Solve the given initial-value problem.

$$\mathbf{X}^{\prime}=\left(\begin{array}{rrr}{1} & {-12} & {-14} \\ {1} & {2} & {-3} \\ {1} & {1} & {-2}\end{array}\right) \mathbf{X}, \quad \mathbf{X}(0)=\left(\begin{array}{r}{4} \\ {6} \\ {-7}\end{array}\right)$$

Calculate the probability P(E). You roll two dice, one red and one green. Losing combinations are doubles (both dice showing the same number) and outcomes in which the green die shows an odd number and the red die shows an even number. The other combinations are winning ones. E is the event that you roll a winning combination.

Occupants in a single space shuttle in orbit feel weightless. Describe a scheme whereby occupants in a pair of shuttles (or even one shuttle and another massive object in orbit) would be able to use a long cable to continuously experience a comfortably normal Earth-like gravity.

What is the correct present tense form of the verb?
Yo ____ (pensar) What is the correct present subjunctive form of the verb? Yo ____ (tener)

Add. Use models if needed: (2x + 3) + (x + 1)

The Lotka–Volterra predator–prey model assumes that, in the absence of predators, the number of prey grows exponentially. If we make the alternative assumption that the prey population grows logistically, the new system is

$$\begin{array}{l}{x^{\prime}=-a x+b x y} \\ {y^{\prime}=-c x y+\frac{r}{K} y(K-y)}\end{array}$$

, where a, b, c, r, and K are positive and $K>a / b$. (a) Show that the system has critical points at (0, 0), (0, K), and $(\hat{x}, \hat{y})$, where $\hat{y}=a / b$ and $c \hat{x}=\frac{r}{K}(K-\hat{y})$. (b) Show that the critical points at (0, 0) and (0, K) are saddle points, whereas the critical point at $(\hat{x}, \hat{y})$ is either a stable node or a stable spiral point. (c) Show that $(\hat{x}, \hat{y})$ is a stable spiral point if $\hat{y}<\frac{4 b K^{2}}{r+4 b K}$. Explain why this case will occur when the carrying capacity K of the prey is large.

For each pair of terms, explain how the meanings of the terms differ. radial symmetry and bilateral symmetry

Match the vocabulary term in column 1 with the most appropriate phrase in column 2.

$$\begin{matrix} \text{Column 1} & \text{Column 2}\\ \text{Independent linear systems} & \text{A. have many solutions}\\ \quad & \text{B. have no solutions}\\ \quad & \text{C. have solutions that can be shown as the intersection of planes}\\ \quad & \text{D. have the same solutions}\\ \quad & \text{E. have unique solutions}\\ \end{matrix}$$

sketch the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. r=4 sin 2θ

What type of crystalline solid do you predict would best suit the following needs? a. a material that can be melted and reformed at a low temperature b. a material that can be drawn into long, thin wires c. a material that conducts electricity when molten d. an extremely hard material that is nonconductive.

Write each number in scientific notation. 0.00465

Solve the problem using a percent equation. 55 is 20% of what number?

Simplify.

$$9 ^ { 3 }$$

The following data represent the height (inches) of boys between the ages of 2 and 10 years.

$$\begin{matrix} \text{Age, x Boy Height, y} & \text{Age, x Boy Height, y} & \text{Age, x Boy Height, y}\\ \text{2 36.1} & \text{5 45.6} & \text{8 48.3}\\ \text{2 34.2} & \text{5 44.8} & \text{8 50.9}\\ \text{2 31.1} & \text{5 44.6} & \text{9 52.2}\\ \text{3 36.3} & \text{6 49.8 } & \text{9 51.3}\\ \text{3 39.5} & \text{7 43.2} & \text{10 55.6}\\ \text{4 41.5} & \text{7 47.9} & \text{10 59.5}\\ \text{4 38.6} & \text{8 51.4}\\ \end{matrix}$$

Treating age as the explanatory variable, determine the estimates of $\beta_{0}$ and $\beta_{1}$. What is the mean height of a 7-year-old boy?