# 4.5.1: Applications to Curves (Exercises)

## Q4.5.1

In Exercises 4.5.1-4.5.8 find a first order differential equation for the given family of curves.

1. (y(x^2+y^2)=c)

2. (e^{xy}=cy)

3. (ln |xy|=c(x^2+y^2))

4. (y=x^{1/2}+cx)

5. (y=e^{x^2}+ce^{-x^2})

6. ({y=x^3+{cover x}})

7. (y=sin x+ce^x)

8. (y=e^x+c(1+x^2))

## Q4.5.2

9. Show that the family of circles [(x-x_0)^2+y^2=1,;-infty

[(x-x_0)^2+y^2=1,; x_0

[(x-x_0)^2+y^2=1,; x_0-1

10. Suppose (f) and (g) are differentiable for all (x). Find a differential equation for the family of functions (y=f+cg) ((c)=constant).

## Q4.5.3

In Exercises 4.5.11-4.5.13 find a first order differential equation for the given family of curves.

11. Lines through a given point ((x_0,y_0)).

12. Circles through ((-1,0)) and ((1,0)).

13. Circles through ((0,0)) and ((0,2)).

## Q4.5.4

14. Use the method Example 4.5.6 (a) to find the equations of lines through the given points tangent to the parabola (y=x^2). Also, find the points of tangency.

1. ((5,9))
2. ((6,11))
3. ((-6,20))
4. ((-3,5))

15.

1. Show that the equation of the line tangent to the circle [x^2+y^2=1 ag{A}] at a point ((x_0,y_0)) on the circle is [y={1-x_0xover y_0}quad ext{if} quad x_0 epm1. ag{B}]
2. Show that if (y') is the slope of a nonvertical tangent line to the circle (A) and ((x,y)) is a point on the tangent line then [(y')^2(x^2-1)-2xyy'+y^2-1=0. ag{C}]
3. Show that the segment of the tangent line (B) on which ((x-x_0)/y_0>0) is an integral curve of the differential equation [y'={xy-sqrt{x^2+y^2-1}over x^2-1}, ag{D}] while the segment on which ((x-x_0)/y_0<0) is an integral curve of the differential equation [y'={xy+sqrt{x^2+y^2-1}over x^2-1}. ag{E}] HINT: Use the quadratic formula to solve (C) for (y'). Then substitute (B) for (y) and choose the (pm) sign in the quadratic formula so that the resulting expression for (y') reduces to the known slope (y'=-x_{0}/y_{0})
4. Show that the upper and lower semicircles of (A) are also integral curves of (D) and (E).
5. Find the equations of two lines through (5,5) tangent to the circle (A), and find the points of tangency.

16.

1. Show that the equation of the line tangent to the parabola [x=y^2 ag{A}] at a point ((x_0,y_0) e(0,0)) on the parabola is [y={y_0over2}+{xover2y_0}. ag{B}]
2. Show that if (y') is the slope of a nonvertical tangent line to the parabola (A) and ((x,y)) is a point on the tangent line then [4x^2(y')^2-4xyy'+x=0. ag{C}]
3. Show that the segment of the tangent line defined in (a) on which (x>x_0) is an integral curve of the differential equation [y'={y+sqrt{y^2-x}over2x}, ag{D}] while the segment on which (xUse the quadratic formula to solve (c) for (y'). Then substitute (B) for y and choose the (pm) sign in the quadratic formula so that the resulting expression for (y') reduces to the known slope of (y'=frac{1}{2y_{0}})
4. Show that the upper and lower halves of the parabola (A), given by (y=sqrt x) and (y=-sqrt x) for (x>0), are also integral curves of (D) and (E).

17. Use the results of Exercise 4.5.16 to find the equations of two lines tangent to the parabola (x=y^2) and passing through the given point. Also find the points of tangency.

1. ((-5,2))
2. ((-4,0))
3. ((7,4))
4. ((5,-3))

18. Find a curve (y=y(x)) through (1,2) such that the tangent to the curve at any point ((x_0,y(x_0))) intersects the (x) axis at ({x_I=x_0/2}).

