For the following exercises, find the principal square root. $\sqrt[5]{-243}$

Find the following limit or state that they do not exist. Assume a, b, c, and k are real numbers. $\lim _{x \rightarrow 3} \frac{x-3}{|x-3|}$

Find the length of the following polar curve. The complete circle r = a sin $\theta$, where a > 0.

Draw the state graph for the machine, and compute the output sequence for the given input sequence. 0011

$$\begin{array}{lllr} \text{Present state} & \text{Next state} & \text{ } & \text{Output}\\ \text{ } & \text{Present input} & \text{ } & \text{ }\\ \text{ } & \text{0} & \text{1} & \text{ }\\ \hline \text{$s_{0}$} & \text{$s_{2}$} & \text{$s_{3}$} & \text{0}\\ \text{$s_{1}$} & \text{$s_{0}$} & \text{$s_{1}$} & \text{1}\\ \text{$s_{2}$} & \text{$s_{1}$} & \text{$s_{3}$} & \text{0}\\ \text{$s_{3}$} & \text{$s_{1}$} & \text{$s_{2}$} & \text{1}\\ \end{array}$$

A friend argues that the location of a petroleum deposit is a sign that marine organisms once lived there. Is your friend correct? Explain.

Find two different trees that have the same list of nodes under a preorder traversal.

Ultrasound, also known as sonography, is a techno logy that uses high frequency sound waves to produce images. It is used in medical applications to help doctors see inside patients' bodies. How do you think the sound waves can be used to image bones, muscles, and other internal structures?

For the following exercises, factor by grouping. $20 w^{2}-47 w+24$

Consider a function $f\left(x_{1}, \ldots, x_{n}\right)$ of n variables. How many different partial derivatives of order r does f possess?

Which of the following renewable energy resources comes from underground? A geothermal energy, B. hydropower, C. wind power, D. solar power.

A friend randomly chooses two cards, without replacement, from an ordinary deck of 52 playing cards. In each of the following situations, determine the conditional probability that both cards are aces. (a) You ask your friend if one of the cards is the ace of spades, and your friend answers in the affirmative. (b) You ask your friend if the first card selected is an ace, and your friend answers in the affirmative. (c) You ask your friend if the second card selected is an ace, and your friend answers in the affirmative. (d) You ask your friend if either of the cards selected is an ace, and your friend answers in the affirmative.

Ruth runs a 100-meter race. For parts (a)-(c), make a graph that illustrates each situation.

a. Ruth's official time is 14 seconds.

b. Ruth trips and falls at the beginning of the race. She loses five seconds, so her official time is 19 seconds.

c. Ruth starts running two seconds before the starting gun. Nobdoy notices, and her official time is 12 seconds.

d. Fill in the blank. To get from the graph in part (a) to the graph in part (b), you can apply the rule $(x, y) \rightarrow (x+ \_\_\_, y)$.

e. Fill in the blank. To get from the graph in part (a) to the graph in part (c), you can apply the transformation $(x, y) \rightarrow) (x+\_\_\_, y)$.

Does it make sense to look for the irreflexive closure of a relation? Why or why not?

Write three-dimensional (wedge-dashed wedge-line) representations for each of the following: (a) $\mathrm{CH}_{3} \mathrm{Cl}$ (b) $\mathrm{CH}_{2} \mathrm{Cl}_{2}$ (c) $\mathrm{CH}_{2} \mathrm{BrCl}$ (d) $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}$