Consider a 0.60-M solution of $\mathrm { HC } _ { 3 } \mathrm { H } _ { \mathrm { 5 } } \mathrm { O } _ { 3 } ,$ lactic acid $\left( K _ { a } = 1.4 \times 10 ^ { - 4 } \right).$ a. Which of the following are major species in the solution? i. $\mathrm { HC } _ { 3 } \mathrm { H } _ { 5 } \mathrm { O } _ { 3 },$ ii. $\mathrm { C } _ { 3 } \mathrm { H } _ { 5 } \mathrm { O } _ { 3 } ^ { - },$ iii. $\mathrm { H } ^ { + },$ iv. $\mathrm { H } _ { 2 } \mathrm { O },$ v. $\mathrm { OH } ^ { - }.$ b. Complete the following ICE table in terms of x, the amount (mol/L) of lactic acid that dissociates to reach equilibrium.

$$\begin{matrix} & \text{$\left[ \mathrm { HC } _ { 3 } \mathrm { H } _ { 5 } \mathrm { O } _ { 3 } \right]$} & \text{$\left[ \mathrm { H } ^ { + } \right]$} & \text{$\left[ \mathrm { C } _ { 3 } \mathrm { H } _ { 5 } \mathrm { O } _ { 3 } ^{-} \right]$}\\ \hline \text{Initial} & \text{\_\_\_\_\_\_\_\_\_\_\_\_} & \text{\_\_\_\_\_\_\_\_\_\_\_\_} & \text{\_\_\_\_\_\_\_\_\_\_\_\_}\\ \text{Change} & \text{\_\_\_\_\_\_\_\_\_\_\_\_} & \text{\_\_\_\_\_\_\_\_\_\_\_\_} & \text{\_\_\_\_\_\_\_\_\_\_\_\_}\\ \text{Equilibrium} & \text{$0.60 - x$} & \text{\_\_\_\_\_\_\_\_\_\_\_\_} & \text{\_\_\_\_\_\_\_\_\_\_\_\_}\\ \end{matrix}$$

c. What is the equilibrium concentration for $\mathrm { C } _ { 3 } \mathrm { H } _ { 5 } \mathrm { O } _ { 3 } ^ { - } ?$ d. Calculate the pH of the solution.

It is midmorning on the cardiac unit where you work, and you are getting a new patient. G.P. is a 60-yearold retired businessman who is married and has 3 grown children. As you take his health history, he tells you that he began feeling changes in his chest about 10 days ago. He has hypertension (HTN) and a 3-year history of angina pectoris. During the past week, he has had frequent episodes of mid-chest discomfort. The chest pain responds to nitroglycerin (NTG), which he has taken sublingually about 8 to 10 times over the past week. During the week, he has also experienced increased fatigue. He states, "I just feel crappy all the time." A cardiac catheterization done several years ago revealed 50% stenosis of the right coronary artery and 50% stenosis of the left anterior descending coronary artery. He tells you that both his mother and his father had coronary artery disease (CAD). He is currently taking amlodipine (Norvasc), metoprolol (Lopressor), atorvastatin (Lipitor), and aspirin $81 \mathrm{mg} /$ day. He is retired and says that he spends his days watching television, with some occasional yard work. He has gained $25 \mathrm{lb}(11.3 \mathrm{~kg})$ since retiring and admits that he is overweight. When you first admit G.P., you place him on telemetry and observe his cardiac rhythm. Within the hour, G.P. converts with intravenous diltiazem (Cardizem) to sick sinus syndrome with long sinus pauses that cause lightheadedness and hypotension. Because G.P.'s dysrhythmia is causing unacceptable symptoms, he is taken to surgery and a permanent DDDR pacemaker is placed and set at a rate of 70. What is the purpose of DDDR pacing?

In this problem you will use your program from last problem to solve the differential equation

$$\frac{d y}{d x}=-\frac{x}{2 y},$$

from first problem , with different values of $\Delta x$, and you'll compare your results with the exact solution.

a) The curve in Quadrant IV of Figure shows the exact particular solution of the differential equation that contains the point (0,-4). Solve the differential equation algebraically. Use your result to calculate y when x=5 and show that this point is on the graph.

b) Use your program from last problem to calculate values of y by Euler's method, starting at the point (0,-4), using $\Delta x=0.5$. Record in a table the y-values for x=0,1,2,3,4,5,6, and 7 .

c) On a photocopy of given figure , plot the values from the table you made in part b) Write some comments on how well Euler's method fits the exact solution.

d) Repeat part b and part c, using $\Delta x=0.1$. Again, record only the calculated y-values for x=0,1,2,3,4,5,6, and 7. Comment on the relative accuracy of Euler's method as smaller values of $\Delta x$ are used)

e) Figure out a time-efficient way to calculate y when x=5, using Euler's method with $\Delta x=0.01$. How closely does this value match the exact value you calculated in part a?

