1. Expand \( (1+x)^{\frac{1}{3}} \) in ascending powers of x up to the term \( x^{3} \). By substituting 0.08 for \( x \) in your result. Obtain an approximation value of the cube root of 5 .

Exercises 1. In each of the following, two open statements \( P(x, y) \) and \( Q(x, y) \) are given, where the domain of both \( x \) and \( y \) is \( Z \). Determine the truth value of \( P(x, y) \Rightarrow Q(x, y) \) for the given values of \( x \) and \( y \). a. \( P(x, y): x^{2}-y^{2}=0 \). and \( Q(x, y): x=y .(x, y) \in\{(1,-1),(3,4),(5,5)\} \). b. \( P(x, y):|x|=|y| \). and \( Q(x, y): x=y .(x, y) \in\{(1,2),(2,-2),(6,6)\} \). c. \( P(x, y): x^{2}+y^{2}=1 \). and \( Q(x, y): x+y=1 \). \( (x, y) \in\{(1,-1),(-3,4),(0,-1),(1,0)\} \). 2. Let \( O \) denote the set of odd integers and let \( P(x): x^{2}+1 \) is even, and \( Q(x): x^{2} \) is even. be open statements over the domain \( O \). State \( (\forall x \in O) P(x) \) and \( (3 y \in O) Q(x) \) in words. 3. State the negation of the following quantified statements.

1. Expand \( (1+x)^{\frac{1}{3}} \) in ascending powers of x up to the term \( x^{3} \). By substituting 0.08 for \( x \) in your result. Obtain an approximation value of the cube root of 5 .

Una sucesión infinita tiene la caracteristica de que: a. El término \( a_{n} \) siempre es el último de la sucesión. b. Cada término \( a_{n} \) tiene un antecesor igual a \( a_{n+1} \) c. Cada término \( a_{n} \) es el último en sucesiones pares. d. Cada término \( a_{n} \) tiene un sucesor igual a \( a_{n+1} \)

j) \( \sin ^{-1}(y)+x^{3}=\sqrt{y} \)

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 \( t=2\pi \)).