defined in Part B. We consider the following algorithm:

   \begin{tabular}{l} Variables:

   Processing:

   &

   $a$ is a real number

   $b$ is a real number

   $a$ takes the value 2

   $b$ takes the value $2\mathrm{e}$

   While $v(b) - v(a) > 10^{-3}$ do:

   $c$ takes the value $(a + b)/2$

   If $v(c) < V/2$, then:

   $a$ takes the value $c$

   Otherwise

   $b$ takes the value $c$

   End If

   End While

   Display $f(c)$

   \hline \end{tabular} Interpret the result that this algorithm allows to display.

We denote $\mathscr { A } ( \theta )$ the region bounded by the lines with equations $t = 10 , t = \theta$, $y = 85$ and the curve $\mathscr { C } _ { f }$ representing $f$. We consider that sterilization is complete after a time $\theta$, if the area, expressed in square units of the region $\mathscr { A } ( \theta )$ is greater than 80.

1. [a.] Justify, using the graph given in the appendix, that we have $\mathscr { A } ( 25 ) > 80$.
2. [b.] Justify that, for $\theta \geqslant 10$, we have $\mathscr { A } ( \theta ) = 15 ( \theta - 10 ) - 75 \int _ { 10 } ^ { \theta } \mathrm { e } ^ { - \frac { \ln 5 } { 10 } t } \mathrm {~d} t$.
3. [c.] Is sterilization complete after 20 minutes?

Let $f$ and $g$ be the functions defined on the set $\mathbb { R }$ of real numbers by
$$f ( x ) = \mathrm { e } ^ { x } \quad \text { and } \quad g ( x ) = \mathrm { e } ^ { - x } .$$
We denote by $\mathscr { C } _ { f }$ the representative curve of function $f$ and $\mathscr { C } _ { g }$ that of function $g$ in an orthonormal coordinate system of the plane.
For every real number $a$, we denote by $M$ the point of $\mathscr { C } _ { f }$ with abscissa $a$ and $N$ the point of $\mathscr { C } _ { g }$ with abscissa $a$.
The tangent line to $\mathscr { C } _ { f }$ at $M$ intersects the $x$-axis at $P$, the tangent line to $\mathscr { C } _ { g }$ at $N$ intersects the $x$-axis at $Q$.
Questions 1 and 2 can be treated independently.

1. Prove that the tangent line to $\mathscr { C } _ { f }$ at $M$ is perpendicular to the tangent line to $\mathscr { C } _ { g }$ at $N$.
2. a. What can be conjectured about the length $PQ$? b. Prove this conjecture.

The plane is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. Consider the points $\mathrm{A}(-1;1)$, $\mathrm{B}(0;1)$, $\mathrm{C}(4;3)$, $\mathrm{D}(7;0)$, $\mathrm{E}(4;-3)$, $\mathrm{F}(0;-1)$ and $\mathrm{G}(-1;-1)$.
The part of the curve located above the x-axis is decomposed as follows:

• the portion located between points A and B is the graph of the constant function $h$ defined on the interval $[-1;0]$ by $h(x) = 1$;
• the portion located between points B and C is the graph of a function $f$ defined on the interval $[0;4]$ by $f(x) = a + b\sin\left(c + \frac{\pi}{4}x\right)$, where $a$, $b$ and $c$ are fixed non-zero real numbers and where the real number $c$ belongs to the interval $\left[0; \frac{\pi}{2}\right]$;
• the portion located between points C and D is a quarter circle with diameter [CE].

The part of the curve located below the x-axis is obtained by symmetry with respect to the x-axis.

1. a. We call $f'$ the derivative function of function $f$. For every real number $x$ in the interval $[0;4]$, determine $f'(x)$. b. We require that the tangent lines at points B and C to the graph of function $f$ be parallel to the x-axis. Determine the value of the real number $c$.
2. Determine the real numbers $a$ and $b$.

Part C

In this part, we consider the function $h$ defined on $\mathbb { R }$ by $$h ( x ) = ( x - 1 ) \mathrm { e } ^ { - 2 x } + 1 .$$ We admit that the function $h$ is differentiable on $\mathbb { R }$. We place ourselves in an orthonormal coordinate system ( O ; I, J). We denote $\mathscr { C } _ { h }$ the representative curve of the function $h$ and $d$ the line with equation $y = x$. We admit that the curve $\mathscr { C } _ { h }$ is above the line $d$ on the interval $[ 0 ; 1 ]$. Let $\mathscr { D }$ be the region of the plane bounded by the curve $\mathscr { C } _ { h }$, the line $d$ and the vertical lines with equations $x = 0$ and $x = 1$. Let $\mathscr { A }$ be the area of $\mathscr { D }$ expressed in square units.

1. On ANNEX 1, shade the region $\mathscr { D }$ and justify that $$\mathscr { A } = \int _ { 0 } ^ { 1 } [ h ( x ) - x ] \mathrm { d } x$$
2. a. Prove that, for all real $x$, $$h ( x ) - x = ( 1 - x ) \left( 1 - \mathrm { e } ^ { - 2 x } \right) .$$ b. We admit that, for all real $x$, $\mathrm { e } ^ { - 2 x } \geqslant 1 - 2 x$. Prove that, for all real $x$ in the interval $[ 0 ; 1 ]$, $$h ( x ) - x \leqslant 2 x - 2 x ^ { 2 } .$$ c. Deduce that $\mathscr { A } \leqslant \frac { 1 } { 3 }$.
3. Let $H$ be the function defined on $[ 0 ; 1 ]$ by $$H ( x ) = \frac { 1 } { 4 } ( 1 - 2 x ) \mathrm { e } ^ { - 2 x } + x$$ We admit that the function $H$ is an antiderivative of the function $h$ on $[ 0 ; 1 ]$. Determine the exact value of $\mathscr { A }$.