You are the 1900- 0700 charge nurse in the intermediate cardiac care unit in a large hospital. One of the patients, R.J., is being cared for by a new graduate nurse under your supervision. R.J. was admitted at 1300 after an auto accident in which he sustained a chest contusion and fractures of the fourth and fifth ribs on his left side. At about 2000 hours, his wife runs up to you at the nurses' station and says, "I think my husband just had a heart attack. Come quick!" She follows you into his room, where you find him still in bed. He is breathing and is cyanotic from the neck up. His pulse is rapid but very weak.

The code team arrives. RJ.'s trauma surgeon is making rounds on your unit when the code is called, and he runs into the room. R.J. is intubated, and the normal saline lock is changed to an IV of lactated Ringer's solution at "wide open." The trauma surgeon recognizes Beck's triad associated with cardiac tamponade and calls for a cardiac needle and syringe. He inserts the needle below the xiphoid process and aspirates 75 mL of unclotted blood.

What is the most likely reason R.J. developed cardiac tamponade?

M.G., a "frequent flier," is admitted to the emergency department (ED) with a diagnosis of heart failure (HF). She was discharged from the hospital 10 days ago and comes in today stating, "I just had to come to the hospital today because I can't catch my breath and my legs are as big as tree trunks." After further questioning, you learn she is strictly following the fluid and salt restriction ordered during her last hospital admission. She reports gaining 1 to 2 pounds (0.5 to 1 kg) every day since her discharge.During the admission interview, the nurse makes a list of the medications M.G. took at home.

Nursing Assessment: Medications Taken at Home

$$\begin{array}{ll}\text { Enalapril (Vasotec) } & 5 \mathrm{mg} \text { PO bid } \\ \text { Pioglitazone (Actos) } & 45 \mathrm{mg} \text { PO every morning } \\ \text { Furosemide (Lasix) } & 40 \mathrm{mg} / \text { day PO } \\ \text { Potassium chloride (K-Dur) } & 20 \mathrm{mEq} / \text { day PO }\end{array}$$

After reviewing M.G.'s medications, the cardiologist writes the following medication orders.

Medication Orders

$$\begin{array}{ll}\text { Enalapril (Vasotec) } & 5 \mathrm{mg} \text { PO bid } \\ \text { Carvedilol (Coreg) } & 3.125 \mathrm{mg} \text { PO twice daily } \\ \text { Metformin (Glucophage) } & 500 \mathrm{mg} \text { twice daily } \\ \text { Furosemide (Lasix) } & 80 \mathrm{mg} \text { intravenous push (IVP) now, then } 40 \mathrm{mg} / \text { day IVP } \\ \text { Potassium chloride (K-Dur) } & 20 \mathrm{mEq} / \text { day PO }\end{array}$$

What lab tests should be ordered for M.G. related to the order for furosemide (Lasix)? Select all that apply.

a. Sodium level

b. Potassium level

c. Magnesium level

d. Coagulation studies

e. Serum glucose level

f. Complete blood count

You are the 1900- 0700 charge nurse in the intermediate cardiac care unit in a large hospital. One of the patients, R.J., is being cared for by a new graduate nurse under your supervision. R.J. was admitted at 1300 after an auto accident in which he sustained a chest contusion and fractures of the fourth and fifth ribs on his left side. At about 2000 hours, his wife runs up to you at the nurses' station and says, "I think my husband just had a heart attack. Come quick!" She follows you into his room, where you find him still in bed. He is breathing and is cyanotic from the neck up. His pulse is rapid but very weak.

The code team arrives. RJ.'s trauma surgeon is making rounds on your unit when the code is called, and he runs into the room. R.J. is intubated, and the normal saline lock is changed to an IV of lactated Ringer's solution at "wide open." The trauma surgeon recognizes Beck's triad associated with cardiac tamponade and calls for a cardiac needle and syringe. He inserts the needle below the xiphoid process and aspirates 75 mL of unclotted blood.

Describe how you titrate a dopamine infusion.

