Free Practice Quiz Question List

7 Differential Equations and Dynamic Models Online Quiz Questions

Use this free practice quiz with 20 questions to review 7 Differential Equations and Dynamic Models, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: An initial condition such as $y(0)=y_0$ , together with a differential equation, forms an initial-value problem.

A
True
B
False

Choose one

1 point

A segment in a slope field tilts upward at a point $(t,y)$ . What does this indicate about a solution passing through that point?

A
The solution is increasing at that location.
B
The solution is decreasing at that location.
C
The solution must be constant everywhere.
D
The solution has reached an equilibrium.

Written response

1 point

In the logistic model $P'=rP\left(1-\frac{P}{K}\right)$ , what term describes the limiting population or resource level $K$ ?

Fill in the blank

1 point

Complete the statement: An initial-value problem consists of a and a .

Choose one

1 point

For the autonomous equation $y'=y(4-y)$ , which listed equilibrium is stable for nearby positive solutions?

A
$y=1$
B
$y=2$
C
$y=4$
D
$y=8$

True or false

1 point

True or false: Under the exponential decay model with a positive initial amount, the amount approaches zero but does not reach zero in finite time.

A
True
B
False

Choose all

1 point

Select all conclusions that follow from the logistic model $P'=rP\left(1-\frac{P}{K}\right)$ when $r>0$ and $0<P<K$ .

A
If $0<P<K$ , then the population increases.
B
Growth is fastest when $P=\frac{K}{2}$ .
C
The population grows without bound as $t\to\infty$ .
D
As $t\to\infty$ , the population approaches $K$ .

Written response

1 point

An exponentially growing quantity has proportional growth rate $r=0.2$ per hour. What is its doubling time, rounded to three decimal places? Values within $0.001$ hours are accepted.

Fill in the blank

1 point

Complete the solution procedure for a separable differential equation: first ; then both sides.

Choose one

1 point

At what population level does a logistic model have its fastest growth?

A
$P=0$
B
$P=\frac{K}{2}$
C
$P=K$
D
$P=2K$

Choose all

1 point

For the logistic model with $r>0$ , $K>0$ , and an initial population $P_0>K$ , select all conclusions supported by the model.

A
The population initially decreases.
B
The population approaches $K$ from above.
C
The population increases without bound.
D
The population remains positive while moving toward $K$ .

Open ended

1 point

Solve the initial-value problem $\frac{dy}{dt}=2ty$ , $y(0)=3$ . Show the main separation and integration steps, state the solution, and explain its behavior for $t>0$ .

Choose one

1 point

Which differential equation is the best model when a population grows but available resources cause its growth to slow as the population approaches a carrying capacity $K$ ?

A
$P'=rP$
B
$P'=rP\left(1-\frac{P}{K}\right)$
C
$P'=r+P$
D
$P'=K-P$

Choose one

1 point

What is the primary role of an initial condition in a differential-equation model?

A
It gives the instantaneous rate of change at every time.
B
It specifies the starting state of the system.
C
It identifies all equilibrium solutions.
D
It determines the carrying capacity.

Choose one

1 point

In a slope field for $y'=f(t,y)$ , what does an upward-tilting segment at a point indicate about a solution passing through that point?

A
The solution has reached an equilibrium.
B
The derivative is negative at that point.
C
The derivative is positive at that point.
D
The solution must be constant from that point onward.

Choose one

1 point

For the autonomous equation $y'=y(4-y)$ , what qualitative behavior is expected for a solution with $0<y(0)<4$ ?

A
It increases toward $y=4$ .
B
It decreases toward $y=0$ .
C
It remains constant at its initial value.
D
It decreases without bound.

Choose one

1 point

Why is the differential equation $\frac{dy}{dt}=2ty$ separable?

A
It is linear because the coefficient of $y$ is constant.
B
It is autonomous because it contains no time variable.
C
It is not solvable because the derivative is first order.
D
It is separable because the right-hand side factors into a function of $t$ and a function of $y$ .

True or false

1 point

True or false: In the logistic model $P'=rP\left(1-\frac{P}{K}\right)$ , growth is fastest when $P=K/2$ .

A
True
B
False

Written response

1 point

What term describes the limiting population or resource level $K$ in a logistic growth model?

Written response

1 point

A substance follows the decay model $A(t)=80e^{-0.12t}$ , where $t$ is measured in hours and $A$ in milligrams. What is the amount after $5$ hours? Give the value in milligrams rounded to one decimal place; the stated tolerance is $0.1$ mg.