(a) Investigate $\lim _{x \rightarrow 0} \cos \frac{\pi}{x}$ by using a table and evaluating the function $f(x)=\cos \frac{\pi}{x}$ at $x=-\frac{1}{2},-\frac{1}{4},-\frac{1}{8},-\frac{1}{10},-\frac{1}{12}, \ldots, \frac{1}{12}, \frac{1}{10}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}$. (b) Investigate $\lim _{x \rightarrow 0} \cos \frac{\pi}{x}$ by using a table and evaluating the function $f(x)=\cos \frac{\pi}{x}$ at $x=-1,-\frac{1}{3},-\frac{1}{5},-\frac{1}{7},-\frac{1}{9}, \ldots, \frac{1}{9}, \frac{1}{7}, \frac{1}{5}, \frac{1}{3}, 1$. (c) Compare the results from (a) and (b). What do you conclude about the limit? Why do you think this happens? What is your view about using a table to draw a conclusion about limits? (d) Use technology to graph f. Begin with the x -window $[-2 \pi, 2 \pi]$ and the y-window [-1,1]. If you were finding $\lim _{x \rightarrow 0} f(x)$ using a graph, what would you conclude? Zoom in on the graph. Describe what you see.

Use a geometric series or the nth-Term Test to determine the convergence or divergence of the series. $\sum_{n=0}^{\infty} 9^{-n}$

You will further explore finding deltas graphically. Use a CAS to perform the following steps: a. Plot the function y=f(x) near the point c being approached. b. Guess the value of the limit L and then evaluate the limit symbolically to see if you guessed correctly. c. Using the value $\varepsilon=0.2$, graph the banding lines $y_1=L-\varepsilon$ and $y_2=L+\varepsilon$ together with the function $f$ near $c$. d. From your graph in part (c), estimate a $\delta>0$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$. Test your estimate by plotting $f, y_1$, and $y_2$ over the interval $0<|x-c|<\delta$. For your viewing window use $c-2 \delta \leq$ $x \leq c+2 \delta$ and $L-2 \varepsilon \leq y \leq L+2 \varepsilon$. If any function values lie outside the interval $[L-\varepsilon, L+\varepsilon]$, your choice of $\delta$ was too large. Try again with a smaller estimate. e. Repeat parts (c) and (d) successively for $\varepsilon=0.1,0.05$, and $0.001$. $f(x)=\frac{x^4-81}{x-3}, \quad c=3$

Which of the following is not a closing entry?

Selected accounts for Kebby Photography at December 31, 2018, follow:

$$\begin{array}{ c } \hspace{0pt}\textbf{Dividends} \end{array}\\$$

$$\begin{array} {r|l}\hline \hspace{25pt}\text{}\hspace{25pt} \text{14,000}&\ \text{}{\hspace{70pt}\text{}\hspace{5pt}}\\ \end{array}$$

$$\begin{array}{ c } \hspace{0pt}\textbf{Salaries Expense} \end{array}\\$$

$$\begin{array} {r|l}\hline \hspace{25pt}\text{}\hspace{25pt} \text{31,800}&\ \text{}{\hspace{70pt}\text{}\hspace{5pt}}\\ \hspace{25pt}\text{}\hspace{25pt} \text{1,400}&\ \end{array}$$

$$\begin{array}{ c } \hspace{0pt}\textbf{Depreciation Expense - Building} \end{array}\\$$

$$\begin{array} {r|l}\hline \hspace{25pt}\text{}\hspace{30pt} \text{7,000}&\ \text{}{\hspace{70pt}\text{}\hspace{5pt}}\\ \end{array}$$

$$\begin{array}{ c } \hspace{0pt}\textbf{Depreciation Expense - Furniture} \end{array}\\$$

$$\begin{array} {r|l}\hline \hspace{25pt}\text{}\hspace{30pt} \text{1,500}&\ \text{}{\hspace{70pt}\text{}\hspace{5pt}}\\ \end{array}$$

$$\begin{array}{ c } \hspace{0pt}\textbf{Supplies Expense} \end{array}\\$$

$$\begin{array} {r|l}\hline \hspace{25pt}\text{}\hspace{30pt} \text{2,700}&\ \text{}{\hspace{70pt}\text{}\hspace{5pt}}\\ \end{array}$$

$$\begin{array}{ c } \hspace{0pt}\textbf{Retained Earnings} \end{array}\\$$

$$\begin{array} {r|l}\hline \hspace{5pt}\text{}\hspace{70pt} \text{}& \text{49,000}{\hspace{25pt}\text{}\hspace{25pt}}\\ \end{array}$$

$$\begin{array}{ c } \hspace{0pt}\textbf{Service Revenue} \end{array}\\$$

$$\begin{array} {r|l}\hline \hspace{5pt}\text{}\hspace{70pt} \text{}& \text{33,000}{\hspace{30pt}\text{}\hspace{20pt}}\\ &\text{\hspace{5pt}4,500}{\hspace{30pt}\text{}\hspace{25pt}}\\ \end{array}$$

Requirements

1. Journalize Kebby Photography's closing entries at December 31, 2018.
2. Determine Kebby Photography's ending Retained Earnings balance at December 31, 2018.

