7.3.6When do I use differential equations?

Differential Equation and Slope Field Applications

7-161.

Since $\frac { d } { d x }[ \operatorname { ln } ( x ) ]=\frac { 1 } { x }$ , then it seems that $\int_{ }^{ }\frac{1}{x}dx$ should equal $\ln(x) + C$ . However, in Chapter 6, you learned that $\int\frac { 1 } { x }dx = \ln|x| + C$ . Why is this? Use a slope field for $\frac { d y } { d x }=\frac { 1 } { x }$ to investigate why the absolute value is necessary. Write down your observations.

7-162.

POPULATION EXPLOSION!

In 2008 Depedete’s population was approximately $800,000$ people. If the population of Depedete grows at a rate of $5\%$ per year, then Depedete can expect to increase by $40,000$ people in 2009. In 2010, the increase should be $5\%$ of its new population ( $840,000$ ), which is $42,000$ . Therefore, each year as the population changes, the rate of change in people per year, $\frac { d P } { d t }$ , changes.

Write an equation that represents $\frac { d P } { d t }$ , the rate of change of the population with respect to time.
Study the slope field at right for Depedete. The slope of each tangent line represents the rate of growth for $P$ . Examine the tangent lines for $P = 800,000$ . Why do they all have the same slope?
Place your paper over the slope field. If $P(0) = 800,000$ , draw the particular solution for $P$ given this initial condition. What type of function is $P\left(t\right)$ ?
Use implicit integration to write an equation for $P(t)$ .
The slopes of $P$ are not the same for each value of $t$ , yet depend only on the values of $t$ . Explain why. Hint: Think about the role of the constant of integration in this problem versus other problems.
Write equation that will estimate future populations of Depedete if the city grows at a rate of $3.5\%$ per year. Use this equation, and the fact that the 2008 population was $800,000$ to estimate the population in the year 2099.

7-163.

Set up a differential equation to express the information in each situation below. Define the variables you use.

The rate at which the population of a city changes varies directly with its current population.
The rate of change of the volume of water in a tank is proportional to the difference between the amount entering and the amount leaving.

7-164.

POMP AND CIRCUMSTANCE

Today is graduation day and Winnie awakes foggy from a dream at 8:30 a.m. Struggling to be alert, she makes herself a cup of coffee. She remembers that she has to be out of the house at 9:30 a.m. to make it to school on time. When she tastes the cup of coffee, she burns her mouth. “No wonder,” she says after testing its temperature, “this coffee is $205^\circ\text{F}$ !” She puts the coffee on the counter and runs off to get ready.

Forty-five minutes later, she returns to find her coffee lukewarm (in fact, it is $98^\circ\text{F}$ ). “At least it’s not room temperature,” she thinks, since she knows the room temperature is $68^\circ\text{F}$ . She leaves the coffee to continue doing her hair.

Later, with her hair up and her shoes on, she is finally ready! However, she cannot see the clock without her glasses! One last swig of coffee reveals it is now cold: $82^\circ\text{F}$ . Winnie is worried that she is going to be late to her graduation. Will she be late? Use a differential equation to determine your answer.

7-165.

Stingray populations grow based on the differential equation below, where $P$ is the population (in thousands) and $t$ is time in years. Use the graph of the slope field at right to complete parts (a) through (c) below. Homework Help ✎

$\frac { d P } { d t } = 0.026P(12.5 − P)$

If there are $3000$ stingrays for time $t = 0$ , sketch a curve representing the population of stingrays.
What if the original population when $t = 0$ is $10,000$ stingrays? Draw this population curve and decide if its rate of growth is the same or different as that in part (a).
What if the original population when $t = 0$ is $17,000$ stingrays? Draw this population curve and decide if its rate of growth is the same or different as those in parts (a) and (b).

7-166.

Evaluate each integral below. Show your steps. If you use $u$ -substitution, be sure to change the bounds of integration. Homework Help ✎

$\int _ { \pi / 4 } ^ { 0 } \sqrt { \operatorname { sin } ( x ) } \cdot \operatorname { cos } ( x ) d x$

$\int \frac { 1 } { \operatorname { sin } ( x ) } \operatorname { cos } ( x ) d x$

$\int \frac { 3 x ^ { 4 } - 4 x ^ { 2 } - 11 x + 6 } { x ^ { 2 } } d x$

$\int \frac { 1 } { | x | \sqrt { x ^ { 2 } - 1 } } d x$

$\int _ { 1 } ^ { e } \frac { 1 } { x } \operatorname { cos } ( \operatorname { ln } ( x ) ) d x$

$\int _ { - 1 } ^ { 3 } x ^ { 2 } ( x ^ { 3 } - 8 ) d x$

7-167.

Explain why a differential equation has infinitely many solutions. Homework Help ✎

7-168.

A Ferris wheel, $50$ feet in diameter, revolves at a rate of $2$ radians per minute. How fast is a passenger moving vertically when the passenger is $15$ feet higher than the center of the Ferris wheel and is rising? Homework Help ✎

7-169.

During the semester, Arezo’s daily calculus grade at time, $t$ (in days), is given by $G(t) = 0.008t^2 + 22 \sin( \frac { t } { 5 } )+ 45$ . She will receive $$50$ if she passes calculus, which requires an average grade of $60\%$ or better. If her semester grade (over $18$ weeks) is the average grade during that time period, should she celebrate? Homework Help ✎

$\$

7-170.

CHECK FOR UNDERSTANDING: SLOPE FIELDS

When you draw a curve based on a slope field, what are you finding? What is its relationship with the equation that formed the slope field to begin with? Homework Help ✎

7-171.

Multiple Choice: The normal line to the curve represented by the equation $y = x^2 + 6x + 4$ at the point $(–2, –4)$ also intersects the curve at $x =$ Homework Help ✎

$−6$

$−\frac { 9 } { 2 }$

$-\frac { 7 } { 2 }$

$−3$

$-\frac { 1 } { 2 }$

Calculus Third Edition