19. Find all curves (y=y(x)) such that the tangent to the curve at any point ((x_0,y(x_0))) intersects the (x) axis at (x_I=x^3_0).

20. Find all curves (y=y(x)) such that the tangent to the curve at any point passes through a given point ((x_1,y_1)).

21. Find a curve (y=y(x)) through ((1,-1)) such that the tangent to the curve at any point ((x_0,y(x_0))) intersects the (y) axis at (y_I=x^3_0).

22. Find all curves (y=y(x)) such that the tangent to the curve at any point ((x_0,y(x_0))) intersects the (y) axis at (y_I=x_0).

23. Find a curve (y=y(x)) through ((0,2)) such that the normal to the curve at any point ((x_0,y(x_0))) intersects the (x) axis at (x_I=x_0+1).

24. Find a curve (y=y(x)) through ((2,1)) such that the normal to the curve at any point ((x_0,y(x_0))) intersects the (y) axis at (y_I=2y(x_0)).

## Q4.5.5

In Exercises 4.5.25-2.5.29 find the orthogonal trajectories of the given family of curves.

25. (x^2+2y^2=c^2)

26. (x^2+4xy+y^2=c)

27. (y=ce^{2x})

28. (xye^{x^2}=c)

29. ({y={ce^xover x}})

## Q4.5.6

30. Find a curve through ((-1,3)) orthogonal to every parabola of the form [y=1+cx^2 onumber] that it intersects. Which of these parabolas does the desired curve intersect?

31. Show that the orthogonal trajectories of [x^2+2axy+y^2=c onumber] satisfy [|y-x|^{a+1}|y+x|^{a-1}=k. onumber]

32. If lines (L) and (L_1) intersect at ((x_0,y_0)) and (alpha) is the smallest angle through which (L) must be rotated counterclockwise about ((x_0,y_0)) to bring it into coincidence with (L_1), we say that (alpha) is the angle from (L) to (L_1); thus, (0lealpha

33. Use the result of Exercise 4.5.32 to find a family of curves that intersect every nonvertical line through the origin at the angle (alpha=pi/4).

34. Use the result of Exercise 4.5.32 to find a family of curves that intersect every circle centered at the origin at a given angle (alpha e pi/2).

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Squeezing a workout in during your busy day becomes a whole lot easier with an assist from a workout app, as it takes the guesswork out of formulating a training plan. And who couldn't use a little help switching up their sweat sessions to stave off boredom and keep your muscles guessing? The best workout apps for women can do all of that and then some.

When it comes to workout apps, there are plenty of options out there for everyone, from yogis to lifters to runners. Many are free to download and offer endless variety when it comes to types of workouts, trainers, scenery, and more.

What&rsquos also great about using a workout app is that you can track your progress. &ldquoBe consistent and spend a little time each week reviewing the data,&rdquo says Pete McCall, CSCS, host of the All About Fitness Podcast and instructor at Equinox in San Diego. &ldquoIt can help you identify when you&rsquore training too hard or not training hard enough&mdashor not hitting the ideal heart rate long enough for significant calorie burn.&rdquo

Here you'll find a list of the 20 top workout apps&mdashall approved by fitness pros and loved by team Women's Health. Each one will give you something to look forward to in your next sweat session&mdashfrom the Monkey Method app, that will push you to master your pullups in no time, to Strava, a running app that will make your next long-distance run feel like a community experience. Sound good? Then, make sure your phone is charged and get to sweating.

## 10 Best Booty Band Workouts

Add a few of these booty band exercises to your next leg day routine to sculpt better curves and build tons of functional strength in your lower body.

### 1. Banded Curtsy Lunges

Lunges are a fantastic workout. Quad, glute, and hip abductor activation is the main benefit of this variation on the classic lunge. With a booty band around your legs, you’ll get even more of a workout.

Wrap the booty band around your legs just above the knees and stand with your feet hip-width apart. Cross your right leg behind your left leg and put your right foot out to one side, then bend your right knee and lower your torso until your right knee touches the floor in a curtsy pose.

