Solve the given equation or inequality graphically. State your answers rounded to two decimals. $\text { (b) } x^{3}+3 x^{2} \geq-x^{2}+3 x+7$

Express each of the following in the equivalent logarithmic form:

(a). $5^4=625$

(b). $4^{-2}=\frac{1}{16}$

($c$). $x^3=3$

(d). $10^w=450$

(e). $3^8=z$

(f). $a^b=T$

Find the derivative of the vector function $\mathbf{r}(t)=\mathbf{i}-\mathbf{j}+e^{4t}\mathbf{k}$

Use a graphing utility to graph the curve represented by the parametric equations. Prolate cycloid: $x=2 \theta-4 \sin \theta, y=2-4 \cos \theta$

Jaime le ha escrito a su amiga Lupe contándole lo que hizo durante las vacaciones de verano. Completa esta parte de la carta con una de estas palabras de ¡Así lo decimos!

acampar - escalar - navegamos a vela

aficionado - fracturé - pescamos

camping - lago - velero

• Querida Lupe

Todo nos fue muy bien hasta que me __________ al bajar del velero. Mi viaje fue más corto de lo esperado, pero ¡fue una experiencia inolvidable!

Hasta pronto,

Jaime

Find a set of parametric equations of the line. (There are many correct answers.) Passes through (2, 3, 4) and is perpendicular to $3 x+2 y-z=6$

Solve the system

$$A \mathbf { x } = \mathbf { b }$$

using the given LU factorization of A.

$$A = \left[ \begin{array} { r r r r } { 1 } & { 4 } & { 3 } & { 0 } \\ { - 2 } & { - 5 } & { - 1 } & { 2 } \\ { 3 } & { 6 } & { - 3 } & { - 4 } \\ { - 5 } & { - 8 } & { 9 } & { 9 } \end{array} \right] =\left[ \begin{array} { r r r r } { 1 } & { 0 } & { 0 } & { 0 } \\ { - 2 } & { 1 } & { 0 } & { 0 } \\ { 3 } & { - 2 } & { 1 } & { 0 } \\ { - 5 } & { 4 } & { - 2 } & { 1 } \end{array} \right] \times \left[ \begin{array} { r r r r } { 1 } & { 4 } & { 3 } & { 0 } \\ { 0 } & { 3 } & { 5 } & { 2 } \\ { 0 } & { 0 } & { - 2 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 1 } \end{array} \right] , \mathbf { b } = \left[ \begin{array} { r } { 1 } \\ { - 3 } \\ { - 1 } \\ { 0 } \end{array} \right]$$

You are planning a party and want to know how many cans of soda to buy. A genie offers to tell you either the mean number of cans guests will drink or the median number of cans. Which measure of center should you ask for? Why? To make your answer concrete, suppose that there will be 30 guests and the genie will tell you either $\overline{x}=5$ cans or M=3 cans. How many cans should you have on hand?

Find a set of parametric equations of the line. (There are many correct answers.) Passes through (2,3,4) and is perpendicular to the plane given by 3x + 2y - z = 6.

Use a graphing utility to graph the curve defined by the given parametric equations. $x=\sin t+\cos t, y=\sin t-\cos t$

Find the velocity vector for the function $\mathbf { r } ( t ) = \left\langle e ^ { t } , e ^ { - t } , 0 \right\rangle$.

Find parametric equations for the line.
The line through $(2,3,0)$ perpendicular to the vectors $A=i+2j+3 k$ and $B=3 i+4 j+5k$

Use a contact goniometer, illustrated in Figure 1.12, to measure the angle between adjacent faces on the quartz crystals on display in the lab.

a. Are the angles about the same for each quartz specimen, or do they vary from one sample to another?

b. Write a generalization that describes how the angle between crystal faces relates to the size and/or shape of the sample.

Find the velocity, acceleration, and speed of a particle with the given position function r(t)=2sqrt(2)t*i+e^(2t)*j+e^(-2t)*k.