Integrate f over the given region.

$$f ( x , y ) = x ^ { 2 } + y ^ { 2 }$$

over the triangular region with vertices (0, 0), (1, 0), and (0, 1)

Without graphing, decide.

a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point?

b. How many solutions does the system have? See the discussed examples.

$$\left\{\begin{array}{l}x+y=4 \\ x+y=3\end{array}\right.$$

Sketch the region of integration, and write an equivalent double integral with the order of integration reversed.

$$\int_0^2 \int_0^{4-y^2} y d x d y$$

How do the following genomic features of prokaryotic organisms compare with those of eukaryotic organisms? How do they compare among eukaryotes?
a. Genome size
b. Number of genes
c. Gene density (bp/gene)
d. Number of exons

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ be a random sample from a normal distribution. Use the statement to prove that $S^{2}$ is consistent for $\sigma^{2}$.

The volume V of an elastic container for air—like a tire or a balloon—is related to the temperature T and to the pressure P of the air inside the container. The formula $V = k \cdot \frac { T } { P }$ is one way of expressing that relationship. The number k > 0 is a constant depending on the units of measurement for volume, temperature, and pressure. How will volume of a balloon change if: i. pressure remains constant, but temperature increases? ii. pressure remains constant, but temperature decreases? iii. temperature remains constant, but pressure increases? iv. temperature remains constant, but pressure decreases?

Match each expression below to a set of steps I - V.

a. $2\cdot x-2$

b. $2\cdot (x+5)$

c. $2\cdot (x-2)$

d. $5\cdot (x+3)$

e. $3x+5$

I. Choose any number. Subtract 2. Multiply it by 2.

II. Choose any number. Multiply it by 2. Subtract 2.

III. Choose any number. Add 3. Multiply it by 5.

IV. Choose any number. Add 5. Multiply it by 3.

V. Choose any number. Multiply it by 3. Add 5.

Syntenic genes can assort independently. Explain this observation.

Verify the Divergence Theorem for the vector field and region.

$$\mathbf { F } ( x , y , z ) = \langle x , 0,0 \rangle,$$

the region

$$x ^ { 2 } + y ^ { 2 } \leq z \leq 4$$

To make a golden rectangle, you should a. inscribe a rectangle in a circle. b. draw a rectangle so that the ratio of the long side to the short side is the golden ratio. c. draw a rectangle so that the ratio of the diagonal to the short side is the golden ratio.

Sales by quarter for last year and the first three quarters of this year were as follows.

$$\begin{array}{lrrrc} && \text{ Quarter } \\ \hline &\text { I } & \text { II } & \text { III } & \text { IV } \\ \hline \text { Last year } & \$ 23,000 & \$ 27,000 & \$ 18,000 & \$ 9,000 \\ \text { This year } & 19,000 & 24,000 & 15,000 & \\ \hline \end{array}$$

Using a procedure that you develop that captures the change in demand from last year to this year and also the seasonality in demand, forecast expected sales for the fourth quarter of this year.

Use complete sentences to answer the following questions. Why are illegal drugs dangerous to the user?

Described the relationship of the two graphed lines.

Consider electrons accelerated to a total energy of 20.0 GeV in the 3.00-km-long Stanford Linear Accelerator. What is the y factor for the electrons?