The following tabled values show the height above the ground of a point on a bicycle wheel as the bicycle is wheeled along a flat surface.

$$\begin{matrix} \text{Distance travelled (cm)} & \text{0} & \text{20} & \text{40} & \text{60} & \text{80} & \text{100} & \text{120} & \text{140} & \text{160} & \text{180} & \text{200}\\ \text{Height above ground (cm)} & \text{0} & \text{6} & \text{23} & \text{42} & \text{57} & \text{64} & \text{59} & \text{43} & \text{23} & \text{7} & \text{1}\\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ \text{Distance travelled (cm)} & \text{220} & \text{240} & \text{260} & \text{280} & \text{300} & \text{320} & \text{340} & \text{360} & \text{380} & \text{400}\\ \text{Height above ground (cm)} & \text{5} & \text{27} & \text{40} & \text{55} & \text{63} & \text{60} & \text{44} & \text{24} & \text{9} & \text{3}\\\end{matrix}$$

Is the data periodic? If so, estimate: (i) the equation of the principal axis (ii) the maximum value (iii) the period (iv) the amplitude.

J.E. is a 34 -year-old Filipino man who presents to the emergency department reporting shortness of breath, dizziness, and a rapid fluttering in his chest. He states that he occasionally has had the same symptoms but they usually pass if he sits or lies down to rest. But this time resting has not helped the symptoms to subside. J.E. has a past medical history of appendectomy and rheumatic fever as a child. His current vital signs are BP $164 / 78, \mathrm{P} 160, \mathrm{R} 28, \mathrm{~T} 98.1^{\circ} \mathrm{F}$ (36.7 $\left.{ }^{\circ} \mathrm{C}\right)$. His oxygen saturation is 94% on room air. J.E. is referred to a cardiothoracic surgeon to decide the best course of action for treating his mitral valve stenosis. At this time, he is still experiencing atrial fibrillation, with a heart rate in the 160 s.

Which drugs do you expect to be started at this time? Explain your rationale.

a. Amiodarone infusion and a heparin infusion

b. IV push metoprolol and enoxaparin (Lovenox) subcutaneous injections

c. A one-time oral dose of diltiazem (Cardizem) and subcutaneous heparin

d. Lidocaine bolus, followed by an infusion, and subcutaneous heparin injections

It is midmorning on the cardiac unit where you work, and you are getting a new patient. G.P. is a 60-yearold retired businessman who is married and has 3 grown children. As you take his health history, he tells you that he began feeling changes in his chest about 10 days ago. He has hypertension (HTN) and a 3-year history of angina pectoris. During the past week, he has had frequent episodes of mid-chest discomfort. The chest pain responds to nitroglycerin (NTG), which he has taken sublingually about 8 to 10 times over the past week. During the week, he has also experienced increased fatigue. He states, "I just feel crappy all the time." A cardiac catheterization done several years ago revealed 50% stenosis of the right coronary artery and 50% stenosis of the left anterior descending coronary artery. He tells you that both his mother and his father had coronary artery disease (CAD). He is currently taking amlodipine (Norvasc), metoprolol (Lopressor), atorvastatin (Lipitor), and aspirin $81 \mathrm{mg} /$ day. He is retired and says that he spends his days watching television, with some occasional yard work. He has gained $25 \mathrm{lb}(11.3 \mathrm{~kg})$ since retiring and admits that he is overweight. When you first admit G.P., you place him on telemetry and observe his cardiac rhythm. Within the hour, G.P. converts with intravenous diltiazem (Cardizem) to sick sinus syndrome with long sinus pauses that cause lightheadedness and hypotension.

What risks does the new rhythm pose for G.P.? Explain the reasons for your answers.

We can obtain formulas involving half-angles from the double-angle formulas. To find an expression for $\sin \frac{1}{2} v$, we solve $\cos 2 u=1-2 \sin ^2 u$ for $\sin ^2 u$ to obtain

$$\sin ^2 u=\frac{1-\cos 2 u}{2} .$$

Taking the square root gives:

$$\sin u=\pm \sqrt{\frac{1-\cos 2 u}{2}}$$

and finally we make the substitution $u=\frac{1}{2} v$ to get:

$$\sin \frac{1}{2} v=\pm \sqrt{\frac{1-\cos v}{2}} .$$

(a) Show that $\cos \frac{1}{2} v=\pm \sqrt{\frac{1+\cos v}{2}}$.

(b) Show that $\tan \frac{1}{2} v=\pm \sqrt{\frac{1-\cos v}{1+\cos v}}$. The sign in these formulas depends on the value of $v / 2$. If $v / 2$ is in quadrant I then the sign of $\sin v / 2$ is $+$, the sign of $\cos v / 2$ is $+$, and the $\operatorname{sign}$ of $\tan v / 2$ is $+$.

(c) If $v / 2$ is in quadrant II find the signs for $\sin v / 2$, for $\cos v / 2$, and for $\tan v / 2$.

(d) If $v / 2$ is in quadrant III find the signs for $\sin v / 2$, for $\cos v / 2$, and for $\tan v / 2$.

