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Home Question Bank Math Pre Calculus Archive 2025 January 07

Pre-calculus Questions from Jan 07,2025

Browse the Pre-calculus Q&A Archive for Jan 07,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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enl ' Calculer lestross premiers termes de lasmite (Un) dslescas suivants. \( L_{1} \) ( 1) \( U_{n}=3 \)-4nin \( \epsilon(\mathbb{N} \) 2) \( U_{n}=\frac{1}{\sqrt{n-5}} ; n \geqslant 6 \) 3) \( \left\{\begin{array}{l}U_{0}=1 \\ L_{n+1}=3 u_{n}+1 .\end{array} \quad(n \in \mathbb{N})\right. \) en2, Etndier la monotonie de ( \( U_{n} \) ) ds les cas suivants \( 3 \left\lvert\, y U_{n}=\frac{n^{2}+1}{n+1}\right. \) e) \( U_{n}=3 n^{2}-2 \quad(n \in(\mathbb{W}) \). ex3r soit \( \left(U_{n}\right) \) une suite numérique définie par. \( \forall n \in \mathbb{N}_{r} \cdot U_{n}=-7 n+1 \). 0,22 1) Calculer \( L_{0}, L_{1} \) et \( U_{20} \). 2) Vérifier que ( \( U_{n} \) ) est une suite arithmétique en déterninoat sa raisun. Sketch on the same set of axes the graphs of \( f(x)=-2 x^{2}-4 x+6 \) and \( g(x)=-2 \cdot 2^{x-1}+1 \) Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s). Determine the end behavior of tl 20. \( y=-7 x^{3}+8 x^{2}+x \) 23. \( y=8 x^{11}-2 x^{9}+3 x^{6}+4 \) 26. \( y=x^{4}-7 x^{2}+3 \) 29. \( y=-x^{3}-x^{2}+3 \) Determine the end behavior of the graph of each polynomial function. \( \begin{array}{lll}\text { 20. } y=-7 x^{3}+8 x^{2}+x & \text { 21. } y=-3 x+6 x^{2}-1 & \text { 22. } y=1-4 x-6 x^{3}-15 x^{6} \\ \begin{array}{lll}\text { 23. } y=8 x^{11}-2 x^{9}+3 x^{6}+4 & \text { 24. } y=-x^{5}-15 x^{7}-4 x^{9} & \text { 25. } y=-3-6 x^{5}-9 x^{8} \\ \text { 26. } y=x^{4}-7 x^{2}+3 & \text { 27. } y=-8 x^{7}+16 x^{6}+9 & \text { 28. } y=-14 x^{6}+11 x^{5}-11 \\ \text { 29. } y=-x^{3}-x^{2}+3 & \text { 30. } y=x^{3}-14 x-4 & \text { 31. } y=5-17 x^{7}+9 x^{10}\end{array}\end{array}> \) Esercizio 5. Sia data su \( \mathbb{R}^{3} \) la forma quadratica \[ Q(x, y, z)=-2 x^{2}+2 x y+y^{2}-4 x z-z^{2} . \] (a) Scrivere l'espressione della forma bilineare polare \( b \) di \( Q \) (b) Calcolare rango e segnatura di \( Q \) (c) Calcolare una base di Sylvester di \( Q \) (d) Trovare, se esistono, due sottospazi distinti di \( \mathbb{R}^{3} \) su cui la restrizione di \( Q \) è definita positiva. 7. An arithmetic and a geometric sequence are combined to form the pattern, which is given by: \( P_{n}=x ; \frac{1}{3} ; 2 x ; \frac{1}{9} ; \frac{1}{27} \) (a) Write down the next TWO terms of the pattern. (b) Determine the general term \( \left(T_{n}\right) \) for the odd terms of this pattern. (4) Write down your answer in terms of \( x \). (c) Calculate the value of \( P_{26} \). (d). If \( \quad \sum_{n=1}^{21} P_{n}=33,5 \), determine the value of \( x \). 7. An arithmetic and a geometric sequence are combined to form the pattern, which is given by: \( P_{n}=x ; \frac{1}{3} ; 2 x ; \frac{1}{9} ; \frac{1}{27} \) (a) Write down the next TWO terms of the pattern. (b) Determine the general term \( \left(T_{n}\right) \) for the odd terms of this pattern. (4) Write down your answer in terms of \( x \). (c) Calculate the value of \( P_{26} \). (d). If \( \quad \sum_{n=1}^{21} P_{n}=33,5 \), determine the value of \( x \). lemas: en las siguientes funciones determina, gráfico, domino y rango de cada una, de las funcione etando los intervalos en cada caso. \[ \begin{array}{l} f(x)=x-5 \text { en }-2<x 4 \\ f(x)=\sqrt{x-3} \text { en }-4>x>3 \\ f(x)=\sqrt{x^{2}-9}-4 \text { en } 3<x<5 \\ f(x)=\sqrt{x^{2}-25} \text { en } 5<x<8 \\ f(x)=\sqrt{9-x^{2}} \quad \text { en }-3<x<3 \end{array} \] \[ f(x)=\left\{\begin{array}{ccc} x & \text { si } & x<0 \\ x^{2} & \text { si } & 0 \leq x \leq 1 \\ 2-x & \text { si } & x>1 \end{array}\right. \] g) \( g(x)=\left\{\begin{array}{ccc}1 & \text { si } & x<0 \\ -x & \text { si } & 0 \leq x \leq 1 \\ x-2 & \text { si } & x>1\end{array}\right. \) \[ f(x)=\left\{\begin{array}{ccc} x^{2} & \text { si } & x<0 \\ x+1 & \text { si } & 0 \leq x \leq 1 \\ x^{2} & \text { si } & x>1 \end{array}\right. \] i) \( f(x)=\left\{\begin{array}{ccc}-x^{2} & \text { si } & x \leq 1 \\ 3 x & \text { si } & x>1\end{array}\right. \) olema: Dadas las siguientes expresiones algebraicas, explica si es función y da su gráfico \[ \begin{array}{l} y=2+\sqrt{x^{2}-9} \\ y=4-\sqrt{x+3} \\ y=2-\sqrt{16-x^{2}} \end{array} \] 2.- \( \quad y=-5 x+3 \) 4.- \( y=\sqrt{x+5} \) 6.- \( \quad x^{2}-(y-3)^{2}=16 \) dema: demuestra si para \( \mathrm{x}=-1, \mathrm{x}=1 \) y \( \mathrm{x}=5 \), las soluciones son correctas en las siguientes funciones. \[ \begin{array}{l} f(x)=2 x+1 \\ f(x)=\frac{1}{x} \\ f(x)=3 x^{2} \\ f(x)=\frac{3}{2 x-1} \\ f(x)=\sqrt{1-x} \end{array} \] sol: \( 3,-1,1,-9 \) sol: \( 2, \quad 0,-1,44 \) sol: 3, 3, 75 \[ \text { sol: } 3,-1,-3, \quad-\frac{3}{11} \] sol: \( 0, \sqrt{2}, 1, \sqrt{6} \) 14. ¿Qué tipo de gráfica tienen las funciones exponenciales? a) Recta. b) Parábola. c) Curva exponencial. 15. ¿Qué tipo de gráfica tienen las funciones logarítmicas? a) Recia. b) Parábola. c) Curva logarítmica. 11. If \( f(x)=\log \left(\frac{1-x}{1+x}\right) \) then prove that \( f(x)+f(y)=f\left(\frac{x+y}{1+x y}\right. \)
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