A real process may be nearly adiabatic if it occurs over a very short time. How does the short time span help the process to be adiabatic?

Consider the system

$$\mathbf{y}^{\prime}=\left(\begin{array}{ccc} 1 & -1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right) \mathbf{y}$$

. (a) Find the eigenvalues and eigenvectors. Use your numerical solver to sketch the half-lines generated by the exponential solutions $\mathbf{y}(t)=C_{i} e^{\lambda_{i} t} v_{i}$ for i=1, 2, and 3. (b) The six half-line solutions found in part (a) come in pairs that form straight lines. These three lines, taken two at a time, generate three planes. In turn these planes divide phase space into eight octants. Use your numerical solver to add solution trajectories with initial conditions in each of the eight octants. (c) Based on your phase portrait, what would be an appropriate name for the equilibrium point in this case?

Use the Distributive Property to write each expression as an equivalent algebraic expression. $5(x+y)$

Find each quotient. $-105 \div 15$

Solve. Check your solution. $6(m-2)=12$

A prisoner is told that he will be released at a time chosen uniformly at random within the next 24 hours. Let T denote the time that he is released. What is the hazard function for T ? For what values of t is it smallest and largest? If he has been waiting for 5 hours, is it more likely that he will be released in the next few minutes than if he has been waiting for 1 hour?

A coin is dropped from a hot-air balloon that is 300 m above the ground and rising at 10.0 m/s upward. For the coin, find (a) the maximum height reached, (b) its position and velocity 4.00 s after being released, and (c) the time before it hits the ground.

Solve Ax = b, where A and b are given by

$$A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right], \ \mathbf{b}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right]$$

Calculate the final speed of a 110-kg rugby player who is initially running at 8.00 m/s but collides head-on with a padded goalpost and experiences a backward force of $1.76 \times 10^{4}\ \mathrm{N}$ for $5.50 \times 10^{-2}\ \mathrm{s}.$

Let $T: \mathbb{R}^{2} \longrightarrow \mathbb{R}^{2}$ be the transformation that performs a horizontal stretching by a factor of 3, followed by a vertical contraction by a factor of 2, followed by a reflection through the x axis. a. Find the matrix of T relative to the standard basis. b. Apply the transformation to the triangle with vertices (1, 0), (3, 0), and (2, 2), and give a sketch of the result. c. Find the matrix relative to the standard basis that reverses T.

The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer dives from it, starting just before the swimmer steps on the board until just after his feet leave it.

Find the value of the constant k such that the function is a probability density function on the indicated interval. $f(x)=k ;[1,4]$

Suppose a fireworks shell explodes, breaking into three large pieces for which air resistance is negligible. How is the motion of the center of mass affected by the explosion? How would it be affected if the pieces experienced significantly more air resistance than the intact shell?

A 63.0-kg sprinter starts a race with an acceleration of $4.20\ \mathrm{m} / \mathrm{s}^{2}.$ What is the net external force on him?