Based on the following table, which shows crashworthiness ratings for several categories of motor vehicles. In all of these exercises, take X as the crash-test rating of a small car, Y as the crash-test rating for a small SUV, and so on as shown in the table.

$$\begin{matrix} & \text{} & \text {} & \text{Overall Frontal Crash-Test Rating}\\ & \text{Number Tested} & \text{3 (Good)} & \text{2 (Acceptable)} & \text{1 (Marginal)} & \text{0 (Poor)}\\ \text{Small Cars X} & \text{16} & \text{1} & \text{11} & \text{2} & \text{2}\\ \text{Small SUVs Y} & \text{10} & \text{1} & \text{4} & \text{4} & \text{1}\\ \text{Medium SUVs Z} & \text{15} & \text{3} & \text{5} & \text{3} & \text{4}\\ \text{Passenger Vans U} & \text{13} & \text{3} & \text{0} & \text{3} & \text{7}\\ \text{Midsize Cars V} & \text{15} & \text{3} & \text{5} & \text{0} & \text{7}\\ \text{Large Cars W} & \text{19} & \text{9} & \text{5} & \text{3} & \text{2}\\ \end{matrix}$$

Compute the probability distributions and expected values of Z and V. Based on the results, which of the two types of vehicle performed better in frontal crashes?

Find the slope of the line passing through ( - 3, 4) and (- 5, - 2).

In this problem we consider the general case—that is, with $\theta$ dependence—of the vibrating circular membrane of radius c:

$$\begin{array}{l}{a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}\right)=\frac{\partial^{2} u}{\partial t^{2}}, \quad 0<r<c, t>0} \\ {u(c, \theta, t)=0, \quad 0<\theta<2 \pi, t>0}\end{array}$$

$$\begin{array}{l}{u(r, \theta, 0)=f(r, \theta), \quad 0<r<c, 0<\theta<2 \pi} \\ {\left.\frac{\partial u}{\partial t}\right|_{t=0}=g(r, \theta), \quad 0<r<c, 0<\theta<2 \pi}\end{array}$$

. (a) Assume that $u=R(r) \Theta(\theta) T(t)$ and the separation constants are $-\lambda \text { and }-\nu$ Show that the separated differential equations are

$$\begin{aligned} T^{\prime \prime}+a^{2} \lambda T=0, & \Theta^{\prime \prime}+\nu \Theta=0 \\ r^{2} R^{\prime \prime}+r R^{\prime}+\left(\lambda r^{2}-\nu\right) R &=0 \end{aligned}$$

. (b) Let $\lambda=\alpha^{2}$ and $\nu=\beta^{2}$ and solve the separated equations in part (a). $\lambda=\alpha^{2}$ and $\nu=\beta^{2}$ problem. (d) Use the superposition principle to determine a multiple series solution. Do not attempt to evaluate the coefficients.

Find an equation in standard form for the ellipse that satisfies the given conditions. Foci (±2, 0) major axis length 10

Suppose that a function f(x, y, z) is differentiable at the point (3, 2, 1) and L(x, y, z)=x-y+2z-2 is the local linear approximation to f at (3, 2, 1). Find f(3, 2, 1),

$$f _ { x } ( 3,2,1 ) , f _ { y } ( 3,2,1 )$$

, and

$$f_z$$

(3, 2, 1).

You are a personal trainer, and your clients' goals are listed below. Give each one a different piece of advice, using familiar commands.

1. "Quiero adelgazar".
2. "Quiero tener músculos bien definidos".
3. "Quiero quemar grasa".
4. "Quiero respirar sin problemas ".
5. "Quiero correr un maratón".
6. "Quiero aumentar un poco de peso".

Find the product or quotient. Write your answer using exponents.

$$4 ^ { 8 } \cdot 4 ^ { 9 }$$

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. If

$$A _ { 1 } ( D ) A _ { 2 } ( D )$$

annihilates F(x), then either

$$A _ { 1 } ( D )$$

annihilates F(x) or

$$A _ { 2 } ( D )$$

annihilates F(x).

Graph each function.

$$y = \frac { 1 } { x - 4 } + 2$$

Write the number in scientific notation: 0.000105

Explain how the drag on a given smokestack could be the same in a 2-mph wind as in a 4-mph wind. Assume the values of $\rho$ and $\mu$ are the same for each case.

Write each expression in simplest form. Assume that all variables are positive. $\left(-32 y^{15}\right)^{\frac{1}{5}}$

Find the quotient: (-[1/5]/[7/5]).

identify any outliers. 16 9 11 12 8 10 12 13 11 10 24 9 2 15 7