Return to the starting position and then repeat the same motion with your left leg. Make sure you do the same number of reps on each leg so that you develop muscles on both legs evenly.

Target your gluteus medius with this move. The gluteus medius is vital for stability and standard lunges and squats don’t give it much of a workout.

### 2. Clamshells

Many people enjoy this exercise as part of a warm-up or cooldown. It’s also simple enough to do during a home workout or at your desk on a lunch break.

The hips get a good workout during clamshells, as do both the gluteus maximus and gluteus medius, which is the main mover and rotator of the hip. To get into the starting position for clamshells, you need to get on the ground and lie on your side. Stretch out the arm that’s on the ground for a more comfortable position.

Bend your knees and keep your feet in line with your spine. Place the booty band around your legs above the knee. Lift the top knee as far as you can without letting your foot or hips get out of position, then bring the top knee back down.

Repeat this motion 10 – 15 times and then roll over so the other leg is on top and repeat the same number of reps.

### 3. Donkey Kicks

You might feel a bit awkward performing this move at first, but the action they give your glute muscles is well worth it. Donkey kicks are also kind of a full-body exercise because they also involve the shoulders and core muscles to keep the body balanced.

Get down on all fours and make sure to keep your hands directly underneath your shoulders. Stack your wrists to prevent injury. Rather than staying on your toes as you would in a push-up, support your lower body weight with your knees.

Keep a flat back and look down slightly so your neck stays in alignment. Lift your right foot up without losing the 90° angle of your knee. Continue lifting the right foot until the thigh is level with your spine, then return it to the starting position.

Complete all the reps on your right side before switching to the left.

### 4. Ankle Jumping Jacks

Wrap a booty band around your ankles to make jumping jacks more challenging. It’s a great way to get some mild cardio into your leg day routine without risking your gains.

Your glutes, quads, hip flexors and abductors, hamstrings, and calves all get attention with this exercise. They’re also fairly simple to do. Forget the arm raises of traditional jumping jacks, though, because the action is all in the lower body with this banded variation.

Stand with your feet hip-width apart and get into a 25% squat position. Tuck your elbows and keep your hands in a fist in front of your chest. Jump and move your feet out to each side a short distance – no need to try and do a split, but stretch that booty band.

Land on your heels and then jump again to return to the starting position. Repeat as many times as you’re comfortable with.

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Banded plank kickbacks train core strength and lower body simultaneously.

### 5. Banded Plank Kickbacks

Improve your core strength and balance with this move, which requires strength in your upper body as well as your lower body. You’ll have to get into the plank position, which is the same one you use for push-ups.

Lift your left foot until it’s level with your spine and then return it to the ground. Repeat the same movement with your right foot. Continue until you’ve done 20 reps on each foot.

If you want to build some upper body strength on leg day, insert a push-up between kickback reps. Just make sure you don’t overtrain your arms.

Single-leg glute bridges are a much more challenging alternative to regular glute bridges.

### 6. Single-Leg Glute Bridge

Glute bridges might seem too simple for the main part of your routine, but they actually challenge your gluteus maximus, obliques, and biceps quite a bit, especially in the single-leg variant.

To get into the starting position, lie flat on your back with the booty band wrapped around your leg just above the knee. Lift your dominant foot up in the air and keep the sole of the other foot flat on the ground for stability.

Leave your arms at your sides with your palms resting on the ground. You won’t need your arms for support in this exercise. Lift your hips so that your lower back is off the ground and there’s a straight line going from your neck to the knee of your non-dominant leg.

Hold that position for several seconds, then sink back to the starting position by lowering your hips. Continue for 10 reps before switching sides and repeating the same movement.

### 7. Banded Walks

This very simple exercise is a great butt workout and you can also hold dumbbells or kettlebells if you want to add some upper-body activity to this lower-body workout. The booty band should be wrapped around both legs across the thighs.

Stand with your feet shoulder-width apart. If you aren’t holding any extra weight in your hands, you can place them on your hips. Take a step out to the side with your left foot and then move your right foot to the place where the left foot just was.

