Given the function $f ( x ) = \left\{ \begin{array} { l l } \mathrm { e } ^ { x } , & x \leqslant 0 , \\ \ln x , & x > 0 , \end{array} \right.$ and $g ( x ) = f ( x ) + x + a$. If $g ( x )$ has 2 zeros, then the range of $a$ is A. $( 0 , + \infty )$ B. $[ 0 , + \infty )$ C. $[ - 1 , + \infty )$ D. $[ 1 , + \infty )$
Let $f ( x ) = \begin{cases} 2 ^ { -x } & x \leq 0 \\ 1 & x > 0 \end{cases}$. Then the range of $x$ satisfying $f ( x + 1 ) < f ( 2 x )$ is A. $( - \infty , - 1 ]$ B. $( 0 , + \infty )$ C. $( - 1,0 )$ D. $( - \infty , 0 )$
Given the constraints $\left\{ \begin{array} { l } x + 2 y - 5 \geq 0 , \\ x - 2 y + 3 \geq 0 , \\ x - 5 \leq 0 , \end{array} \right.$ the minimum value of $z = x + y$ is \_\_\_\_.
5. The sum of the zeros of the function $f ( x ) = \left\{ \begin{array} { l } 6 ^ { x } - 2 , x > 0 , \\ x + \log _ { 6 } 12 , x \leq 0 \end{array} \right.$ is A. $-1$ B. $1$ C. $-2$ D. $2$
5. The graph of function $f ( x ) = \frac { \sin x + x } { \cos x + x ^ { 2 } }$ on $[ - \pi , \pi ]$ is approximately A.[Figure] B.[Figure] C.[Figure] D.[Figure]
The graph of the function $y = \frac { 2 x ^ { 3 } } { 2 ^ { x } + 2 ^ { - x } }$ on $[ - 6,6 ]$ is approximately A. [graph A] B. [graph B] C. [graph C] D. [graph D]
7. The graph of the function $y = \frac { 2 x ^ { 3 } } { 2 ^ { x } + 2 ^ { - x } }$ on $[ - 6,6 ]$ is approximately A. [Figure] B. [Figure] C. [Figure] D. [Figure]
11. Regarding the function $f ( x ) = \sin | x | + | \sin x |$, there are four conclusions: (1) $f ( x )$ is an even function (2) $f ( x )$ is monotonically increasing on the interval $\left( \frac { \pi } { 2 } , \pi \right)$ (3) $f ( x )$ has 4 zeros on $[ - \pi , \pi ]$ (4) The maximum value of $f ( x )$ is 2 The numbers of all correct conclusions are A. (1)(2)(4) B. (2)(4) C. (1)(4) D. (1)(3)
11. Regarding the function $f ( x ) = \sin | x | + | \sin x |$ , there are four conclusions: (1) $f ( x )$ is an even function (2) $f ( x )$ is monotonically increasing on the interval $\left( \frac { \pi } { 2 } , \pi \right)$ (3) $f ( x )$ has 4 zeros on $[ - \pi , \pi ]$ (4) The maximum value of $f ( x )$ is 2 The numbers of all correct conclusions are A. (1)(2)(4) B. (2)(4) C. (1)(4) D. (1)(3)
Let $a \in \mathbb{R}$. If there exists a function $f ( x )$ with domain $\mathbb{R}$ that satisfies both ``for any $x _ { 0 } \in \mathbb{R}$, the value of $f \left( x _ { 0 } \right)$ is either $x _ { 0 } ^ { 2 }$ or $x _ { 0 }$'' and ``the equation $f ( x ) = a$ has no real solutions'', find the range of $a$ as $\_\_\_\_$
Regarding the function $f ( x ) = \sin x + \frac { 1 } { \sin x }$ , there are four propositions: (1) The graph of $f ( x )$ is symmetric about the $y$-axis. (2) The graph of $f ( x )$ is symmetric about the origin. (3) The graph of $f ( x )$ is symmetric about the line $x = \frac { \pi } { 2 }$ . (4) The minimum value of $f ( x )$ is 2 . The sequence numbers of all true propositions are $\_\_\_\_$ .
