Calculus Application Problem DUE SOON

Calculus Application Problem #3 Name __________________________________________

The Deriving Dead! Due at the beginning of class ______________________

Introduction: Imagine that you are one of many people at a “party” and that, unknown to everyone else, one person was bitten by a zombie on the way to the party! How quickly will the “zombiepocalypse” spread, and what are the chances that you will leave the party as a zombie? The objective of this activity is to create a mathematical model that describes the spread of a disease (such as a zombie virus) in a closed environment, and then apply calculus concepts to this mathematical model.

Collecting the Data:

Let’s collect some data from an activity that will simulate the spread of a communicable disease over a period of time, divided into “stages”.

The number of people in our “closed environment” is ________________

1

1

131211109876543210

Number of Total Infected

Number of Newly Infected

Stage Number

Applying Calculus to the Data:

1. Using the data from the chart, make a scatterplot of the "Stage Number" (in L1) vs. the "Number of Total Infected" (in L2). Sketch the scatterplot below. Connect the data points to create a continuous function for Y(t).

2. Using the data that was collected in the activity, answer the following questions about the derivative function Y’(t), which represents the instantaneous rate of change of the number of infected at any stage. Consider the domain to be [ 0 , 13 ].

a. When, if ever, is Y’(t) positive? ____________________________________

b. When, if ever, is Y’(t) negative? ___________________________________

c. When, if ever, is Y’(t) increasing? ____________________________________

d. When, if ever, is Y’(t) decreasing? ____________________________________

e. From your answers above, sketch a graph of Y’(t) below.

f. The t-value where Y’(t) changes from increasing to decreasing is the inflection point on Y(t). According to the data in the chart, this occurs when t = _________, and the corresonding “y-value”

is ________. (Note: We will check this later in the problem!)

Finding a Logistic Function that Models the Data

3. Since the data (should) appear to be a model for a logistic function, we need to find a function in the form:

Y(t ) = c

1 + a ⋅ e− b⋅t ,

where t represents the stage number and Y(t) represents the total number of infected people in stage t.

Therefore, we need to find values for the three constants a, b, and c. The value of c should be easy. For our activity,

c = _________

To find a, use the initial point ( 0 , 1 ). Substitute this ordered pair, with the value of c into our logistic model and solve for a. Show your work below.

a = _______________

To find b, the last constant in the model, we need another ordered pair. Let’s use an ordered pair near the middle of the data, say during Stage #7.

Record this ordered pair: ( 7 , _________)

Substitute these values into the equation (with the values of a and c), and solve for b. Show your work below. Write your answer exactly, then round the answer to four decimal places. Then write the function Y(t).

b = _____________________ = __________ The final equation is: Y(t) = __________________________ (exact answer) (decimal)

Of course, graph it to see how it fits the scatterplot. If it doesn’t fit fairly well, find your mistake!

4. To find a better fitting model, we will let the calculator find the logistic function that best fits all of our data points. Using the regression capabilities of the calculator, find the logistic function (B:Logistic), that best fits the data. Round a and c to the nearest whole number, and the value of b to 4 decimal places.

Notes: 1. The value of c in the logistic regression equation may not be the “real value” of c, but it should be close!

2. The values of a and b may not be close to the values we found, because, remember, we only used two data values to find our equation. The regression equation is using all data points.

The regression equation is: Y(t) = _________________________________

Calculus and Logistic Functions:

Note: Use the regression equation (in #4 above) for the remainder of the activity. On your calculator, you may want to turn off the scatterplot and delete the equation we found (not the regression equation), if you have not already done so.

Your work on the remainder of this problem should be completed on a separate sheet of paper! These are important parts of this activity! NEATLY, show all of your work.

5. Analytically, using derivative properties, find the rule for Y’(t), the derivative of Y(t).

Hints:

1. Use the Quotient Rule when finding the derivative Y’(t). 2. When you are finished, the derivative should be in the form:

Y '(t) = CONSTANT( ) ⋅ e SOMETHING( )

SOMETHING ELSE( )2

When finished, you can check your answer by letting:

Y1 = the logistic function Y(x)

Y2 = nDeriv(Y1,X,X)

Y3 = your answer.

Turn off (do not delete) Y1, set a Window, and graph Y2 and Y3. If you have found the derivative correctly Y2 and Y3 should be the same!

6. Using your answer to #5 above, evaluate Y’(6), and interpret your answer. Be sure to include units in your interpretation!

7. Notice that the graph of a logistic curve is concave up (the slope is increasing) at first, then changes to concave down (where the slope is decreasing). At the inflection point, where the concavity changes, the slope is the largest. This point, called the “point of diminishing returns”, is very important in many applications of logistic growth functions.

As was discussed in class, the inflection point of a logistic function in the form Y(t ) = c

1 + a ⋅ e− b⋅t will occur at

the point where the y-value is equal to c/2.

For the final part of the activity, we (you) are going to verify this analytically.

We know that wherever a function Y(t) has an inflection point, then the second derivative Y’’(t) = 0.

(Don’t worry, you are not going to have to find the second derivative Y’’(t). This is very messy, but you can do it if you want, and I’d be glad to check it!)

It can be shown that the t-value of the inflection point of the logistic function Y(t ) = c

1 + a ⋅ e− b⋅t will occur when

a ⋅ e− b⋅t − 1 = 0

Solve this equation for t. Your answer for t will be in terms of the variables a and b. (Do not use values for a and b!) Be sure to show all of your work neatly!

Once you have found the t-value of the inflection point (in terms of a and b), substitute this expression in for t in the function Y and simplify. You should begin by writing :

Y( ____ ) = c

1 + a ⋅ e− b⋅( ___ )

Again, show all of your work, step by step, neatly. When you are finished simplifying the expression, your result should be c/2!

Two hints! These are algebra hints involving the function ln(x) and the function ex .

- By applying properties of logarithms, the expression ln 1 x

⎛ ⎝⎜

⎞ ⎠⎟ = ________________________, and,

when simplified, is equal to _______________________. Although it is not necessary, you may want to use this property.

- An expression in the form eln( f ( x )) = ___________________. (If you don’t know, evaluate eln( 3.75) on your calculator. This should help you remember this property!

8. Conclusion: From your work on #7 above, complete the following statement.

“A logistic function in the form Y(t ) = c

1 + a ⋅ e− b⋅t has an inflection point at the ordered pair ( _____ , ______ ).”

Now, using the regression equation of the logistic function Y(t), not the data collected in class, what is the inflection point of Y(t). State your answer as an ordered pair and show (explain) your work.

When you are finished with #5 - #8 on the seperate sheet of paper, staple that to this activity. If it is not done NEATLY, we will not grade it!