10.2.1Are you feeling sick?

The Cootie Lab

10-103.

CATCHING COOTIES

Oh no! Cooties are spreading around the school and nothing can stop them. In fact, this cootie epidemic threatens the success of the prom, which will occur $10$ days from now. After your class has remained cootie-free, a student came today with cooties and is infecting another student at this very moment! From now on, each infected student will have a chance to infect another member of the class each day.

Will your whole class be infected before the prom? Discuss what a graph of the number of infected students over time will look like.

10-104.

THE COOTIE LAB

Let’s assume the incubation period is one day and, once infected, you stay infected for two weeks. Your teacher will randomly pick the first person who will have cooties at $t = 0$ . On each “day” thereafter, each infected person will randomly infect someone else. Record data for who is sick each day. This problem can be completed using the Cootie Lab Data eTool (Desmos).

After everyone gets cooties, put the following data into your calculator. In one column, put the number of days ( $0, 1, 2, 3$ , etc.) and in another put the number of people with cooties on each day. Then, plot the data in a well-suited window.
The behavior of this data is unlike any curve you have studied before (linear, exponential, direct or inverse variation, etc.). It is called a logistic curve, and many calculators have the capability to fit a logistic curve to data points. If your calculator can fit a logistic curve, do that now. If not, plot the data on graph paper and sketch a function that will best fit the data.
Describe the shape of the logistic curve. What are its maximum and minimum? Where does it increase or decrease? Where is it concave up or concave down? Does the graph appear to have symmetry?
Now sketch the graph of a logistic differential equation that corresponds with the logistic curve you sketched in part (b). That is, sketch a graph of the rate of change, $\frac { d y } { d x }$ , where $y$ represents the number of infected students and t represents time in days. Describe the shape of this graph. What important features does it have?

10-105.

Let $\frac { d y } { d x }= 2y(3 - y)$ where y is a function of $x$ .

Sketch the slope field for the differential equation.
Sketch a particular solution at $(0, 2)$ .
Describe any important characteristics such as asymptotes, maxima, minima, or points of inflection from part (b) that you observe.
What is $\lim\limits _ { x \rightarrow \infty } y ( x )$ ?
In part (d) of problem 10-104, you sketched a logistic differential equation that represented the rate that students were affected with cooties in your class. Does $\frac { d y } { d x }= 2y(3 - y)$ represent a logistic differential equation? Explain?

10-106.

At $t = 0$ , there are $100$ peacocks in a bird sanctuary. The rate of change of peacocks grows at a rate jointly proportional to the product of the peacock’s population and its change in population. Write a differential equation to model this situation. Let $k$ represent the constant of proportionality.

10-107.

Decide if each of the following series converges or diverges. State the tests you used. Homework Help ✎

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n } { 2 n + 5 }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n ! } { 5 ^ { n } }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \operatorname { ln } ( \frac { 1 } { n } )$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { \operatorname { cos } ( n ) } { 2 ^ { n } }$

$\displaystyle\sum _ { n = 0 } ^ { \infty } \frac { ( - 5 ) ^ { n } } { 4 ^ { n } }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n ^ { 4 } + 6 }$

10-108.

Review Ying’s technique for manipulating infinite expressions in problem 9-45. Then, use this strategy to evaluate the infinite “nested radical” $\sqrt { 3 + \sqrt { 3 + \sqrt { 3 + \ldots } } }$ Homework Help ✎

10-109.

Suppose $S = \displaystyle\sum _ { n = 1 } ^ { \infty } a _ { n }$ is a convergent series and $a_n ≠ 0$ for any $n$ . Explain why the series $T = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { a _ { n } }$ must diverge. Homework Help ✎

10-110.

Compute without a calculator Let $r = \frac { 1 } { \theta }$ . Homework Help ✎

Sketch $r$ without the help of your calculator.
Write another polar equation with the exact same graph over all $θ$ .

10-111.

Evaluate each of the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow \infty } \frac { x ^ { 3 } } { e ^ { x } }$

$\lim\limits _ { x \rightarrow 0 } \frac { \operatorname { sin } ^ { - 1 } ( x ) } { x ^ { 2 } }$

$\lim\limits _ { x \rightarrow 3 } \frac { x ^ { n } - 3 ^ { n } } { x - 3 }$

$\lim\limits _ { x \rightarrow 1 ^ { - } } \frac { \operatorname { ln } ( 1 - x ) } { \operatorname { cot } ( \pi x ) }$

$\lim\limits _ { x \rightarrow \infty } \frac { x ^ { 2 } } { 2 ^ { x } }$

10-112.

Calculate the area bounded by the curve $y =\sqrt { x }$ , the line tangent to the curve at $x = 9$ , and the $y$ -axis. Homework Help ✎

10-113.

Let $x_1(t) = t + \cos(t)$ and $x_2(t) = t + \sin(t)$ describe the positions along the $x$ -axis of two particles, for $0 ≤ t ≤ 3$ . 10-113 HW eTool (Desmos). Homework Help ✎

At what time(s) over this interval do the particles have the same velocity?
What are the positions of the particles at these times?

10-114.

For the function defined below, determine the values of $a$ and $b$ such that $f$ is both continuous and differentiable for all values of $x$ . Justify your answer. Homework Help ✎ $f ( x ) = \left\{ \begin{array} { l l } { a x ^ { 2 } } & { \text { for } x \leq 1 } \\ { b x + 2 } & { \text { for } x > 1 } \end{array} \right.$

10-115.

A rocket, rising vertically, is being tracked by a radar station $3$ miles away from the rocket’s launch pad. How fast is the rocket rising when it is $5$ miles above the ground and the distance between the rocket and the radar station is increasing at a rate of $3000$ miles per hour? Homework Help ✎

Calculus Third Edition