Calculate $\Delta G^{\circ}$ at 45 degree C for reactions for which

(a) $\Delta H^{\circ}=293 \mathrm{~kJ} ; \Delta S^{\circ}=-695 \mathrm{~J} / \mathrm{K}$

(b) $\Delta H^{\circ}=-1137 \mathrm{~kJ} ; \Delta S^{\circ}=0.496 \mathrm{~kJ} / \mathrm{K}$

(c) $\Delta H^{\circ}=-86.6 \mathrm{~kJ} ; \Delta S^{\circ}=-382 \mathrm{~J} / \mathrm{K}$

Find all prime ideals of $\mathbb{Z} \times \mathbb{Z}$. Find all maximal ideals of $\mathbb{Z} \times \mathbb{Z}$.

An airplane leaves Los Angeles bound for Caracas, Venezuela, flying at an average rate of 500 mph. At the same time, another airplane leaves Caracas bound for Los Angeles, averaging 550 mph. If the airports are 3,675 miles apart, when will the air traffic controllers have to make the pilots aware that the planes are passing each other?

Respond with true or false to each of the following assertions. Be prepared to justify your answer.

If $f$ is differentiable and increasing and if $d x=\Delta x>0$, then $\Delta y>d y$.

Factor by grouping. $5 x^{3}-10 x^{2}+3 x-6$

Show that $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln n}$ diverges.

(b) By considering the grouping of terms

$$\left(\frac{1}{2\ln 2}\right) + \left(\frac{1}{3\ln 3}+\frac{1}{4\ln 4}\right) + \left(\frac{1}{5\ln 5}+\frac{1}{6\ln 6}+\frac{1}{7\ln 7}+\frac{1}{8\ln 8}\right)+\cdots.$$

Find the exact value of the expression.

$$\newline\cos\frac{\pi}{16}\cos\frac{3\pi}{16}-\sin\frac{\pi}{16}\sin\frac{3\pi}{16}$$

Dylan sells his car in his freshman year and puts $7000\textrm{ dollar}$ in an account that earns $5\%$ interest compounded quarterly. The balance in the account after $n$ quarters is

$$A_n=7000\left( 1+\frac{0.05}{4}\right)^n \qquad n=1,2,3,\dots$$

Calculate $A_{12}$. What does $A_{12}$ represent?

Using a phasor diagram, determine $\mathbf{V}$ when $\mathbf{V}=\mathbf{V}_{1}-\mathbf{V}_{2}+\mathbf{V}_{3}^{*}$ and $\mathbf{V}_{1}=3+j 3$, $\mathbf{V}_{2}=4+j 2$, and $\mathbf{V}_{3}=-3-j 2$. (Units are volts.)

Let A={1, 2, 3, 4}, B={3, 4, 5, 6}, and C={5, 6, 7, 8}. Find each of the following sets: a.

$$A \triangle B$$

, b.

$$B \Delta C$$

, c.

$$A \triangle C$$

, d.

$$( A \Delta B ) \Delta C$$

Complete the following x: -4, -2, 0, 2, 4, 8 y: 2, 0, 4, 4, 6, 8 by determining the slopes (if possible) in the slope field at the given points. dy/dx = 2x - y

Identify the slope, $m$, and the $y$-intercept, $b$, for each line. Then graph.

$$x=-3$$

Express the following in phasor form: a. $\sqrt{2}(160) \sin \left(\omega t+30^{\circ}\right)$ b. $\sqrt{2}\left(25 \times 10^{-3}\right) \sin \left(157 t-40^{\circ}\right)$ c. $100 \sin \left(\omega t-90^{\circ}\right)$

How many terms has expression $4 c+7 \frac{1}{2}$?