The graph of the function $y = \left( 3 ^ { x } - 3 ^ { - x } \right) \cos x$ on the interval $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is approximately: (see figures A, B, C, D)
The graph of the function $f ( x ) = \left( 3 ^ { x } - 3 ^ { - x } \right) \cos x$ on the interval $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is approximately [see figures A, B, C, D in the original paper].
The figure on the right is the approximate graph of one of the following four functions on the interval $[ - 3,3 ]$ . The function is A. $y = \frac { - x ^ { 3 } + 3 x } { x ^ { 2 } + 1 }$ B. $y = \frac { x ^ { 3 } - x } { x ^ { 2 } + 1 }$ C. $y = \frac { 2 x \cos x } { x ^ { 2 } + 1 }$ D. $y = \frac { 2 \sin x } { x ^ { 2 } + 1 }$
Let $f(x)$ be the function obtained by shifting $y = \cos\left(2x + \frac{\pi}{4}\right)$ to the left by $\frac{\pi}{6}$ units. The number of intersection points of $y = f(x)$ and $y = \frac{1}{2}x - \frac{1}{2}$ is A. $1$ B. $2$ C. $3$ D. $4$
Let $f ( x ) = a ( x + 1 ) ^ { 2 } - 1 , g ( x ) = \cos x + 2 a x$. When $x \in ( - 1,1 )$, the curves $y = f ( x )$ and $y = g ( x )$ have exactly one intersection point. Then $a =$ A. $- 1$ B. $\frac { 1 } { 2 }$ C. 1 D. 2
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = (x^2 - 3)e^x + 2$, then A. $f(0) = 0$ B. When $x < 0$, $f(x) = -(x^2 - 3)e^{-x} - 2$ C. $f(x) < 0$ if and only if $x > \sqrt{3}$ D. $x = -1$ is a local maximum point of $f(x)$
(1) [4 marks] For each of the functions $f _ { 1 }$ and $f _ { 2 }$, specify the maximum domain and the zero. $$f _ { 1 } : x \mapsto \frac { 2 x + 3 } { x ^ { 2 } - 4 } \quad f _ { 2 } : x \mapsto \ln ( x + 2 )$$ (2) [3 marks] Specify the term of a function defined on $\mathbb { R }$ whose graph has a horizontal tangent at the point (2|1), but no extremum. (3) [5 marks] Given is the function $f$ defined on $\mathbb { R }$ with $f ( x ) = - x ^ { 3 } + 9 x ^ { 2 } - 15 x - 25$. Prove that $f$ has the following properties: (1) The graph of $f$ has slope $-15$ at the point $x = 0$. (2) The graph of $f$ has the $x$-axis as a tangent at the point $A ( 5 \mid f ( 5 ) )$. (3) The tangent $t$ to the graph of the function $f$ at the point $B ( - 1 \mid f ( - 1 ) )$ can be described by the equation $y = - 36 x - 36$. (4) [3 marks] The figure shows the graph $G _ { f }$ of a function $f$ defined on $\mathbb { R }$ with the inflection point $W ( 1 \mid 4 )$. Using the figure, determine approximately the value of the derivative of $f$ at the point $x = 1$. Sketch the graph of the derivative function $f ^ { \prime }$ of $f$ into the figure; in doing so, pay particular attention to the location of the zeros of $f ^ { \prime }$ and the approximate value determined for $f ^ { \prime } ( 1 )$. [Figure] For each value of $a$ with $a \in \mathbb { R } ^ { + }$, a function $f _ { a }$ is given by $f _ { a } ( x ) = \frac { 1 } { a } \cdot x ^ { 3 } - x$ with $x \in \mathbb { R }$. (5a) [2 marks] One of the two figures shows a graph of $f _ { a }$. Specify which figure this is. Justify your answer. [Figure] Fig. 1 [Figure] Fig. 2 (5b) [3 marks] For each value of $a$, the graph of $f _ { a }$ has exactly two extrema. Determine the value of $a$ for which the graph of the function $f _ { a }$ has an extremum at the point $x = 3$. Given is the function $f : x \mapsto 2 \cdot \left( ( \ln x ) ^ { 2 } - 1 \right)$ defined on $\mathbb { R } ^ { + }$. Figure 1 shows the graph $G _ { f }$ of $f$. [Figure] Fig. 1
