Graph and determine the magnitudes of the following complex numbers. See the earlier example.

3+5i

For this exercise, determine the first five terms of the given sequence below.

$$a_n=n^2-2 n$$

A CFD model of steady two-dimensional incompressible flow has printed out the values of velocity potential $\phi(x, y)$, in $\mathrm{m}^2 / \mathrm{s}$, at each of the four corners of a small $10$-cm-by$10$-$\mathrm{cm}$ cell, as shown in Fig. P4.73. Use these numbers to compute the resultant velocity in the center of the cell and its angle $\alpha$ with respect to the $x$ axis.

For the following melting points, indicate what might be concluded regarding the purity of the sample:

a. $120-122^{\circ} \mathrm{C}$
b. $46-60^{\circ} \mathrm{C}$
c. $147^{\circ} \mathrm{C}(\mathrm{dec})$
d. $162.5-163.5^{\circ} \mathrm{C}$

Find the first five terms of the sequence.

$$a_1=-3, a_{n+1}=a_n+n^2$$

For many travelers, the cost of getting to an airport needs to be factored into the travel expense. The table at the right lists the cost of six different ways to get from the center of a major city to the nearest airport. A researcher polled travelers to determine the number of people who paid each price on a given day. Variables a, b, c, d, e, and f represent the frequencies of the costs.

$$\begin{array}{c|c} \text { Cost to Airport } & \text { Frequency } \\ \hline \$ 25 & \mathrm{a} \\ \$ 38 & \mathrm{b} \\ \$ 42 & \mathrm{c} \\ \$ 60 & \mathrm{d} \\ \$ 65 & \mathrm{e} \\ \$ 70 & \mathrm{f} \end{array}$$

Write an algebraic expression for the percentile rank of $70.

As the moon revolves around the earth, the side that faces the earth is usually just partially illuminated by the sun. The phases of the moon describe how much of the surface appears to be in sunlight. An astronomical measure of phase is given by the fraction F of the lunar disc that is lit. When the angle between the sun, earth, and moon is

$$\theta \left( 0 \leq \theta \leq 360 ^ { \circ } \right)$$

, then

$$F = \frac { 1 } { 2 } ( 1 - \cos \theta )$$

Determine the angles

$$\theta$$

that correspond to the following phases: (a) F = 0 (new moon) (b) F = 0.25 (a crescent moon) (c) F = 0.5 (first or last quarter) (d) F = 1 (full moon)

What is the basis for the separation of proteins by the following techniques?

(a) gel-filtration chromatography

(b) affinity chromatography

(c) ion-exchange chromatography

(d) reverse phase HPLC

Find a potential function f for the field

$$\mathbf { F }.$$

$$\mathbf { F } = ( y \sin z ) \mathbf { i } + ( x \sin z ) \mathbf { j } + ( x y \cos z ) \mathbf { k }$$

Use the figure below to answer. $\triangle \mathrm{A} B C \equiv \triangle D E F$.

$$\overline{EF} \cong\blacksquare$$

Simplify the complex fraction.

$$\frac{\frac{7}{x+h}-\frac{7}{x}}{h}$$

Graph and determine the magnitudes of the following complex numbers. See the earlier example.

-1+3i

"Simplify $5 \log_2 k − 8 \log_2 m + 10 \log_2 n$.

$A$. $7 \log_2 (k − m + n)$
$B$. $7 \log_2 kn$ over $m$
$C$. $\log_2 50 kn$ over $8 m$
$D.$ $\log_2 k$ to the fifth power $n$ to the tenth power/ $m$ to the eighth power"

What point charges, all having the same magnitude, would you place at the corners of a square (one charge per comer), so that both the electric field and the electric potential (assuming a zero reference value at infinity) are zero at the center of the square? Account for the fact that the charge distribution gives rise to both a zero field and a zero potential.