Precálculo preguntas y respuestas

Find the magnitude and positive direction angle of the vector \( \langle-2 \sqrt{3},-2\rangle \). The magnitude of the vector is (Simplify your answer.) The direction angle is \( \square^{\circ} \). (Simplify your answer. Use angle measures greater than or equal to 0 and less than 360. .)

For the following rectangular equation, give the equivalent polar equation and sketch its graph. 10 of 22 \( x-y=8 \) The polar form of the rectangular equation is \( \square \). Typestio (Use a comma to separate answers as needed. Type each answer only once.)

On considère les fonctions \( f \) et \( g \) définies par: \[ f(x)=\sqrt{x+2} \text { et } g(x)=\frac{3 x}{2 x-1} \] Et soient \( \left(C_{f}\right) \) et \( \left(C_{g}\right) \) les courbes respectives des fonctions \( f \) et \( g \) dans un repère orthonormé \( (O ; \vec{l} ; \vec{\jmath}) \). 1) Déterminer \( D_{f} \) et \( D_{g} \). 2) Montrer que \( A(-1 ; 1) \) et \( B(2 ; 2) \) sont des points aux courbes \( \left(C_{f}\right) \) et \( \left(C_{g}\right) \). 3) Dresser les tableaux de variations des fonctions \( f \) et \( g \). 4) Construire les courbes \( \left(C_{f}\right) \) et \( \left(C_{g}\right) \) sur le repère (O; \( \vec{l} ; \vec{\jmath}) \). 5) Résoudre graphiquement les inéquations: \[ \sqrt{x+2}-\frac{3 x}{2 x-1}<0 \text { et } \frac{3 x}{2 x-1} \sqrt{x+2} \leq 0 \]

Complete the following simplification. \( \begin{array}{l} \frac{6 \operatorname{cis}\left(45,000^{\circ}\right)}{\operatorname{cis}\left(45,000^{\circ}\right)}=6 \mathrm{cis} \\ = \\ \begin{aligned} \frac{6 \operatorname{cis}\left(45,000^{\circ}\right)}{\operatorname{cis}\left(45,000^{\circ}\right)} & =6 \operatorname{cis}(\square) \\ & =\square\end{aligned}\end{array} \)

Ricardo shot an arrow from ground level at the foot of a tower. The height of the arrow \( y \), in feet, was a function of the number of seconds, \( t \), since Ricardo shot the arrow: \[ y(t)=-16(t-4)^{2}+256 \] Ricardo then climbed to the top of the tower and shot another arrow in the same way. The arrow shot from the tower was in the air 1 second longer than the arrow shot from ground level. How tall was the tower in feet?

Ricardo shot an arrow from ground level at the foot of a tower. The height of the arrow \( y \), in feet, was a function of the number of seconds, \( t \), since Ricardo shot the arrow: \[ y(t)=-16(t-4)^{2}+256 \] Ricardo then climbed to the top of the tower and shot another arrow in the same way. The arrow shot from the tower was in the air 1 second longer than the arrow shot from ground level. How tall was the tower in feet?

\[ f(x)=\left\{\begin{array}{ll}|x+1|, & -2 \leq x \leq 0 \\ |x-1|, & 0<x \leq 2\end{array}\right. \] (a) The letter of the alphabet that most resembles the graphth of \( f \) is (b) Is \( f \) an even function?

\[ f: \mathbb{R} \rightarrow \mathbb{R}, \quad f(x)=18+22 \cos \left(\frac{2 \pi}{44}(x-7)\right) \text {. } \] Sea la función restricción \( g=\left.f\right|_{[51,95]} \). La imagen de la función \( g \) es rango \( (g)=[A, B] \). (a) Encuentre el valor de \( A \). (b) Encuentre el valor de \( B \). La imagen inversa de 18 satisface \( g^{-1}(18)=\{C, D\} \) con \( C<D \). (c) Encuentre el valor de \( C \). (d) Encuentre el valor de \( D \). Considere la función restricción \( h=\left.f\right|_{[51 . E]} \), donde \( E \) es una con- stante a ser determinada. (e) Encuentre máximo valor posible de \( E \in \mathbb{R} \) de manera que \( h \) sea inyectiva.

3. En el presente problema trabajamos con restricciones de funciones. Sea la función \[ f: \mathbb{R} \rightarrow \mathbb{R} . \quad f(x)=18+22 \cos \left(\frac{2 \pi}{44}(x-7)\right) \text {. } \] Sea la función restricción \( g=\left.f\right|_{[51,95]} \). La imagen de la función \( g \) es rango \( (g)=[A, B] \). (a) Encuentre el valor de \( A \). (b) Encuentre el valor de \( B \). La imagen inversa de 18 satisface \( g^{-1}(18)=\{C, D\} \) con \( C<D \). (c) Encuentre el valor de \( C \). (d) Encuentre el valor de \( D \).

\( P(t)=\frac{520}{1+6 e^{-0.18 t}} \) Find the initial population size of the species and the population size after 9 years. Round your answers to the nearest whole number as necessary.