Let \( l \) be the length of a diagonal of a rectangle whose sides have lengths \( x \) and \( y \), and assume that \( x \) and \( y \) vary with time. If \( x \) increases at a constant rate of \( \frac{1}{4} \mathrm{ft} / \mathrm{s} \) and \( y \) decreases at a constant rate of \( \frac{1}{8} \) \( \mathrm{ft} / \mathrm{s} \), how fast is the size of the diagonal changing when \( x=1 \mathrm{ft} \). and \( y=4 \mathrm{ft} \) ? Answer:

4. Considere a função definida por: \[ g(x)=\left[\begin{array}{l}x^{2}+2 \text { se } x<0 \\ 3 x-1 \text { se } 0 \leq x \leq 2 \\ 4 x-3 \text { se } x>2\end{array}\right. \] \( \begin{array}{lll}\text { a) } g(1) & \text { b) } g(2) & \text { calcule: } \\ g(3)\end{array} \)

(3) \( \operatorname{Lim}_{x \rightarrow \infty} \frac{\sqrt[3]{x^{3}-2 x^{2}+3}}{2 x+1} \)

Equation of the parabola Vertex:((471,125) Point:(0,0) \( 0=a(0-471)^{2}+125 \) \( 0=a(-471)^{2}+125 \) \( 0=a(221,841)+125 \) \( 0=221841 \mathrm{a}+125 \) \( \frac{-125}{221841}=\frac{221841}{221841} \)

1. \( x^{2}+5 x+6=0 \)

Resolver: \( 5-\frac{\frac{3}{4}+\frac{1}{2}-\frac{2}{5}}{2-\frac{\frac{1}{2}+\frac{3}{4}}{5+\frac{2}{1-\frac{1}{3}}}} \)