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Home Question Bank Math Pre Calculus Archive 2024 December 30

Pre-calculus Questions from Dec 30,2024

Browse the Pre-calculus Q&A Archive for Dec 30,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Which of the following is a property of the graph of an expo- nential growth function of the form \( f(x)=a b^{x} \) ? select all that apply. A The graph is a curve that increases more rapidly for larger values of \( x \). B The graph has a \( y \)-intercept at \( (0, a) \). C The graph has an \( x \)-intercept at \( (b, 0) \). D The graph has a horizontal asymptote at \( y=1 \). On considère la suite \( \left(U_{n}\right)_{n} \) définie par: \( U_{0}=-\frac{3}{4} \) ct \( U_{n+1}=\frac{2 U_{n}-1}{2 U_{n}+5} \) 1) a) vérifier que \( (\forall n \in \mathbb{N}) \quad U_{n+1}=1-\frac{6}{2 U_{n}+5} \) b) prouver que \( (\forall n \in \mathbb{N})-1<U_{n}<-\frac{1}{2} \) 2) étudier la monotonie de la suite \( \left(U_{n}\right)_{n} \) 3) on pose \( V_{n}=\frac{2 U_{n}+1}{U_{n}+1} \) a) montrer que \( \left(V_{n}\right)_{n} \) est une suite géométrique b) déterminer \( U_{n} \) en fonction de \( n \) Suites EX/1 \( \left(u_{n}\right)_{n} \) une suite telle que \( :\left\{\begin{array}{l}u_{0}=2 \\ u_{n+1}=\frac{2 u_{n}-3}{4-u_{n}}\end{array}\right. \) Pour tout \( n \) de \( \mathbb{N} \) on pose \( v_{n}=\frac{u_{n}-3}{u_{n}+1} \) 1. montrer que \( \forall n \in \mathbb{N} \quad-1 \leq u_{n} \leq 3 \) 2. étudier la monotonie de \( \left(u_{n}\right)_{n} \) 3. a) montrer que \( \left(v_{n}\right)_{n} \) est géométrique b) calculer \( u_{n} \) en fonction de \( n \) c) déterminer: \( S_{n}=\sum_{k=0}^{n} v_{k} \) et \( P_{n}=\prod_{k=0}^{n} v_{k} \) en fonction de \( n \) The equation of a hyperbola is given by \( f(x)=\frac{3}{x-7}-4 \). Write down the equation of the new function that is formed when \( f \) is transformed as follows: \( \begin{array}{ll}3.1 & \text { Shift two units to the left } \\ 3.2 & \text { Shift } 3 \text { units up } \\ 3.3 & \text { Shift } 1 \text { unit right and } 2 \text { units down } \\ 3.4 & \text { The equation of the new hyperbola has new asymptotes at } x=-4 \text { and } y=-1\end{array} \) (1) On considère la suite numérique \( \left(U_{n}\right) \) définie par: \( U_{0}=\frac{1}{3} \) et \( U_{n+1}=\frac{1}{3-U_{n}} \) pour tout \( n \) de \( N \). 1. Calculer \( U_{1} \) et \( U_{2} \). 2. Montrer par récurrence que pour tout \( n \) de \( N: U_{n}<1 \). (a) Montrer que pour tout \( n \) de \( N: U_{n+1}-U_{n}=\frac{\left(U_{n}-1\right)^{2}}{3-U_{n}} \) (b) En déduire la monotonie de \( \left(U_{n}\right) \). 4. On considère la suite numérique \( \left(V_{n}\right) \) définie par: \( V_{n}=\frac{1}{1-U_{n}} \) pour tout \( n \) de \( N \). (a) Montrer que \( \left(V_{n}\right) \) est une suite arithmétique en déterminant sa raison et son premier terme. (b) Écrire \( V_{n} \) en fonction de \( n \) puis déduire que \( U_{n}=\frac{n+1}{3+n} \) pour tout \( n \) de \( N \). (c) Calculer la somme : \( S_{8}=V_{0}+V_{1}+\ldots+V_{8} \) etch on the same set of axes the graphs of \( f(x)=-2 x^{2}-4 x+6 \) and \( g(x)=-2 \cdot 2^{x-1}+1 \) early indicate all intercepts with the axes, turning point(s) and asymptote(s). etch on the same set of axes the graphs of \( f(x)=-2 x^{2}-4 x+6 \) and \( g(x)=-2 \cdot 2^{x-1}+1 \) early indicate all intercepts with the axes, turning point(s) and asymptote(s). \( \left. \begin{array} { l } { \log _ { 3 } 2 = a } \\ { \log _ { 3 } 5 = b } \end{array} \right. \left\{ \begin{array} { l } { \log _ { 25 } 16 = ? } \end{array} \right. \) Graph the function \[ f(x)=\left\{\begin{array}{ll}\lfloor x\rfloor, & x \geq 0 \\ \lceil x\rceil, & x<0\end{array}\right. \] What is the y-intercept of the exponential function \( f(x) = 2^x \)?
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