Is it possible for a $2 \times 3$ matrix to equal a $3 \times 2$ matrix? Explain.

Sketch the graph of the equation using extrema, intercepts, symmetry, and asymptotes. y = 1 - 1/x

solve the logarithmic equation algebraically. Approximate the result to three decimal places. ln√x+8 = 3

Solve the given initial-value problem.

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

Find T(t) and N(t) at the given point. r(t)=

$$\frac { 1 } { 2 } t ^ { 2 } \mathbf { i } + \frac { 1 } { 3 } t ^ { 3 } \mathbf { j }$$

t=1

Meredith wants to go on a trip this summer that costs $375. She paid a deposit of$125 and will save an additional $25 per week to pay for the trip. The equation 125 + 25w = 375 can be used to find the number of weeks Meredith will need to save. Describe the steps needed to solve the equation.

Neglecting air resistance and the weight of the propellant, determine the work done in propelling a five-ton satellite to a height of (a) 100 miles above Earth and (b) 300 miles above Earth.

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.

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