You complete your assessment of G.S. You note shortness of breath (SOB), fine crackles throughout all lung fields posteriorly and in both lower lobes anteriorly, and coarse crackles over the large airways.

It is time to administer furosemide (Lasix) $40$ mg IV push (IVP). What effect, if any, will furosemide have on G.S.'s breath sounds?

Let $f(x)=\frac{1}{x}, x \neq 0$. The graph y = f(x) is transformed to the graph y = g(x) by a translation of

$$\left(\begin{array}{c}{-2} \\ {0}\end{array}\right)$$

followed by a reflection in the x-axis. (i) Find an expression for g(x). (ii) Write down the equations of the asymptotes of y = g(x). (iii) Find the y-intercept of y = g(x). (iv) Sketch y = g(x) showing the features you have found.

Use rectangles to find lower and upper sums for the areas between the graphs of each of the following functions and the x-axis for $0 \leqslant x \leqslant 1$. Use values of n = 5, 10, 50, 100, 500, 1000, and 10000. Give your answer to 5 decimal places in each case. (i) $y=x^{3}$ (ii) y = x (iii) $y=x^{\frac{1}{2}}$ (iv) $y=x^{\frac{1}{3}}$

For each polynomial function f, (i) Determine the end behavior of f. (ii) Determine the zeros of f. State the multiplicity of each zero. Determine whether the graph of f crosses or only touches the axis at each x-intercept. (iii) Find the x- and y-intercepts of the graph of f. (iv) Use test numbers to find the intervals over which the graph of f is above or below the x-axis. (v) Sketch the graph of y = f(x). $f ( x ) = - x ^ { 3 } ( x - 2 ) ^ { 2 }$

a) Factor each expression. i) $x^{4}-1$ ii) $x^{4}-16$ iii) $x^{5}-1$ iv) $x^{5}-32$ b) Use the results of part a) to predict a pattern for factoring $x^{n}-a^{n}$. c) Use your pattern from part b) to factor $x^{6}-1$. Verify your answer by expanding. d) Factor each expression. i) $x^{4}-625$ ii) $x^{5}-243$

Show that the general solution of the Cauchy–Euler equation (i) in ${x > 0}$ is

$$\begin{gather*} y = c_{1} e^{\lambda_{1}t} + c_{2} e^{\lambda_{2}t} = c_{1}x^{\lambda_{1}} + c_{2}x^{\lambda_{2}} , \end{gather*}$$

if $\lambda_{1}$ and $\lambda_{2}$ are real and $\lambda_{1}$ $\neq$ $\lambda_{1}$,

$$\begin{gather*} y = c_{1} e^{\lambda_{1}t} + c_{2} e^{\lambda_{1}t} = c_{1}x^{\lambda_{1}} + c_{2}x^{\lambda_{1}} \ \ln x, \end{gather*}$$

if $\lambda_{1}$ = $\lambda_{2}$,

$$\begin{align*} y & = e^{\mu t} [c_{1} \cos(vt) + c_{2} \sin(vt)] & = x^\mu [c_{1} \cos(v \ln x) + c2 sin(v \ln x)] \end{align*}$$

if $\lambda_{1}$ = $\mu$ + iv and $\lambda_{2}$ = $\mu$ - iv.

The position at time $t$ of an object moving in the $x y$-plane is given by $\mathbf{r}(t)=\left\langle t^2+3 t, 4 t^2\right\rangle$, where distance is measured in feet and time $t$ in seconds. Find the average velocity of the object over the following time periods. (i) $[2,4]$ (ii) $[2,3]$ (iii) $[2,2.5]$ (iv) $[2,2.1]$

a. Sodium reacts with excess oxygen to form sodium peroxide, $\mathrm{Na}_2 \mathrm{O}_2$.

$$2 \mathrm{Na}+\mathrm{O}_2 \longrightarrow \mathrm{Na}_2 \mathrm{O}_2$$

Calculate the maximum mass of sodium peroxide formed when $4.60 \mathrm{~g}$ of sodium is burnt in excess oxygen.

$\left(A_r\right.$ values: $\mathrm{Na}=23.0, \mathrm{O}=16.0)$

b. Tin $(IV)$ oxide is reduced to tin by carbon. Carbon monoxide is also formed.

$$\mathrm{SnO}_2+2 \mathrm{C} \longrightarrow \mathrm{Sn}+2 \mathrm{CO}$$

Calculate the mass of carbon that exactly reacts with $14.0 \mathrm{~g}$ of tin $(IV)$ oxide. Give your answer to $3$ significant figures.

$(A_r$ values: $\left.\mathrm{C}=12.0, \mathrm{O}=16.0, \mathrm{Sn}=118.7\right)$

Consider the following four titrations (i-iv): i. 150 mL of $0.2\ \mathrm { M }\ \mathrm { NH } _ { 3 } \left( K _ { \mathrm { b } } = 1.8 \times 10 ^ { - 5 } \right) \text { by } 0.2\ \mathrm { M }\ \mathrm { HCl }$ ii. 150 mL of 0.2 M HCl by 0.2 M NaOH iii. 150 mL of 0.2 M HOCl $\left( \mathrm { K } _ { \mathrm { a } } = 3.5 \times 10 ^ { - 8 } \right)$ by 0.2 M NaOH. iv. 150 mL of 0.2 M HF $\left( \mathrm { K } _ { \mathrm { a } } = 7.2 \times 10 ^ { - 4 } \right)$ by 0.2 M NaOH a. Rank the four titrations in order of increasing pH at the halfway point to equivalence (lowest to highest pH). b. Rank the four titrations in order of increasing pH at the equivalence point. c. Which titration requires the largest volume of titrant (HCl or NaOH) to reach the equivalence point?

For each of the following functions: (i) Find and classify all points where $f^{\prime}(x)=0$ (ii) Find and classify all points of inflection (iii) Find intervals where the function is increasing or decreasing (iv) Find intervals where the function is concave up or down (v) Sketch the function showing the features you have found. $f(x)=3 x^{4}+4 x^{3}-2$

The total cost C(x), in hundreds of dollars, to produce x jars of mayonnaise is given by $C(x)=0.000003 x^{3}+4 x+300$. a. Calculate the average cost per jar over the following intervals: i. [100, 100.1] ii. [100, 100.01] iii. [100, 100.001] iv. [100, 100.0001] b. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.

Consider the geometric series $\Sigma_{k=1}^5 27\left(-\frac{1}{3}\right)^{k-1}$ a. Calculate the following sums. i. $S_2$ ii. $S_3$ iii. $S_4$ iv. $S_5$ b. Plot the sums $S_1$ (which is 27), $S_2$, $S_3$, $S_4$, and $S_5$ on a number line. How is $S_n$ related to $S_{n-1}$ and $S_{n-2}$?

Let f(x) = ln x.

(a) Calculate the rate of change of f(x) over the following intervals:

(i) $1 \leq x \leq 2$

(ii) $11 \leq x \leq 12$

(iii) $101 \leq x \leq 102$

(iv) $501 \leq x \leq 502$

(b) Use your answers to guess the value approached by the rate of change of f(x)=ln x over an interval $n \leq x \leq n+1$ as $n$ grows large.

For each function, i) determine equations for the asymptotes ii) determine the y-intercepts iii) sketch a graph iv) describe the increasing and decreasing intervals v) state the domain and range a) $f(x)=\frac{1}{x^{2}+6 x+5}$ b) $g(x)=\frac{1}{x^{2}-5 x-24}$ c) $h(x)=-\frac{1}{x^{2}-6 x+9}$ d) $k(x)=-\frac{2}{x^{2}+5}$