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Home Question Bank Math Calculus Archive 2024 December 21

Calculus Questions from Dec 21,2024

Browse the Calculus Q&A Archive for Dec 21,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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A particle moves in a straight line such that its velocity v at the moment t is \( \mathrm{v}=\left(12-\frac{1}{2} \mathrm{t}^{2}\right) \mathrm{cm} . / \mathrm{sec} . \), if the particle starts its motion when it is at a distance 2 cm to the right of point \( (\mathrm{O}) \), find its distance from \( (\mathrm{O}) \) after 6 seconds from the starting of motion. The population of a certain species in a protected area can be modeled by the function \( P(t) = 200 + 50 \sin(t) \), where \( t \) is measured in years. Determine the total population increase over one complete cycle (from \( t=0 \) to \( t=2\pi \)). Exercise 3: (6pts) On cc nsidere la suite \( \left(U_{n}\right)_{n \in \mathbb{N}} \) et la fonction f definie par: \( \left\{\begin{array}{l}U_{0}=\frac{1}{2} \\ U_{n+1}=f\left(U_{n}\right) \quad \forall n \in \mathbb{N}\end{array}\right. \) 1) Etudier les variation de f om \( \left[0_{3}\right]_{(1,5 p t s)} \) (2) Montrer que \( : \forall n \in \mathbb{N}: 0<U_{n}<1 \) (3) Etudier les variation de la suite \( \left(U_{n}\right)_{n \in \mathbb{N}} \quad(1,5 \mathrm{pts}) \) (4) En deduire que la suite \( \left(U_{n}^{r}\right)_{n \in \mathbb{N}} \) estr convergente et calculer sa limite EXERCICE 4. Déterminer le produit de convolution des fonctions : \( f(x)=\left\{\begin{array}{rll}1 & \text { si } & 0 \leq x \leq 2 \\ 0 & \text { ailleurs }\end{array}, g(x)=\left\{\begin{array}{rr}x-1 & \text { si } 1 \leq x \leq 3 \\ 0 & \text { ailleurs }\end{array}\right.\right. \) 4. Найти \( d y \), если \( y=\arcsin e^{-x} \). 5. Исследовать функцию \( y=\frac{4 x}{(x+1)^{2}} \) и построить ге график. Desarrollo 1. Resuelve los siguientes ejercicios . \( -y^{\prime \prime}-8 y^{\prime}+20 y=100 x^{2}-26 x e^{x} \) \( 10 \quad \lim _ { x \rightarrow 0 } \frac { \sin 3 x } { x } \) n considere la fonction \( f \) définie pour tout rél \( x \) par: \( f(x)=\left(-3 x^{2}\right)+x-3 \). n donne \( f^{\prime}(-1)=7 \). éterminer une équation de la tangente à la courbe représentative de \( f \) au point d'abscisse -1 . ne équation de la tangente est On considère la fonction \( f \) définie pour tout réel \( x \) par: \( f(x)=x^{3}-3 x^{2}+5 x+4 \). On donne \( f^{\prime}(2)=5 \). Déterminer une équation de la tangente à la courbe représentative de \( f \) au point d'abscisse 2 . Une équation de la tangente est Evaluate \( \lim _{x \rightarrow 0} \frac{3^{x}-2^{x}}{4^{x}-3^{x}} \)
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