People do this move in two different ways from this point. You can continue moving to the left for ten steps and ten change your direction to move right ten steps back to the starting point. If space is limited, take a step to the left and then back to the right and repeat this back-and-forth for ten steps in each direction.

### 8. Fire Hydrants

This strangely-named booty workout pushes your glutes and thigh muscles in a direction that many other exercises don’t. It’s a great bodyweight exercise that creates tons of gluteus maximus activation and will help give you the sculpted booty curves you’re looking for.

It’s crucial to keep your knees at 90° throughout this exercise. Your back should also remain straight the whole time.

Lift your left leg out to the side without changing the angle of your knee. It won’t reach the height of your spine but raise it as high as you can. Lower back to the starting position.

From here, you can continue all your reps on that same leg or you can alternate if you have the balance to do so.

Helpful Hint: Our Beautiful You Fitplan will give you huge booty gains!

Banded squats are a great way to train your legs without the joint stress that comes with weighted squats.

### 9. Banded Squats

You can do this exercise with a barbell over your shoulders or you can just use your body weight. The resistance from the booty band will be enough to give your glutes and leg muscles a great workout without additional weight, especially if you’re just starting out.

Put the booty band around your legs above the knees and stand with your feet shoulder-width apart. If you aren’t using additional weight, you can let your hands go out in front of you for better balance. Lean forward slightly but maintain a straight back.

Bend both knees so that your torso lowers and then stick your butt out like you were going to take a seat in a chair. Continue lowering until your thighs are parallel with the floor. Hold the lowest position for some isometric gains and then return to the starting position by pushing through your heels.

### 10. Banded Bulgarian Split Squat

A variation on the classic squat, this move employs the booty band in a slightly different way. Rather than wrapping it around your legs, you’re going to step on one end and hold the other end of the loop in your hands.

Find an elevated surface that’s a couple of feet high. If you’re doing a home workout, the couch is a great option. Stand two feet or so in front of your chosen elevated platform and put one foot behind you so that the toes are resting on the elevated surface.

Wrap the booty band around your forward foot and then straighten up with the other end of the loop in your hands so that it’s stretched out. Now go through a squat motion by bending your forward knee and sticking your butt out, lowering your body until your forward thigh is parallel with the ground.

Return to the starting position and repeat for 10 reps. Remember to switch feet positions to get an even workout.

The further out your foot is, the more your glutes will be activated. Move it closer to the elevated surface to give your quads more attention. The Bulgarian split squat is a great squat variation for building balance and coordination in addition to great booty gains.

## Practical Econometrics and Data Science

Of course, you are encouraged to find any other data, which may be interesting to you. The data that will be used in this section is chosen for its ease of access.

### 5.3.1 Datasets

Below we list some select datasets, which you can analyse.

#### 5.3.1.1 Dataset 1

We want to evaluate whether it is better to drive by an auto mobile or via public transportation.

The dataset can be loaded in both R and Python :

#### 5.3.1.2 Dataset 2

The dataset: new titanic data (previous, much smaller version titanic).

Let’s say that we are interested in estimating whether a passenger will Survive based on their age, gender, economic status and other factors.

The dataset can be loaded in both R and Python :

#### 5.3.1.3 Dataset 3

we want to evaluate whether a customer will choose coke , or pepsi.

The dataset can be loaded in both R and Python :

#### 5.3.1.4 Dataset 4

The aim is to predict which customers will default on their credit card debt.

The dataset can be loaded in both R and Python :

Note: it appears that this dataset is artificial. Be aware that this may lead to unexpected coefficient signs. Nevertheless, it is always interesting to analyse data, which challenge your assumptions for variables signs and significance.

#### 5.3.1.5 Dataset 5

The aim is to predict whether a woman will decide to return to the labor force, inlf .

The dataset can be loaded in both R and Python :

The following tasks are universal for all datasets. This is in order to highlight that in practical applications you will usually need to carry out similar steps and ask (yourself) similar general questions when working with any kind of data.

