Use the notation $\mathbf{r}=\langle x, y\rangle$ and $r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}$. Show that $\nabla(\ln r)=\frac{\mathbf{r}}{r^{2}}$.

What does the kinetic theory say about the motion of atoms?

Identify the causes of human risks for a business.

All other conditions being equal, why is the induced current greater in claypan soil than in topsoil? (a) Claypan soil has a higher concentration of magnetic ions compared to topsoil. (b) Claypan soil is partly metallic in composition. (c) Claypan soil has greater density than loose topsoil. (d) The clay is closely packed, moist, and a better electrical conductor than loose, dry topsoil.

Solve each equation by completing the square. $x^{2}=3 x+4$ _____.

For each pair of terms, explain how the meanings of the terms differ. hunting and poaching

Adenine and guanine are members of a class of molecules known as purines; they have two rings in their structure. Thymine and cytosine, on the other hand, are pyrimidines, and have only one ring in their structure. Predict which have larger dispersi on forces in aqueous solution , the purines or the pyrimidines.

An object moving at constant speed $v$ around a circle of radius $r$ has an acceleration $a$ directed toward the center of the circle. The SI unit of acceleration is $\text{m/s}^2$. (a) Use dimensional analysis to find $a$ as a function of $v$ and $r$. (b) Of the speed is increased 10.0%, by what percentage does the radial acceleration increases?

Sewell is doing a problem regarding the area of pairs of squares. Sewell says that the expression $(x+1)^2$ will be greater than $(x-1)^2$ for all values of x because x + 1 will always be greater than x - 1. Why is he correct when the expressions are areas of squares? Is he correct for any real x outside this model?

$$1+\dfrac{\cos^2 x}{\sin x-1}$$

The following data represent the revenue R (in dollars) received from selling x wristwatches at Wilk’s Watch Shop.

$$\begin{matrix} \text{Wristwatches, x} & \text{Revenue, R}\\ \hline \text{0} & \text{0}\\ \text{25} & \text{2340}\\ \text{40} & \text{3600}\\ \text{50} & \text{4375}\\ \text{90} & \text{6975}\\ \text{130} & \text{8775}\\ \text{160} & \text{9600}\\ \text{200} & \text{$10,000$}\\ \text{220} & \text{9900}\\ \text{250} & \text{9375}\\ \end{matrix}$$

Find the average rate of change of revenue from x = 25 to x = 130 wristwatches.

Four service clubs are each asked to enlist 2 of their members to help with the Fall Review for high school students. The clubs are Alpha Club with 22 members; Beta Club, 19 members; Gamma Club, 25 members; and Zeta Club, 14 members. In how many ways can the helpers be selected?

Relate the symptoms of type 1 diabetes to its cause and the effect on blood glucose regulation.

A millisecond (ms) is a unit of time used to describe how long it takes a computer to complete an operation. 1 second = 1,000 ms. If a computer can execute 35 operations per millisecond, how many operations can it complete in one minute? (60 seconds per minute)