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 \( t=2\pi \)).

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

QUESTION 4 It is given that the asymptotes of \( f(x)=\frac{6}{x+p}+q \) intersect at \( (4 ; 3) \). \( 4.1 \quad \) Write down the equation of \( f \). \( 4.2 \quad \begin{array}{l}\text { Sketch the graph of } f \text {, clearly showing all the intercepts with the axes and any } \\ \text { asymptotes. }\end{array} \) \( 4.4 \quad \begin{array}{l}g \text { is one of the axes of symmetry of } f \text { and it is a decreasing function. Determine the } \\ \text { equation of } g \text {. } \\ 4.5 \\ (-3 ; 2) \text { is a point on } f \text {. Determine the coordinates of the image of this point after } \\ \text { reflection in } g \text {. }\end{array} \)

(b) Find the angle between \( \overline{\mathbf{a}} \) and \( \overline{\mathbf{b}} \) where \( -6 \bar{a}+2 \bar{b}=2 i+j-8 k \) and \( 2 \bar{a}-\bar{b}=3 i-4 j+k \) (c) show that the line \( \frac{x-1}{2}=\frac{y-2}{-6}=\frac{z-1}{7} \quad \) is parallel to the tangent vector of the

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 \( a_{n+1} \)

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.