An example with one of the datasets is provided in section 5.4. As before, some comments are provided within the example to highlight the additional possible insights and additional questions and possible further difficulties that can be identified from the dataset.

We stress that the tasks and questions are there to give you some general steps (which are mostly in a logical order, which you would follow in real-world applications, though some are arranged to follow the chapter ordering) of the modelling process as a whole - it is not a rule that you always need to examine the scatterplots of every variable, or always run specific tests when carrying out your analysis. However, they may help you support any arguments/insights that you may discover from the modelling process.

Below are the tasks that you should carry out for the datasets:

Note: Take (80\%) of the data as the training set and fit your model on this data. Hold the remaining (20\%) of the data as the test set, which you can then use to check the out-of-sample characteristics of your model.

#### 5.3.2.1 Exercise Set 1

(Reminder: use the training set)

Postulate, what kind of model(-s) would you need to specify to model the dependent variable (there may be more than one):

• A linear regression?
• A logistic regression?
• A probit regression?
• A multinomial logit regression?
• A regression for count data (e.g. Poisson regression)?
• How the independent variables relate to the dependent variable and to one-another? Do you notice any relationships?
• What variables would you include in your model? (Note: do not include any polynomial or interaction terms just yet)
• What signs do you expect them to have?

Estimate one or more model(-s) based on your answers in the previous tasks. Are there any insignificant variables? Are the signs as you would expect?

Are there any collinear variables? If so, remove the multicollinearity, if it is meaningful to do so.

#### 5.3.2.2 Exercise Set 2

(Reminder: use the training set)

Include polynomial and/or interaction terms in your model. Explain your motivation for selecting these variables and their signs.

Calculate the predicted values for various combinations of values:

• For a continuous data variable (X_) - create new, equally spaced, values ( ilde_ > ilde_) , for (j = 1. M) , with ( ilde_ = min (X_)) and ( ilde_ = max (X_)) . Select an arbitrary value (M) so that the predicted probability plot chart is readable.
• For a discrete/categorical variable - select some cases to compare the probabilities. For example two curves: when (X_ = 1) vs when (X_ = 0) .
• Plot the (95\%) confidence bounds for the predictions of the two cases.

select one continuous and one discrete variable and treat the remaining variable values as fixed. In other words, calculate the predicted values with the new value of a single variable, ceteris paribus. You can do this by first selecting a random observation and duplicating it in a new dataset. Then replace the continuous or discrete variable with its new values. Repeat this for the continuous and discrete variable cases.

remember that confidence intervals are not the same as prediction intervals!

1. If you are estimating the probability, select the cutoff prediction probability depending on the confusion matrix results:
• use the default (0.5) cutoff value
• try to select an alternative (hopefully optimal) cutoff value
• Do the default and the optimal cutoff probability values differ?
2. Examine the (ROC) curve.

#### 5.3.2.3 Exercise Set 3

Provide an interpretation for a few (not necessarily all) variables included in your model. Hint: it may be worthwhile to examine the interpretation of the parameters themselves, as well as the partial effects.

Write down the fitted model.

Can you check some linear restrictions?

Compare the model results between the training and test sets - is your model adequate for this new data?

## Verify in the browser

Once the container starts, connect to the running container using http://localhost in the example shown. Type that URL into your browser, and you should see the running site.

Some VPN or proxy software may prevent you from navigating to your site. You can temporarily disable it to make sure your container is working.

The sample directory on GitHub contains a PowerShell script that executes these commands for you. Open a PowerShell window, change directory to your solution directory, and type:

The command above builds the image, displays the list of images on your machine, and starts a container.

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## Create an Epi Curve

This Quick Learn lesson will take approximately 10 minutes to complete.

When you are finished, you will be able to create an epidemic curve, or &ldquoepi curve.&rdquo

You can move through this lesson by using the NEXT and BACK icons below.

### What is an Epi Curve?

An epi curve is a visual display of the onset of illness among cases associated with an outbreak.

