One or more initial conditions are given for the differential equation in the this exercise. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant solutions on the ty-graph whether they are mentioned or not.

The graph of $z=-\frac{1}{2} y \ln (y / 30)$ has relative maximum point at $y \approx 11.0364$ and y-intercept at y-30; y(0)-1, y(0)=20, and y(0)=40. Sketch the solutions to the Gompertz growth equation

$$\frac{d y}{d t}=-\frac{1}{2} y \ln \frac{y}{30} .$$

Use these data to draw a qualitative phase diagram for $\mathrm{H}_2$ Does $\mathrm{H}_2$ sublime at $0.05 \mathrm{~atm}$? Explain.\

$$\begin{array}{ll}\mathrm{mp} \text { at } 1 \mathrm{~atm}: & 13.96 \mathrm{~K} \\ \mathrm{bp} \text { at } 1 \mathrm{~atm}: & 20.39 \mathrm{~K} \\ \text { Triple point: } & 13.95 \mathrm{~K} \text { and } 0.07 \mathrm{~atm} \\ \text { Critical point: } & 33.2 \mathrm{~K} \text { and } 13.0 \mathrm{~atm} \\ \text { Vapor pressure of solid at } 10 \mathrm{~K}: & 0.001 \mathrm{~atm}\end{array}$$

How do the three subtypes of juvenile rheumatoid arthritis differ?

The amino acid glycine $$ \left( \mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } \right) $$ can participate in the following equilibria in water: $$ \begin{array} { c } { \mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } + \mathrm { H } _ { 2 } \mathrm { O } \rightleftharpoons } \\ { \mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COO } ^ { - } + \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \quad K _ { \mathrm { a } } = 4.3 \times 10 ^ { - 3 } } \end{array} $$ $$ \begin{array} { l } { \mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } + \mathrm { H } _ { 2 } \mathrm { O } \rightleftharpoons } \\ { \quad ^+ \mathrm { H } _ { 3 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } + \mathrm { OH } ^ { - } \quad K _ { \mathrm { b } } = 6.0 \times 10 ^ { - 5 } } \end{array} $$ (a) Use the values of $$ K_a $$ and $$ K_b $$ to estimate the equilibrium constant for the intramolecular proton transfer to form a zwitterion: $$ \mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } \rightleftharpoons ^ { + } \mathrm { H } _ { 3 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COO } ^ { - } $$ (b) What is the pH of a 0.050M aqueous solution of glycine? (c) What would be the predominant form of glycine in a solution with pH 13? With pH 1?

Belltown Athletic Supply (BAS) makes game jerseys for athletic teams. The F. C. Kitsap soccer club has offered to buy 100 jerseys for the teams in its league for $15 per jersey. The team price for such jerseys normally is$18, an 80% markup over BAS’s purchase price of $10 per jersey. BAS adds a name and number to each jersey at a variable cost of$2 per jersey. The annual fixed cost of equipment used in the printing process is $6,000, and other fixed costs allocated to jerseys are$2,000. BAS makes about 2,000 jerseys per year, so the fixed cost is $4 per jersey. The equipment is used only for printing jerseys and stands idle 75% of the usable time.

The manager of BAS turned down the offer, saying, “If we sell at$15 and our cost is $16, we lose money on each jersey we sell. We would like to help your league, but we can’t afford to lose money on the sale.”

1. Compute the amount by which the operating income of BAS would change if it accepted F. C. Kitsap’s offer.
2. Suppose you were the manager of BAS. Would you accept the offer? In addition to considering the quantitative impact computed in requirement 1, list two qualitative considerations that would influence your decision—one qualitative factor supporting acceptance of the offer and one supporting rejection.

How do you know that Yahweh is a main character in the play?

(1) He is fully developed.

(2) He reveals many traits.

(3) He changes during the story.

(4) He is central to the story.

Modify methionine to show its zwitterion form.

The amino acid glycine $\left(\mathrm{H}_2 \mathrm{~N}-\mathrm{CH}_2-\mathrm{COOH}\right)$ can participate in the following equilibria in water:

$$\begin{aligned} & \mathrm{H}_2 \mathrm{~N}-\mathrm{CH}_2-\mathrm{COOH}+\mathrm{H}_2 \mathrm{O} \rightleftharpoons \\ & \mathrm{H}_2 \mathrm{~N}-\mathrm{CH}_2-\mathrm{COO}^{-}+\mathrm{H}_3 \mathrm{O}^{+} \quad \mathrm{K}_{\mathrm{q}}=4.3 \times 10^{-3} \\ & \end{aligned}$$

Use the values of $K_a$ and $K_b$ to estimate the equilibrium constant for the intramolecular proton transfer to form a zwitterion:

$$\mathrm{H}_2 \mathrm{~N}-\mathrm{CH}_2-\mathrm{COOH} \rightleftharpoons{ }^{+} \mathrm{H}_3 \mathrm{~N}-\mathrm{CH}_2-\mathrm{COO}^{-}$$

What assumptions did you need to make?

The amino acid glycine $\left( \mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } \right)$ can participate in the following equilibria in water: $\mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } + \mathrm { H } _ { 2 } \mathrm { O } \rightleftharpoons \mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COO } ^ { - } + \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \quad K _ { \mathrm { a } } = 4.3 \times 10 ^ { - 3 }$ $\mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } + \mathrm { H } _ { 2 } \mathrm { O } \rightleftharpoons ^+ \mathrm { H } _ { 3 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } + \mathrm { OH } ^ { - } \quad K _ { \mathrm { b } } = 6.0 \times 10 ^ { - 5 }$ Use the values of $K_a$ and $K_b$ to estimate the equilibrium constant for the intramolecular proton transfer to form a zwitterion: $\mathrm { H } _ { 2 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COOH } \rightleftharpoons ^ { + } \mathrm { H } _ { 3 } \mathrm { N } - \mathrm { CH } _ { 2 } - \mathrm { COO } ^ { - }$

Phosgene was used as a poisonous gas in World War I. At high temperatures it decomposes as follows:

$$\mathrm{COCl}_2(\mathrm{~g}) \rightleftharpoons \mathrm{CO}(\mathrm{g})+\mathrm{Cl}_2(\mathrm{~g})$$

with $K_c=4.6 \times 10^{-3}$ at $800 \mathrm{~K}$. a. A sample of $7.80 \mathrm{~g}$ of $\mathrm{COCl}_2$ is placed in a $1.00-\mathrm{L}$ reaction vessel and heated to $800 \mathrm{~K}$. What are the equilibrium concentrations of all of the species? b. What fraction of $\mathrm{COCl}_2$ has decomposed? c. If $3 \mathrm{~g}$ of carbon monoxide is inserted into the reaction vessel, what qualitative effect would this have on the fraction of $\mathrm{COCl}_2$ that has decomposed?

Use your calculator to write the following in the form $e^{k}$ where k is correct to 4 decimal places: 0.006

Find the area of each triangle. Round answers to two decimal places. $A=70^{\circ}, B=60^{\circ}, c=4$

Use your calculator to write the following in the form $e^{k}$ where k is correct to 4 decimal places: 1500

Use a calculator to evaluate each expression. Express angles in radians rounded to two decimal places. $\sec ^{-1} 1.4$