(1a) [2 marks] Given is the function $f : x \mapsto \frac { x ^ { 2 } + 2 x } { x + 1 }$ with maximal domain $D _ { f }$. State $D _ { f }$ and the zeros of $f$. (1b) [3 marks] Give a term of a rational function $h$ that has the following properties: The function $h$ is defined on $\mathbb { R }$; its graph has the line with equation $y = 3$ as a horizontal asymptote and intersects the y-axis at the point ( $0 \mid 4$ ). Given is the function $g : x \mapsto \frac { 4 } { x }$ defined on $\mathbb { R } ^ { + }$. Figure 1 shows the graph of $g$. [Figure] Fig. 1 (2a) [2 marks] Calculate the value of the integral $\int _ { 1 } ^ { e } g ( x ) \mathrm { dx }$ (2b) [3 marks] Determine graphically the point $x _ { 0 } \in \mathbb { R } ^ { + }$ for which the following holds: The local rate of change of $g$ at the point $x _ { 0 }$ equals the average rate of change of $g$ on the interval $[ 1 ; 4 ]$. The graph $G _ { f }$ of the polynomial function $f$ defined on $\mathbb { R }$ has a horizontal tangent only at the point $x = 3$ (see Figure 2). Consider the function $g$ defined on $\mathbb { R }$ with $g ( x ) = f ( f ( x ) )$. [Figure] Fig. 2 (3a) [2 marks] Using Figure 2, state the function values $f ( 6 )$ and $g ( 6 )$. (3b) [3 marks] According to the chain rule, $g ^ { \prime } ( x ) = f ^ { \prime } ( f ( x ) ) \cdot f ^ { \prime } ( x )$. Use this and Figure 2 to determine all points where the graph of $g$ has a horizontal tangent. Given are the functions $f _ { a }$ defined on $\mathbb { R }$ with $f _ { a } ( x ) = a \cdot e ^ { - x } + 3$ and $a \in \mathbb { R } \backslash \{ 0 \}$. (4a) [1 marks] Show that $f _ { a } ^ { \prime } ( 0 ) = - a$ holds. )$} Consider the tangent to the graph of $f _ { a }$ at the point $\left( 0 \mid f _ { a } ( 0 ) \right)$. Determine those values of $a$ for which this tangent has a positive slope and also intersects the x-axis at a point whose x-coordinate is greater than $\frac { 1 } { 2 }$. Given is the function $f : x \mapsto 2 \cdot \sqrt { 10 x - x ^ { 2 } }$ defined on $[ 0 ; 10 ]$. The graph of $f$ is denoted by $G _ { f }$. Subtask Part B a (2 marks) Determine the zeros of $f$. (for verification: 0 and 10) The graph $G _ { f }$ has a horizontal tangent at exactly one point. Determine the coordinates of this point and justify that it is a maximum point. (for verification: $f ^ { \prime } ( x ) = \frac { 10 - 2 x } { \sqrt { 10 x - x ^ { 2 } } } ;$ y-coordinate of the maximum point: 10 ) The graph $G _ { f }$ is concave down. One of the following terms is a term of the second derivative function $f ^ { \prime \prime }$ of $f$. Determine whether this is Term I or Term II without calculating a term of $f ^ { \prime \prime }$. I $\quad f ^ { \prime \prime } ( x ) = \frac { 50 } { \left( x ^ { 2 } - 10 x \right) \cdot \sqrt { 10 x - x ^ { 2 } } } \quad$ II $\quad f ^ { \prime \prime } ( x ) = \frac { 50 } { \left( 10 x - x ^ { 2 } \right) \cdot \sqrt { 10 x - x ^ { 2 } } }$ Show that for $0 \leq x \leq 5$ the equation $f ( 5 - x ) = f ( 5 + x )$ is satisfied by appropriately transforming the terms $f ( 5 - x )$ and $f ( 5 + x )$. Use this to justify that the graph $G _ { f }$ is symmetric with respect to the line with equation $x = 5$. State the maximal domain of the term $f ^ { \prime } ( x ) = \frac { 10 - 2 x } { \sqrt { 10 x - x ^ { 2 } } }$. Determine $\lim _ { x \rightarrow 0 } f ^ { \prime } ( x )$ and interpret the result geometrically. State $f ( 8 )$ and sketch $G _ { f }$ in a coordinate system taking into account the previous results. Consider the tangent to $G _ { f }$ at the point $( 2 \mid f ( 2 ) )$. Calculate the angle at which this tangent