2) \( f(x)=\frac{x^{2}+x}{3 x^{2}-12} \)

2. (2.5 pts) Demuestra que para todo número natural \( n \), \[ 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{n+1}} \geq \sqrt{n+1} \] ¿A partir de qué número \( n \) se empieza a cumplir la desigualdad estricta \( > \), sin el igual?

Graph \( m(w)=-2 \cdot 2^{w} \)

The exponential model \( A=177.2 e^{0.013 t} \) describe the population, \( A_{1} \) of a country in millions, \( t \) years after 2003 . Use the model to answer the following questions: a) what was the population of the country in 2003 ? b) By what percentage is the papulation of the country increasing each year?

Given two points for an exponential function, 1. Use the two points to find the growth rate, k . Write an exponential model for each point, then solve this system of two equations for k . 2. Use either point with the k you found to find the initial amount at time zero, \( \mathrm{A}_{\mathrm{o}} \). 3. Doubling time is when the amount is \( 2^{*} \mathrm{~A}_{\mathrm{o}} \). 4. Use the values of k and \( \mathrm{A}_{\mathrm{o}} \) to calculate the amount for a given time or to find the time to reach a specific amount in the future. The count in a bacteria culture was 600 after 20 minutes and 1900 after 30 minutes. Assuming the count grows exponentially. You may enter the exact value or round to 2 decimal places. What was the initial size of the culture? Find the doubling period. Find the population after 105 minutes. When will the population reach 14000 .

Given \( f(x)=\frac{-2}{x}-2 \), Answer the following questions: .1 Write down the equations of the asymptotes of \( f(x) \). 2 Draw a neat sketch graph of \( f(x) \), clearly indicate the asymptotes and the intercepts with the axes. .3 Write down the domain and range of \( f(x) \).