(e) If $v / 2$ is in quadrant IV find the $\operatorname{signs}$ for $\sin v / 2$, for $\cos v / 2$, and for $\tan v / 2$

Complete the given requirements.

a. State the chain rule in each form.

i. $d y / d x$

ii. $f(x)$

iii. Ustng the outside and inside functions

b. Prove the chain rule. What assumption must you make about the quantity $\Delta u$ so that the proof is valid?

c. Differentiate $f(x)=\left(x^2-4\right)^3$ in the two ways described and show that the two answers are equivalent.

i. Use the chain rule.

ii. Expand the binomial power first.

d. Differentiate.

i. $f(x)=\cos \left(x^3\right) \quad$

ii. $g(x)=\sin 5 x$

iii. $h(x)=\cos ^6 x \quad$

iv. $k(x)=\sin 3$

e. If $f^{\prime}(x)=12 \cos 3 x$, find $f^{\prime \prime}(x)$ and $f(x)$. What special name is given to $f^{\prime \prime}(x)$ ? What special name is given to $f(x)$ in relation to $f^{\prime}(x)$ ?

f. Shark Problem: The welght of a great white shark, $W$, in pounds, is given approximately by $W=0.6 x^3$, where $x$ is the length of the shark in feet. Suppose that a baby shark is growing at the rate of $d x / d t=0.4 \mathrm{ft} /$ day. How heavy is the shark when it is $2 \mathrm{ft}$ long? $10 \mathrm{ft}$ long? At what rate is it gaining weight (pounds per day) when it is these lengths? Explain how the chain rule allows you to calculate these answers.

The formula that determines the perimeter of a square p (in inches) in terms of the length of the side of a square s (in inches) is p = 4s. a. Deflne a function f that determines the perimeter of a square f(s) in terms of the squares side length (or when the square's side length s is known). b. Write a formula that determines a square's side length (in inches) in terms of the square's perimeter (in inches) (or when the square's perimeter is known). c. Define a function h that determines a square's side length h(p) in terms of the square's perimeter. (or when the square's perimeter is known). d. Use function composition to evaluate i) h(f(2.5)), ii) f(h(10)), iii) h(f(7)), iv) f(h(28)). e. How are the functions, f and h related? How are the input (independent) and output (dependent) quantities of f and g related? f. A function g that undoes or reverses the function process or rule of another function k is said to be the inverse of the function k. We express this symbolically by writing $g=k^{-1}$ and say that the Function k and the function $k^{-1}$ (or g) are inverses of one another. i) T or F: In part e above $f=h^(-1)$ ii) T or F: in part e above $h = f^1$.

A dietician prepare s a diet specifying the amounts a patient should eat of four basic food groups: group I, meats; group II, fruits and vegetables; group III, breads and starches; group IV, milk products. Amounts are given in "exchanges" that represent 1 oz (meat), 1/2 cup (fruits and vegetables), 1 slice (bread), 8 oz (milk) , or other suitable measurements. (a) The number of "exchanges" for breakfast for each of the four food groups, respectively, are 2, 1, 2, and 1; for lunch, 3, 2, 2, and 1; and for dinner, 4, 3, 2, and 1. Write a $3 \times 4$ matrix using this information. (b) The amounts of fat, carbohydrates, and protein (in appropriate units) in each food group, respectively, are as follows. Fat: 5, 0, 0, 10; Carbohydrates: 0, 10, 15, 12; Protein: 7, 1, 2, 8. Use this information to write a $4 \times 3$ matrix. (c) There are 8 calories per exchange of fat, 4 calories per exchange of carbohydrates, and 5 calories per exchange of protein. Summarize these data in a $3 \times 1$ matrix.

The base of a 20-meter tower is at the origin; the base of a 20-meter tree is at (0, 20, 0). The ground is flat and the z axis points upward. The following parametric equations describe the motion of six projectiles each launched at time t=0 in seconds.

(I) $\vec{r}(t)=\left(20+t^2\right) \vec{k}$

(II) $\vec{r}(t)=2 t^2 \vec{j}+2 t^2 \vec{k}$

(III) $\vec{r}(t)=20 \vec{i}+20 \vec{j}+\left(20-t^2\right) \vec{k}$

(IV) $\vec{r}(t)=2 t \vec{j}+\left(20-t^2\right) \vec{k}$

(V) $\vec{r}(t)=(20-2 t) \vec{i}+2 t \vec{j}+(20-t) \vec{k}$

(VI) $\vec{r}(t)=t \vec{i}+t \vec{j}+t \vec{k}$

Which projectile is not launched from somewhere on the tower and hits the tree? Where and when does it hit the tree?