You can learn a lot about an outbreak from an epi curve, such as

• The outbreak's time trend, that is, the distribution of cases over time
• Outliers, that is, cases that stand apart from the overall pattern
• General sense of the outbreak's magnitude
• The most likely time period of exposure

### X and Y Axes

An epi curve is represented by a graph with two axes that intersect at right angles.

The horizontal x-axis is the date or time of illness onset among cases.

The vertical y-axis is the number of cases.

Each axis is divided into equally spaced intervals, although the intervals for the two axes may differ.

### Cases

Cases (represented by boxes) are placed along the x-axis according to the date when their earliest symptoms began.

If symptoms for more than one case began on the same date (or time), the boxes are stacked one above the other.

### Time Intervals

The unit of time for the x-axis is based on the incubation period of the disease under investigation and the length of time over which cases are distributed.

As a general rule, the unit of time is set at roughly 1/4 (0.25) of the average incubation period for the illness under investigation.

Average incubation period
multiplied by .25 = Intervals

### Time Intervals, continued

If the incubation period of an illness is short, the interval on the epi curve might be indicated in hours or even minutes.

The incubation period for infection with E. coli O1257:H7 is usually 3&ndash4 days. In this case, you would probably use an interval of one day on the x-axis.

4 days multiplied by 0.25 = 1 day
intervals (4 x 0.25 = 1)

### Determining a Title

An epi curve should have a title that describes it, including the type of illness, the place where the outbreak occurred, and the time period.

Cases of [insert illness name] by date of onset, [insert location and time period]

Cases of E. coli O157:H7 by Date of Onset, Epps, Louisiana, February, 2012

Seven cases of hepatitis A have been reported in Homer, Virginia. Construct the epi curve from the line list below.

Select the column heading that you would use to label the x-axis of the epi curve.

• Case
• 1
• 2
• 3
• 4
• 5
• 6
• 7

• Age
• 42
• 31
• 23
• 46
• 54
• 49
• 31

• Sex
• M
• M
• M
• M
• M
• M
• M
• Date of
Onset
• Dec 13
• Jan 5
• Jan 6
• Jan 8
• Jan 9
• Jan 10
• Jan 20
• Date of
Diagnosis
• Dec 26
• Jan 12
• Jan 8
• Jan 16
• Jan 12
• Jan 13
• Jan 20

Exercise 1: Incorrect

That is incorrect.
The correct answer is Date of Onset.

In an epi curve, the date of illness onset is on the x-axis.

The number of cases should go on the y-axis.

Exercise 1: Incorrect

That is incorrect.
The correct answer is Date of Onset.

In an epi curve, the date of illness onset is on the x-axis.

The number of cases should go on the y-axis.

Exercise 1: Incorrect

That is incorrect.
The correct answer is Date of Onset.

In an epi curve, the date of illness onset is on the x-axis.

The number of cases should go on the y-axis.

Exercise 1: Correct

That is correct.
The correct answer is Date of Onset.

In an epi curve, the date of illness onset is on the x-axis.

The number of cases should go on the y-axis.

Exercise 1: Incorrect

That is incorrect.
The correct answer is Date of Onset.

In an epi curve, the date of illness onset is on the x-axis.

The number of cases should go on the y-axis.

Given that the average incubation period for hepatitis A is 28-30 days, what intervals would you use on the x-axis?

Exercise 2: Incorrect

Given the average incubation period for hepatitis A is 28-30 days, and the unit of time is roughly one-quarter the average incubation period, you would use an interval of 7 days on the x-axis.

Exercise 2: Correct

Given the average incubation period for hepatitis A is 28-30 days, and the unit of time is roughly one-quarter the average incubation period, you would use an interval of 7 days on the x-axis.

Exercise 2: Incorrect

Given the average incubation period for hepatitis A is 28-30 days, and the unit of time is roughly one-quarter the average incubation period, you would use an interval of 7 days on the x-axis.

What is the appropriate title for the epi curve?

