Solve a simple fist order separable differential equation \[ \frac{d y}{d x}=x y, \]

The radioactive substance cesium-137 has a half-life of 30 years. The amount \( A(t) \) (in grams) of a sample of cesium-137 remaining after \( t \) years is given by the following exponential function. \[ A(t)=621\left(\frac{1}{2}\right)^{\frac{t}{30}} \] Find the initial amount in the sample and the amount remaining after 100 years. Round your answers to the nearest gram as necessary. Initial amount: \( \square \) grams Amount after 100 years: \( \square \) grams

Use an appropriate test to determine whether the following series converges. \( \sum_{k=1}^{\infty} \frac{7}{\sqrt[3]{k}} \) Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer.)

3) \( f(x)=x^{3+\sqrt{2}}+\sqrt[4]{x^{3}} \)

1.60 iano \( f(x), g(x) \) due funzioni derivabili per \( x>a \). Dimostrare che \( f(x) \leq g(x) \), per ogni \( x>a \), se valgono le due condizioni: \[ f^{\prime}(x) \geq g^{\prime}(x), \quad \forall x>a ; \quad \lim _{x \rightarrow+\infty}[f(x)-g(x)]=0 \text {. } \] [Applichiamo il teorema di Lagrange alla differenza \( f-g \) nell' intervallo \( [x, b] \), con \( a<x<b \). Esiste \( \xi \in(x, b) \) per cui \( [f(b)-g(b)]-[f(x)-g(x)]=\left[f^{\prime}(\xi)-g^{\prime}(\xi)\right](b-x) \geq 0 \) Al limite, per \( b \rightarrow+\infty \), otteniamo la tesi \( ] \)

Exercice 3 : Suites numériques (4 points) On considère la suite numérique \( \left(v_{n}\right) \) définie par: 1-a) Démontrer par récurrence que, pour tout entier na- turel \( n: 0<v_{n}<3 \). b) Démontrer que, pour tout entier naturel \( n \) : \( v_{n+1}-v_{n}=\frac{\left(3-v_{n}\right)^{2}}{6-v_{n}} \). c) Démontrer que la suite \( \left(v_{n}\right) \) est convergente. 2. On considère la suite \( \left(w_{n}\right) \) définie par: \( w_{n}=\frac{1}{v_{n}-3} \). a) Démontrer que \( \left(w_{n}\right) \) est une suite arithmétique de raison \( -J / 3 \). b) En déduire l'expression de \( \left(w_{n}\right) \), puis celle de \( \left(v_{n}\right) \) en fonction de \( n \). c) Déterminer la limite de la suite \( \left(v_{n}\right) \).