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Home Question Bank Math Calculus Archive 2024 December 05

Calculus Questions from Dec 05,2024

Browse the Calculus Q&A Archive for Dec 05,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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28 Multiple Choice 1 point Evaluate this indefinite integral: \( \int \sqrt{ } x+(1) /(2 \sqrt{ } x) d x \) 1. Aplicando el criterio de la razón, determine si la serie dada, converge o diverge \( \sum_{n=0}^{\infty} \frac{4^{n}}{3^{n}+1} \) 2. Determine os limites a seguir: \( \begin{array}{ll}\text { a) } \lim _{x \rightarrow+\infty} \frac{3 x^{2}+5 x-4}{x^{3}+7 x} & \text { b) } \lim _{x \rightarrow-\infty} \frac{x^{5}+3 x^{2}+2 x}{x^{3}-7 x^{2}}\end{array} \) 25. Multiple Choice 1 point The function \( f \) has a first derivative given by \( f^{\prime}(x)=5 x^{4}-12 x^{2} \). Which of the following is one of the three \( x \) coordinates that are inflection points of the graph? 0 1.54919 The function fhas no inflection points -1.54919 25 Multiple Choice 1 point The function \( f \) has a first derivative given by \( f^{\prime}(x)=5 x^{4}-12 x^{2} \). Which of the following is one of the three \( x \) coordinates that are inflection points of the graph? Exercice 2 Soit \( g \) la fonction définie sur \( \mathbb{R} \backslash\{1\} \) par \( g(x)=\frac{e^{x}}{1-x} \). 1. Déterminer \( g^{\prime}(x) \), puis montrer que \( g^{\prime \prime} \) a pour expression \( g^{\prime \prime}(x)=\frac{e^{x}\left(x^{2}-4 x+5\right)}{(1-x)^{3}} \). 2. En déduire la convexité de \( g \) et les abscisses des éventuels points d'inflexion de la courbe de \( g \). 3. Déterminer une équation de la tangente à la courbe représentative de \( g \) au point d'abscisse 0 . 4. En déduire que, pour tout \( x<1 \), on a \( e^{x} \geqslant-2 x^{2}+x+1 \). 25 Multiple choice 1 point The function fhas a first derivative given by \( f^{\prime}(x)=5 x^{4}-12 x^{2} \). Which of the following is one of the three \( x \) coordinates that are inflection points of the graph? 0 1.54919 The function fhas no inflection points -1.54919 a) \( \lim _{x \rightarrow 3^{-}} f(x)= \) b) \( \lim _{x \rightarrow 3^{+}} f(x)= \) c) \( \lim _{x \rightarrow 3^{-}} f(x)= \) d) \( \lim _{x \rightarrow 5^{-}} f(x)= \) e) \( \lim _{x \rightarrow 5^{+}} f(x)= \) f) \( \lim _{x \rightarrow 5} f(x)= \) Let \( r \) be a function whose second derivative exists on an open interval I. Which of the following could be a true statement? \[ \text { If } f^{\prime \prime}(x)<0 \text { for all } x \text { in } I \text {, then the graph of } f \text { is concave upward. } \] \[ \text { If } f^{\prime \prime}(x)<0 \text { for all } x \text { in } I \text {, then the graph of } f \text { is neither concave up nor concave down. } \] If \( f^{\prime \prime}(x)<0 \) for all \( x \) in \( I \), then the graph of \( f \) is concave downward. 23 Multiple Choice 1 point The function \( f \) is continuous for \( -3 \leq x \leq 2 \) and differentiable for \( -3<x<2 \). If \( f(-3)=3 \), and \( f(2)=-2 \), which of the following statements could be false? There exists \( c \), where \( -3<c<2 \), such that \( f(c)=1 \) There exists \( c \), where \( -3<c<2 \), such that \( f^{\prime}(c)=-1 \) There exists \( c \), where \( -3<c<2 \), such that \( f^{\prime}(c)=6 \) There exists \( c \), where \( -3<c<2 \), such that \( f(c)=0 \)
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