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Home Question Bank Math Calculus Archive 2024 November 01

Calculus Questions from Nov 01,2024

Browse the Calculus Q&A Archive for Nov 01,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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2. \( \lim _{x \rightarrow \infty} \frac{x^{2}}{5-x^{3}} \) Soient \( f \) et \( g \) deux applications tel que: \[ f: \mathbb{R} \rightarrow \mathbb{R} \quad ; \quad g:[1 ;+\infty[\rightarrow[2 ;+\infty[ \] \[ x \mapsto f(x)=x^{2}+x+2 \quad ; \quad \mathbf{x} \mapsto g(x)=x+\frac{1}{x} \] 1) a) Montrer que : \( (\forall x \in \mathbb{R}) ; f(-1-x)=f(x) ; \) b) En déduire que l'application \( f \) n'est pas injective. 2) Résoudre dans \( \mathbb{R} \) l'équation : \( f(x)=-\frac{1}{4} \), et déduire que \( f \) n'est pas surjective 3) Montrer que : \( f(\mathbb{R})=\left[\frac{7}{4} ;+\infty[ \right. \) 4) Montrer que \( g \) est une bijection et déterminer sa bijection réciproque. Soient \( f \) et \( g \) deux applications tel que: \[ f: \mathbb{R} \rightarrow \mathbb{R} \quad ; \quad g:[1 ;+\infty[\rightarrow[2 ;+\infty[ \] \[ x \mapsto f(x)=x^{2}+x+2 \quad ; \quad \mathbf{x} \mapsto g(x)=x+\frac{1}{x} \] 1) a) Montrer que : \( (\forall x \in \mathbb{R}) ; f(-1-x)=f(x) ; \) b) En déduire que l'application \( f \) n'est pas injective. 2) Résoudre dans \( \mathbb{R} \) l'équation : \( f(x)=-\frac{1}{4} \), et déduire que \( f \) n'est pas surjective 3) Montrer que : \( f(\mathbb{R})=\left[\frac{7}{4} ;+\infty[ \right. \) 4) Montrer que \( g \) est une bijection et déterminer sa bijection réciproque. RESUELVE LOS SIGUIENTES LIMITES 1. \( \lim _{x \rightarrow \infty} \frac{x}{x-5} \) Se quiere construir un silo (sin incluir la base) en forma de cilin- dro rematado por una semiesfera. El costo de construcción por unidad cuadrada del área superficial es dos veces mayor para la semiesfera que para la pared cilíndrica. Determine las dimensio- nes que se deben usar si el volumen es fijo y el costo de construc- ción debe mantenerse al mínimo. Desprecie el espesor del silo y los desperdicios en la construcción. A conical tank (with vertex down) is 28 ft across the top and 7 ft ft deep. If water is flowing into the tank at the rate of \( 9 \mathrm{ft}^{3} / \mathrm{sec} \), find the rate of change of the depth of the water at the instant when it is 2 \( f t \) deep. Evaluate each of the following limits. If you need to use \( \infty \) or \( -\infty \), enter INFINITY or -INFINITY, respectively. (a) \( \lim _{x \rightarrow-\infty} e^{-x^{5}}=\square \) (b) \( \lim _{x \rightarrow \infty} e^{-x^{7}}=\square \) (c) \( \lim _{x \rightarrow 0^{+}} e^{3 / x}=\square \) (d) \( \lim _{x \rightarrow 0^{-}} e^{2 / x}=\square \) (e) \( \lim _{x \rightarrow \infty} e^{9 / x}=\square \) TAREA \#5, Corte 2 Tema: Integrales dobles 1. Calcular el área determinada por el par de curvas y realizar la gráfica en cada caso. a) \( y=x^{2} ; x+y=2 \) b) \( y=\sin x ; y=\cos x \) en el primer cuadrante c) \( y=\frac{x}{2} ; y=\sqrt{x} \) 5 In this question you must show detailed reasoning. Fig. 12 shows part of the graph of \( y=x^{2}+\frac{1}{x^{2}} \). The tangent to the curve \( y=x^{2}+\frac{1}{x^{2}} \) at the point \( \left(2, \frac{17}{4}\right) \) meets the \( x \)-axis at A and meets the \( y \)-axis at B . Ois the origin. Find the exact area of the triangle OAB . 5 In this question you must show detalled reasoning. Fig. 12 shows part of the graph of \( y=x^{2}+\frac{1}{x^{2}} \). The tangent to the curve \( y=x^{2}+\frac{1}{x^{2}} \) at the point \( \left(2, \frac{17}{4}\right) \) meets the \( x \)-axis at \( A \) and meets the \( y \)-axis at \( B \). O the origin. Find the exact area of the triangle OAB.
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