You are working for a company that creates special magnetic environments. Your new supervisor has come from the financial side of the organization rather than the technical side. He has promised a client that the company can provide a device that will create a magnetic field inside a cylindrical chamber that is directed along the cylinder axis at all points in the chamber and increases in the axial direction as the square of the value of y, where y is the in the axial direction and y = 0 is at the bottom end of the cylinder. Prepare a calculation to show that the field requested by your supervisor and promised to a client is impossible.

Find a vector perpendicular to both $i − 3j + 2k$ and $5i − j − 4k.$

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don’t forget the preliminary test. Use the facts stated above when they apply. $\sum_{n=2}^{\infty} \frac{(-1)^{n}}{n^{2}-n}$

How does the kinetic energy of the particle change as it traverses the velocity selector?
A. The kinetic energy increases.
B. The kinetic energy does not change.
C. The kinetic energy decreases.

Write and solve the Euler equations to make the following integrals stationary. $\int_{x_{1}}^{x_{2}} x d s$

Find (A) $f^{\prime}(x),(B)$ the partition numbers for $f^{\prime}$, and (C) the critical numbers of f. f(x)=|x|

In ideal flow, a liquid of density $850 \mathrm{kg} / \mathrm{m}^{3}$ moves from a horizontal tube of radius 1.00 cm into a second horizontal tube of radius 0.500 cm at the same elevation as the first tube. The pressure differs by $\Delta P$ between the liquid in one tube and the liquid in the second tube. (a) Find the volume flow rate as a function of $\Delta P$. Evaluate the volume flow rate for (b) $\Delta P=6.00 \mathrm{kPa}$ and (c) $\Delta P=12.0 \mathrm{kPa}$.

For the given cost function C(x), find the oblique asymptote of the average cost function $\bar{C}(x)$. $C(x)=10,000+90 x+0.02 x^{2}$

Imagine that you are preparing questions for a Spanish-language game show. You are given topics and must produce two logical and gramatically correct questions about each topic: one using the present tense of the verb saber and one using the present tense of the verb conocer. Use the model to help you write your questions. Modelo: el alfabeto ¿Sabes las letras del alfabeto en espanol? ¿Conoces el alfabeto espanol? el libro Don Quijote de la Mancha ¿? ¿?

Young domestic chickens have BID the ability to orient themselves in the earth's magnetic field. Researchers used a set of two coils to adjust the magnetic field in the chicks' pen. as we discussed earlier. shows the two coils, whose centers coincide, seen edge-on. The axis of coil 1 is parallel to the ground and points to the north; the axis of coil 2 is oriented vertically. Each coil has 43 turns and a radius of $1.0 \mathrm{~m}$. At the location of the experiment, the earth's field had a magnitude of $5.6 \times 10^{-5} \mathrm{~T}$ and pointed to the north, tilted up from the horizontal by $61^{\circ}$. If the researchers wished to exactly cancel the earth's field, what currents would they need to apply in coil 1 and in coil 2?

Find the geodesics on a sphere. Hints: Use spherical coordinates with constant $r = a$. Choose your integration variable so that you can write a first integral of the Euler equation. For the second integration, make the change of variable $w=\cot \theta$. To recognize your result as a great circle, find, in terms of spherical coordinates $\theta$ and $\phi$, the equation of intersection of the sphere with a plane through the origin.

Solve each system by using the inverse of the coefficient matrix.

$$\begin{aligned} &x-2 y+3 z=1\\ &y-z+w=-1\\ &-2 x+2 y-2 z+4 w=2\\ &2 y-3 z+w=-3 \end{aligned}$$

Express the following integrals as B functions, and then, in terms of $\Gamma$ functions. When possible, use $\Gamma$ function formulas to write an exact answer in terms of $\pi, \sqrt{2}$, etc. Compare your answers with computer results and reconcile any discrepancies. $\int_{0}^{\pi / 2} \sqrt{\sin ^{3} x \cos x} d x$

A 2.00-L container has a center partition that divides it into two equal parts. The left side contains 0.0440 mol of $\mathrm{H}_{2}$ gas, and the right side contains 0.0440 mol of $\mathrm{O}_{2}$ gas. Both gases are at room temperature and at atmospheric pressure. The partition is removed, and the gases are allowed to mix. What is the entropy increase of the system?