Hepatitis A outbreak in Virginia

Outbreak in Homer, Virginia

Hepatitis A Cases by Date of Onset in Homer, Virginia, December 2011&ndashJanuary 2012

Exercise 3: Incorrect

The title should include both the name of the disease and the place where the outbreak occurred. The correct title is

Hepatitis A Cases by Date of Onset in Homer, Virginia, December 2011 - January 2012

Exercise 3: Incorrect

The title should include both the name of the disease and the place where the outbreak occurred. The correct title is

Hepatitis A Cases by Date of Onset in Homer, Virginia, December 2011 - January 2012

Exercise 3: Correct

The title should include both the name of the disease and the place where the outbreak occurred. The correct title is

Hepatitis A Cases by Date of Onset in Homer, Virginia, December 2011 - January 2012

Using the following information from a line listing, which date from the Options below represents the Date of Onset for Case 1?

Case Age Sex Date of Onset Date of Diagnosis
1 42 M Dec 13 Dec 26

Exercise 4: Correct

Case Age Sex Date of Onset Date of Diagnosis
1 42 M Dec 13 Dec 26

The date of onset is during the week of 12/7.

Exercise 4: Incorrect

Case Age Sex Date of Onset Date of Diagnosis
1 42 M Dec 13 Dec 26

The date of onset is during the week of 12/7.

Exercise 4: Incorrect

Case Age Sex Date of Onset Date of Diagnosis
1 42 M Dec 13 Dec 26

The date of onset is during the week of 12/7.

Exercise 4: Incorrect

Case Age Sex Date of Onset Date of Diagnosis
1 42 M Dec 13 Dec 26

The date of onset is during the week of 12/7.

Exercise 4: Incorrect

Case Age Sex Date of Onset Date of Diagnosis
1 42 M Dec 13 Dec 26

The date of onset is during the week of 12/7.

Exercise 4: Incorrect

Case Age Sex Date of Onset Date of Diagnosis
1 42 M Dec 13 Dec 26

The date of onset is during the week of 12/7.

Exercise 4: Incorrect

Case Age Sex Date of Onset Date of Diagnosis
1 42 M Dec 13 Dec 26

The date of onset is during the week of 12/7.

Using the following information from a line listing, which date from the Options below represents the Date of Onset for Case 5?

Case Age Sex Date of Onset Date of Diagnosis
5 54 M Jan 9 Jan 12

Exercise 5: Incorrect

Case 5 would be plotted on the week beginning 1/4.

Case Age Sex Date of Onset Date of Diagnosis
5 54 M Jan 9 Jan 12

Exercise 5: Incorrect

Case 5 would be plotted on the week beginning 1/4.

Case Age Sex Date of Onset Date of Diagnosis
5 54 M Jan 9 Jan 12

Exercise 5: Incorrect

Case 5 would be plotted on the week beginning 1/4.

Case Age Sex Date of Onset Date of Diagnosis
5 54 M Jan 9 Jan 12

Exercise 5: Incorrect

Case 5 would be plotted on the week beginning 1/4.

Case Age Sex Date of Onset Date of Diagnosis
5 54 M Jan 9 Jan 12

Exercise 5: Correct

Case 5 would be plotted on the week beginning 1/4.

Case Age Sex Date of Onset Date of Diagnosis
5 54 M Jan 9 Jan 12

Exercise 5: Incorrect

Case 5 would be plotted on the week beginning 1/4.

Case Age Sex Date of Onset Date of Diagnosis
5 54 M Jan 9 Jan 12

Exercise 5: Incorrect

Case 5 would be plotted on the week beginning 1/4.

Case Age Sex Date of Onset Date of Diagnosis
5 54 M Jan 9 Jan 12

### Summary

You have created a basic epi curve. The completed epi curve is below.

Hepatitis A Cases by Date of Onset in Homer, Virginia,
December 2011&ndashJanuary 2012

## Questions

1. Do the graphs of volume of solution vs. moles of solute have the same shape for all three solutions?
2. Is there anything unexpected in the way the volume changes as solute is added to any of the solutions? How might you account for any unexpected behavior that you observe?
3. Are the values of that you calculated in Step 5 all positive? If any of the values of are negative, what does that say about the validity of thinking of as a volume? Can volumes be negative?