True or False. The curvature $\kappa$ of a circle equals the radius of the circle.

If $\mathbf{v}=\langle- 2,-1,3\rangle$ and $\mathbf{w}=\langle 5,4,-2\rangle$, find $3 \mathbf{v}-2 \mathbf{w}$.

Why are throttling devices commonly used in refrigeration and air-conditioning applications?

Simplify the expression

$$x^2+2 x-3 x y-5 x-x^2+5 x y+2 y^2$$

What is flow energy? Do fluids at rest possess any flow energy?

A vertical 12-cm diameter piston–cylinder device contains an ideal gas at the ambient conditions of 1 bar and $24^\circ C.$ Initially, the inner face of the piston is 20 cm from the base of the cylinder. Now an external shaft connected to the piston exerts a force corresponding to a boundary work input of 0.1 kJ. The temperature of the gas remains constant during the process. Determine (a ) the amount of heat transfer, (b ) the final pressure in the cylinder, and (c) the distance that the piston is displaced.

A single sided, single layer DVD has a storage capacity of 4.7 gigabytes. One gigabyte is equal to $10^{9}$ bytes. Write a multiplication expression that represents how many bytes can be stored on the DVD.

Describe in words the set of all points (x, y, z) that satisfy the given statements. x=1 and y=0

Find an equation of the tangent line to each curve at the given value $\theta$. $r=2 \cos (3 \theta)$ at $\theta=\frac{\pi}{6}$

A high school principal is concerned about the drop in the percentage of students who graduate from her school, shown in the following table, compared with the fall in national average from 69.2% in 2006 to 68.8% in 2007.

$$\begin{matrix} \text{Year entered school, t} & \text{1995} & \text{1998} & \text{2001} & \text{2004} & \text{2007}\\ \text{Percent graduating, P} & \text{62.4} & \text{54.1} & \text{48.0} & \text{43.5} & \text{41.8}\\ \end{matrix}$$

Calculate the average rate of change of P for each of the three-year intervals between 1995 and 2007.

Consider a simple ideal Rankine cycle with fixed boiler and condenser pressures. What is the effect of super- heating the steam to a higher temperature on

$$\begin{matrix} \text{Pump work input:} & \text{(a) Increase, (b) decrease (c) remains the same }\\ \text{Turbine work output:} & \text{(a) Increase, (b) decrease (c) remains the same}\\ \text{Heat supplied:} & \text{(a) Increase, (b) decrease (c) remains the same}\\ \text{Heat rejected:} & \text{(a) Increase, (b) decrease (c) remains the same}\\ \text{Cycle efficiency:} & \text{(a) Increase, (b) decrease (c) remains the same}\\ \text{Moisture content at turbine exit:} & \text{(a) Increase, (b) decrease (c) remains the same}\\ \end{matrix}$$

A piston–cylinder device initially contains 0.2 kg of steam at 200 kPa and $300^\circ C.$ Now, the steam is cooled at constant pressure until it is at $150^\circ C.$ Determine the volume change of the cylinder during this process using the compressibility factor and compare the result to the actual value.

The table shows the heights of some United States waterfalls. What is the height of each waterfall?

$$\begin{array}{|l|l|}\hline {\text { Waterfall }} & {\text { Height (ft) }} \\ \hline \text { Bridalveil (California) } & {2^{2} \cdot 5 \cdot 31} \\ \hline \text { Fall Creek (Tennessee) } & {2^{8}} \\ \hline \text { Shoshone (Idaho) } & {2^{2} \cdot 53} \\ \hline\end{array}$$

Determine whether each series converges or diverges. If it converges, find its sum. $\sum_{k=1}^{\infty}\left(4^{1 / 3}\right)^{k}$