The view you have of yourself is known as your..............

Is the opera version of Ile a faithful interpretation of O'Neill's play?

• Prepare a written response to the question, citing specific examples to support your view.

• Identify classmates with opposing viewpoints. Take turns presenting your responses to the class.

• As a class, decide which viewpoint is most compelling.

The trial balances shown below are before and after adjustments for Jack Company at the end of its fiscal year.

Prepare the closing entries for the temporary accounts at August 31.

You will further explore finding deltas graphically. Use a CAS to perform the following steps: a. Plot the function y=f(x) near the point c being approached. b. Guess the value of the limit L and then evaluate the limit symbolically to see if you guessed correctly. c. Using the value $\varepsilon=0.2$, graph the banding lines $y_1=L-\varepsilon$ and $y_2=L+\varepsilon$ together with the function $f$ near $c$. d. From your graph in part (c), estimate a $\delta>0$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$. Test your estimate by plotting $f, y_1$, and $y_2$ over the interval $0<|x-c|<\delta$. For your viewing window use $c-2 \delta \leq$ $x \leq c+2 \delta$ and $L-2 \varepsilon \leq y \leq L+2 \varepsilon$. If any function values lie outside the interval $[L-\varepsilon, L+\varepsilon]$, your choice of $\delta$ was too large. Try again with a smaller estimate. e. Repeat parts (c) and (d) successively for $\varepsilon=0.1,0.05$, and $0.001$. $f(x)=\frac{5 x^3+9 x^2}{2 x^5+3 x^2}, \quad c=0$

Let A be an integral domain. By the closing part of the lesson discussed before , every integral domain can be extended to a "field of quotients." Thus, A[x] can be extended to a field of polynomial quotients, which is denoted by A(x). Note that A(x) consists of all the fractions a(x)/b(x) for a(x) and $b(x) \neq 0$ in A[x], and these fractions are added, subtracted, multiplied, and divided in the customary way.

If A and B are integral domains and $h: A \rightarrow B$ is an isomorphism, prove that h determines an isomorphism.

The data for the following exercise come from a random sample of 40 companies that were part of the S&P 500 and information was current as of May 24, 2013. (Hint: Do not round the mean when calculating the standard deviation.)

Closing stock price (dollars).

$$\begin{array}{rrrr}37.76 & 56.94 & 54.99 & 156.83 \\ 46.04 & 80.58 & 17.35 & 51.33 \\ 28.26 & 55.81 & 49.79 & 45.20 \\ 415.81 & 10.35 & 30.65 & 79.60 \\ 86.21 & 51.62 & 73.83 & 24.87 \\ 34.41 & 47.16 & 61.58 & 95.04 \\ 15.34 & 74.04 & 228.74 & 180.45 \\ 125.45 & 29.19 & 57.31 & 40.06 \\ 125.12 & 13.07 & 45.22 & 53.75 \\ 62.30 & 104.80 & 129.19 & 27.49\end{array}$$

Find the mean and standard deviation of the closing stock price.

Sketch the graph of a function that has the given properties. Identify any critical points, singular points, local maxima and minima, and inflection points. Assume that f is continuous and its derivatives exist everywhere unless the contrary is implied or explicitly stated. $f(-1)=0, f(0)=2, f(1)=1, f(2)=0, f(3)=1$, $\lim _{x \rightarrow \pm \infty}(f(x)+1-x)=0, f^{\prime}(x)>0 \text { on }(-\infty,-1)$, $(-1,0) \text { and }(2, \infty), f^{\prime}(x)<0 \text { on }(0,2)$, $\lim _{x \rightarrow-1} f^{\prime}(x)=\infty, f^{\prime \prime}(x)>0 \text { on }(-\infty,-1)$ and on (1, 3), and f"(x) < 0 on (-1, 1) and on $(3, \infty)$.

Work individually. Read each passage in the chart, and describe its pacing. Make notes about the characteristics of the passage that contribute to its pacing. Discuss your responses with your group. During your discussion, support your ideas by including examples from other parts of the story, as well. One example has been done for you.

The switch has been open a long time before closing at t=0.

a) Find $i_o\left(0^{-}\right)$.

b) Find $i_2\left(0^{-}\right)$

c) Find $i_o\left(0^{+}\right)$

d) Find $i_2\left(0^{+}\right)$.

e) Find $i_o(\infty)$.

f) Find $i_L(\infty)$.

g) Write the expression for $i_L(t)$ for $t \geq 0$.

h) Find $v_L\left(0^{-}\right)$.

i) Find $v_L\left(0^{+}\right)$.

j) Find $v_L(\infty)$.

k) Write the expression for $v_L(t)$ for $t \geq 0^{+}$.

l) Write the expression for $i_o(t)$ for $t \geq 0^{+}$.