10.2.2Exactly how quickly is it spreading?

More Logistic Differential Equations

10-116.

LOGISTIC EQUATIONS

A highly contagious disease will grow at a rate proportional to the product of the number of people infected and the number of people un-infected.

Translate the italicized sentence above into a differential equation, using $y$ (the number of people infected) as a function of $t$ days, and $n$ (the number of students in your class) as the total population. Let $k$ represent the constant of proportionality.
If someone in your class became infected, solve your differential equation from part (a).
Since one person was infected on Day $0$ , we know that $(0,1)$ is on the solution curve. Use this point and one other (from your data in problem 10-86) to solve for all parameters.
Sketch the solution to the differential equation.

10-117.

GRAPH ANALYSIS

Graph your logistic equation from problem 10-116.

Examine the graph of the logistic equation and explain why it behaves as it does. Specifically, use your logistic equation to determine the $y$ -intercept and $\lim\limits _ { t \rightarrow \infty } y ( t )$ .
Graph the derivative of your logistic function. Explain why the general shape of the derivative graph is reasonable.
What are the coordinates of the maximum of the derivative? What do these values mean in the context of the situation? Include units in your explanation.
Using the $t$ -value you found in part (c) find the corresponding $y$ -value on the logistic curve. Does that coordinate make sense, too? Explain.

10-118.

Finding the solution to a logistic differential equation can be tedious. Fortunately, finding the solution is not always necessary! Important information about the solution graph can be found by analyzing the differential equation itself.

Let each of the following differential equations describe the rate that a large popcorn machine pops popcorn, in kernels per hour. Let $P$ represent the number of kernels in the machine at a given time. Without solving the differential equation, answer the following questions:

What is $\lim\limits _ { t \rightarrow \infty } P ( t )$ ? What does that mean in the context of the problem?
When is $P\left(t\right)$ changing the fastest?

How fast is the amount of popcorn changing when it is growing at its fastest rate?

$\frac { d P } { d t } = P ( 8 - \frac { P } { 1000 } )$

$\frac { d P } { d t }= 0.001P(3000 - P)$

10-119.

For a quadratic function, $f(x) = ax^2 + 2x -5$ , $f^\prime(0) = 10$ . What is the value of $a$ ? Homework Help ✎

10-120.

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

$\displaystyle \sum _ { n = 1 } ^ { \infty } n ( \frac { 2 } { 3 } ) ^ { n }$

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

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

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

10-121.

Let $S = \displaystyle \sum _ { n = 1 } ^ { \infty } \frac { a ^ { n } } { n + 1 }$ , where $a$ is a constant. Homework Help ✎

Does $S$ converge if $a = 1$ ? Justify your answer.
Does $S$ converge if $a = -1$ ? Justify your answer.
For what values of $a$ does $S$ converge? Justify your answer.

10-122.

Write an integral expression that will calculate the volume of the solid created when the region formed by $y = e^x$ , $y = e^{-x + 2}$ , and $y = x^2 - 2x + 1$ is rotated about the line $x = 6$ . 10-122 HW eTool (Desmos). Homework Help ✎

10-123.

Use a vector diagram to show that $| \vec { A } + \vec { B } | \leq | \vec { A } | + | \vec { B } |$ . What geometric property justifies this statement? Homework Help ✎

10-124.

Determine the integral that is being approximated with the Riemann sum $\displaystyle\sum _ { k = 1 } ^ { n } \frac { 5 } { n } \sqrt { 4 + \frac { 5 k } { n } }$ . Then, calculate the value of $\lim\limits _ { n \rightarrow \infty } \displaystyle\sum _ { k = 1 } ^ { n } \frac { 5 } { n } \sqrt { 4 + \frac { 5 k } { n } }$ . Homework Help ✎

10-125.

Evaluate each of the following limits. Homework Help ✎

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

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

$\lim\limits _ { x \rightarrow 0 ^ { + } } x \operatorname { ln } ( x )$

$\lim\limits _ { x \rightarrow 8 } \frac { x ^ { 1 / 3 } - 2 } { x - 8 }$

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

10-126.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int \frac { 1 } { x ^ { 2 } + 4 } d x$

$\int \frac { x } { x ^ { 2 } + 4 } d x$

$\int \frac { 1 } { x ^ { 2 } + 4 x } d x$

$\int \frac { x ^ { 2 } } { x ^ { 2 } + 4 } d x$

Calculus Third Edition