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Home Question Bank Math Calculus Archive 2025 January 04

Calculus Questions from Jan 04,2025

Browse the Calculus Q&A Archive for Jan 04,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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\( z ^ { \prime } + 1 = \operatorname { Sin } ( z ) \Rightarrow \frac { d z } { d x } + 1 = \operatorname { Sin } ( z ) \) If \( f(x)=\frac{\sqrt{x+2}}{3-3 x^{2}} \), for which values of \( x \) is \( \begin{array}{ll}1.2 .2 & f(x) \text { non real. } \\ 1.2 .3 & f(x) \text { undefined } \\ 1.2 .4 & f(x)>0\end{array} \) Part A: Let f be the function over \( ]-\infty ; \mathbf{1}[\mathrm{U}] \mathbf{1} ;+\infty\left[\right. \) by \( f(x)=\frac{x^{2}+a x+b}{x-1} \) where \( a \) and \( b \) are two real parameters. We denote by \( (C) \) its representative curve in \( n \) orthonormal system ( \( O ; \vec{i} ; \vec{j} \) ) 1) Write \( f^{\prime}(x) \) in terms of \( a \) and \( b \). 2) Determine \( a \) and \( b \) knowing that \( f \) admits at \( x=3 \) an extremum equal to 4 . Part B: Suppose that \( a=-2 \) and \( b=5 \) and we consider the function \( f \) defined over \( ]-\infty ; 1[U] 1 ;+\infty\left[\right. \) by \( f(x)=\frac{x^{2}-2 x+5}{x-1} \) and Let (C) its representative curve in the orthonormal system \( (O ; \vec{i} ; \vec{j}) \). 1) Calculate \( \lim _{x \rightarrow(1)^{+}} f(x) \) and \( \lim _{x \rightarrow(1)^{-}} f(x) \). Interpret graphically the obtained result. 2) Write \( \mathrm{f}(\mathrm{x}) \) in the form \( \mathrm{f}(\mathrm{x})=a \mathrm{x}+b+\frac{c}{} \), where \( \mathrm{a}, \mathrm{b} \) and c are three integers to be c Considere el movimiento de una particula con función ittinerario dada por \( X(t)=6+5 \cos (\pi / 6 t) \) Donde x estả en metros y t en segundos. a) Determine los instantes cuando la particula está en reposo. b) Determine los instantes en los cuales la aceleración es nula. c) ¿Cuál es la rapidez máxima? ¿En qué instante se alcanza? \( x y + y ^ { 2 } ) d y - x ^ { 2 } d x = 0 \) \( \int \frac { 1 } { 1 + \tan z } \) 125 Encuentra los valores de los números a y b para que las siguientes funciones sean continuas en tod los números reales. a) \( f(x)=\left\{\begin{array}{ll}|2-x| & \text { si } x \leq 2 \\ \frac{x}{b} & \text { si } 2<x \leq 4 \\ a & \text { si } x>4\end{array}\right. \) b) \( f(x)=\left\{\begin{array}{ll}x^{2}+2 b x+4 & \text { si } x<1 \\ a x+5 & \text { si } 1 \leq x \leq 2 \\ \frac{3 x-1}{x} & \text { si } x>2\end{array}\right. \) 122 Halla el valor que debe tomar a para que las siguientes funciones sean continuas. a) \( f(x)=\left\{\begin{array}{ll}3 x-4 & \text { si } x<1 \\ a & \text { si } x=1 \\ \frac{x-3}{2} & \text { si } x>1\end{array}\right. \) b) \( f(x)=\left\{\begin{array}{ll}\frac{x^{2}-4}{x+2} & \text { si } x \neq-2 \\ a & \text { si } x=-2\end{array}\right. \) Determine el volumen del sólido generado al rotar el área bajo la curva \( y = \sin(x) \) desde \( x = 0 \) hasta \( x = \frac{\pi}{2} \) alrededor del eje x. Determinar o volume de um sólido cuja base é a região sob a curva \( y = \frac{1}{x} \) no intervalo de \( x = 1 \) a \( x = 4 \), utilizando seções transversais quadradas.
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