intersects the x-axis. Of the vertices of rectangle ABCD, the point $A ( s \mid 0 )$ with $s \in ] 0 ; 5 [$ and the point $B$ lie on the x-axis, the points $C$ and $D$ lie on $G _ { f }$. The rectangle thus has the line with equation $x = 5$ as its axis of symmetry. Show that the diagonals of this rectangle each have length 10. A water storage tank has the shape of a right cylinder and is filled with water up to a level of 10 m above the tank bottom. If a hole is drilled in the wall of the water storage tank below the water level, water immediately flows out after the hole is completed, hitting the ground at a certain distance from the tank wall. This distance is called the spray distance in the following (see Figure). The dependence of the spray distance on the height of the hole is modeled by the function $f$ considered in the previous subtasks. Here $x$ is the height of the hole above the tank bottom in meters and $f ( x )$ is the spray distance in meters. [Figure] Subtask Part B i (1 mark) The graph $G _ { f }$ passes through the point $( 3,6 \mid 9,6 )$. State the meaning of this statement in the context of the problem. Subtask Part B j (5 marks) Calculate the heights at which the hole can be drilled so that the spray distance is 6 m. Also state the height at which the hole must be drilled so that the spray distance is maximal. Now consider a specific hole in the water storage tank. As water flows out, the volume of water in the tank decreases as a function of time. The function $g : t \mapsto 0,25 t - 25$ with $0 \leq t \leq 100$ describes the temporal development of this volume change. Here $t$ is the time elapsed since the hole was completed in seconds and $g ( t )$ is the instantaneous rate of change of the water volume in the tank in liters per second. Calculate the volume of water in liters that flows out of the container during the first minute after the hole is completed.
The function $h$ is defined on $\mathbb{R}$ by $$h : \mathbb{R} \longrightarrow \mathbb{R}, \quad u \longmapsto u - [u] - 1/2$$ where $[u]$ denotes the integer part of $u$. Carefully draw the graph of the application $h$ on the interval $[-1, 1]$.
We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$. If $f ( x ) = x ^ { 2 }$ for all $x \in [ 0,1 ]$, determine, for all $n \in \mathbb { N } ^ { * }$, the polynomial $B _ { n } ( f )$ and deduce the value of $\left\| B _ { n } ( f ) - f \right\| _ { \infty }$.
We are given four real numbers $a \leqslant b \leqslant c \leqslant d$ such that $a + d = b + c$. Study the variations of the function $x \mapsto |x - a| - |x - b| - |x - c| + |x - d|$; show that it takes positive values. A reasoned argument supported by a graphical representation would be welcome.
Let $z \in \mathbb{C}$. We denote $C_z$ (respectively $\Omega_z$) the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| = 1$ (respectively $|Z(Z-2z)| < 1$). In this question we assume that $z$ is a real number denoted $a$. The curve $C_a$ in polar coordinates $(\rho, \theta)$ in the frame $\mathcal{R}'$ satisfies $$\left(\rho^2 + a^2\right)^2 - 4a^2 \rho^2 \cos^2\theta = 1$$ Simplify this equation when $a = 1$. Study and sketch the shape of the curve $C_1$.
Let $f$ be a function with real values, defined and continuous on $\mathbb{R}^{+}$, and admitting a finite limit at $+\infty$. a) Show that $f$ is bounded on $\mathbb{R}^{+}$. b) Deduce that the function $g$ defined on $\mathbb{R}^{+}$ by $\forall t \in \mathbb{R}^{+}, g(t)=t e^{\gamma t}$ where $\gamma$ is a strictly negative real, is bounded on $\mathbb{R}^{+}$.