In the following exercise, find the value(s) of k that makes the function continuous over the given interval.

$$f(x)=\left\{\begin{aligned} \frac{x^{2}+3 x+2}{x+2}, & x \neq-2 \\ k, & x=-2 \end{aligned}\right.$$

For following Exercise find the value of the constant $k$ that makes the function continuous.

$$h(x)=\left\{\begin{array}{ll}{\frac{3 x^{2}+2 x-8}{x+2}} & {\text { if } x \neq-2} \\ {3 x+k} & {\text { if } x=-2}\end{array}\right.$$

What is the difference between a z score and an area under the graph of a normal probability distribution? Can a z score be negative? Can an area be negative?

The state of Maryland has license plates with three numbers followed by three letters.How many different license plates are possible?

Prove that a necessary and sufficient condition for f(z)=u(x, y)+ iv(x, y) to be continuous at $z=z_0=x_0+i y_0$ is that $u(x, y)$ and $v(x, y)$ be continuous at $\left(x_0, y_0\right)$.

There are 24 flute players and 18 trumpet players in the band. Write the ratio of trumpet players to total number of trumpet players and flute players .

Find the value of the constant $k$ that makes the function continuous:

$$\begin{align*} g(x)=\begin{cases} \dfrac{2x^2-5x-25}{x-5},&\text{ if }x\not=5\\ kx-5,&\text{ if }x=5. \end{cases} \end{align*}$$

You are a home health nurse seeing for the first time P.C., a 64 -year-old divorcee diagnosed with small cell lung cancer about 1 year ago. P.C. was treated with radiation and chemotherapy; however, her oncologist recently informed her that her cancer is not curable because it has spread to her bones and liver. The focus of her treatment will now be primarily palliative. She is confused and somewhat bewildered. She vaguely remembers the term palliative treatment from the discussion of her situation with her oncologist but does not know what it means.

P.C. confides that she has not formally written down her wishes concerning the types of treatments she would or would not want. You recommend she complete advance directives, including a living will and a medical durable power of attorney form. How are advance directives formalized?

Write an equivalent expression without negative exponents.

$$(y-1)^{-1}(y-1)^5$$

a. Write the definitions of continuity at a point and continuity on a closed interval.

b. For the piecewise function in Figure 2$7 \mathrm{~d}$ make a table showing these quantities for $1,2,3,4$, and 5 , or stating that the quantity does not exist.

$$\begin{array}{ll}-f(c) & -\lim _{x \rightarrow c} f(x) \\ \text { - } \lim _{x \rightarrow c^{+}} f(x) & -\lim _{x \rightarrow c} f(x)\end{array}$$

• Continuity or kind of discontinuity

$$f(x)= \begin{cases}2+\frac{2}{|x-1|}, & \text { if } x<3 \text { and } x \neq 2 \\ 1, & \text { if } x=2 \\ 3+2 \cos \frac{\pi}{2}(x-3), & \text { if } 3 \leq x<5 \\ 4-3(x-6)^2, & \text { if } x \geq 5\end{cases}$$

c. Sketch graphs of the functions described.

• Has a removable discontinuity at $x=1$
• Has a step discontinuity at $x=2$
• Has a vertical asymptote at $x=3$
• Has a cusp at $x=4$
• Is continuous at $x=5$
• Has a limit as $x \rightarrow 6$ and a value of $f(6)$, but is discontinuous at $x=6$
• Has a left limit of $-2$ and a right limit of 5 as $x \rightarrow 1$

d. For the piecewise function

$$f(x)= \begin{cases}x^2, & \text { if } x \leq 2 \\ x^2-6 x+k_3 & \text { if } x>2\end{cases}$$

• Sketch the graph of $f$ if $k=10$.
• Show that $f$ is discontinuous at $x=2$ if $k=10$.
• Find the value of $k$ that makes $f$ continuous at $x=2$.

Divide $7/24$ by $35/48$ and reduce the quotient to the lowest fraction.

For an $RLC$ series circuit, the voltage amplitude and frequency of the source are $100 \mathrm{~V}$ and $500 \mathrm{~Hz}$, respectively; $R=500 \Omega$; and $L=0.20 \mathrm{H}$. Find the average power dissipated in the resistor for the following values for the capacitance: $(a)$ $C=2.0 \mu \mathrm{F}$ and $(b)$ $C=0.20 \mu \mathrm{F}$.

When the means of a proportion are equal, each mean is called the mean proportional between the two extremes. Find the positive mean proportional between the following extremes. $\frac{4}{5}$ and $\frac{5}{16}$

Two narrow slits $0.070 \mathrm{~mm}$ apart are illuminated by a very bright $488 \mathrm{~nm}$ light source forming an interference pattern on a screen $4.0 \mathrm{~m}$ away. Calculate the distance between the $m=100$ and $m=101$ lines.