Question 1

Each of the following graphs is from a logistic function   . Which one has the smallest value of b? ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)    E

Question 2

At time   minutes, the temperature of an object is     F. The temperature of the object is changing at the rate given by the differential equation   . Use Euler's Method to approximate the particular solutions of this differential equation at   . Use a step size of   . Round your answer to one decimal place. ​

Accepted Answer

A)  137.1 
B)  139.5 
C)  147.4 
D)  144.6 
E)  132.7 
A)  137.1 
B)  139.5 
C)  147.4 
D)  144.6 
E)  132.7 E

Question 3

The number of bacteria in a culture is increasing according to the law of exponential growth. After 2 hours there are 135 bacteria in the culture and after 4 hours there are 390 bacteria in the culture. Answer the following questions, rounding numerical answers to four decimal places. ​
(i) Find the initial population.
​
(ii) Write an exponential growth model for the bacteria population. Let t represent time in hours.
​
(iii) Use the model to determine the number of bacteria after 8 hours.
​
(iv) After how many hours will the bacteria count be 25,000?
​

Accepted Answer

A)  (i) 46.7341; (ii)   ; (iii) 4,566.8441; (iv) 14.1787 hr 
B)  (i) 48.8841; (ii)   ; (iii) 5,941.5613; (iv) 16.4067 hr 
C)  (i) 46.7341; (ii)   ; (iii) 3,254.11; (iv) 11.8442 hr 
D)  (i) 52.5141; (ii)   ; (iii) 8,693.0147; (iv) 18.5179hr 
E)  (i) 54.0741; (ii)   ; (iii) 11,345.4782; (iv) 20.2973 hr 
A)  (i) 46.7341; (ii)   ; (iii) 4,566.8441; (iv) 14.1787 hr 
B)  (i) 48.8841; (ii)   ; (iii) 5,941.5613; (iv) 16.4067 hr 
C)  (i) 46.7341; (ii)   ; (iii) 3,254.11; (iv) 11.8442 hr 
D)  (i) 52.5141; (ii)   ; (iii) 8,693.0147; (iv) 18.5179hr 
E)  (i) 54.0741; (ii)   ; (iii) 11,345.4782; (iv) 20.2973 hr C

Question 4

Write and solve the differential equation that models the following verbal statement. Evaluate the solution at the specified value of the independent variable, rounding your answer to four decimal places: ​
The rate of change of Q is proportional to Q. When   ,   and when   ,   . What is the value of Q when   ?
​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

Determine whether the function   is homogeneous and determine its degree if it is. ​

Accepted Answer

A)  homogeneous, the degree is 3 
B)  homogeneous, the degree is 4 
C)  not homogeneous 
D)  homogeneous, the degree is 2 
E)  homogeneous, the degree is 1 
A)  homogeneous, the degree is 3 
B)  homogeneous, the degree is 4 
C)  not homogeneous 
D)  homogeneous, the degree is 2 
E)  homogeneous, the degree is 1

Question 6

The logistic function   models the growth of a population. Determine when the population reaches one-half of the maximum carrying capacity. Round your answer to three decimal places.

Accepted Answer

A)  0.402 
B)  3.000 
C)  0.677 
D)  7.500 
E)  2.000 
A)  0.402 
B)  3.000 
C)  0.677 
D)  7.500 
E)  2.000

Question 7

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. ​   ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 8

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ​   ,   ​

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 9

The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is   years, what is the annual rate? Round your answer to two decimal places. ​

Accepted Answer

A)  7.30 % 
B)  7.70 % 
C)  13.71 % 
D)  6.10 % 
E)  8.70 %
​ 
A)  7.30 % 
B)  7.70 % 
C)  13.71 % 
D)  6.10 % 
E)  8.70 %
​

Question 10

The logistic function   models the growth of a population. Identify the value of k.

Accepted Answer

A)  1.7 
B)  2.2 
C)  0.2 
D)  20 
E)  0.5 
A)  1.7 
B)  2.2 
C)  0.2 
D)  20 
E)  0.5

Question 11

Match the logistic differential equation and initial condition with the graph of its solution shown below. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

Use integration to find a general solution of the differential equation   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 13

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. ​   ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 14

Find the general solution of the differential equation   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 15

A 300-gallon tank is half full of distilled water. At time   , a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 6 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 4 gallons per minute. At the time the tank is full, how many pounds of concentrate will it contain? Round your answer to two decimal places.

Accepted Answer

A)    lbs 
B)    lbs 
C)    lbs 
D)    lbs 
E)    lbs 
A)    lbs 
B)    lbs 
C)    lbs 
D)    lbs 
E)    lbs

Question 16

Solve the differential equation. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 17

A 300-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time   distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as   . ​

Accepted Answer

A)  30 
B)  56 
C)  55 
D)  0 
E)  1 
A)  30 
B)  56 
C)  55 
D)  0 
E)  1

Question 18

Find the orthogonal trajectories of the family   . ​   ​

Accepted Answer

A)      
B)      
C)      
D)      
E)      
A)      
B)      
C)      
D)      
E)

Question 19

Find the function   passing through the point   with the first derivative   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 20

A conservation organization releases 20 wolves into a preserve. After 2 years, there are 35 wolves in the preserve. The preserve has a carrying capacity of 125. Determine the time it takes for the population to reach 80.

Accepted Answer

A)  9.020 years 
B)  4.875 years 
C)  6.259 years 
D)  3.692 years 
E)  7.884 years 
A)  9.020 years 
B)  4.875 years 
C)  6.259 years 
D)  3.692 years 
E)  7.884 years

Each of the following graphs is from a logistic function . Which one has the smallest value of b?

At time minutes, the temperature of an object is F. The temperature of the object is changing at the rate given by the differential equation . Use Euler's Method to approximate the particular solutions of this differential equation at . Use a step size of . Round your answer to one decimal place.

Write and solve the differential equation that models the following verbal statement. Evaluate the solution at the specified value of the independent variable, rounding your answer to four decimal places: The rate of change of Q is proportional to Q. When , and when , . What is the value of Q when ?

Determine whether the function is homogeneous and determine its degree if it is.

The logistic function models the growth of a population. Determine when the population reaches one-half of the maximum carrying capacity. Round your answer to three decimal places.

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ,

The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is years, what is the annual rate? Round your answer to two decimal places.

The logistic function models the growth of a population. Identify the value of k.

Match the logistic differential equation and initial condition with the graph of its solution shown below.

Use integration to find a general solution of the differential equation .

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.

Find the general solution of the differential equation .

Solve the differential equation.

A 300-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as .

Find the orthogonal trajectories of the family .

Find the function passing through the point with the first derivative .

A conservation organization releases 20 wolves into a preserve. After 2 years, there are 35 wolves in the preserve. The preserve has a carrying capacity of 125. Determine the time it takes for the population to reach 80.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Exam 6: Differential Equations

Each of the following graphs is from a logistic function . Which one has the smallest value of b? ​

Determine whether the function is homogeneous and determine its degree if it is. ​

The logistic function models the growth of a population. Determine when the population reaches one-half of the maximum carrying capacity. Round your answer to three decimal places.

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. ​ ​ ​

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ​ , ​

The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is years, what is the annual rate? Round your answer to two decimal places. ​

The logistic function models the growth of a population. Identify the value of k.

Match the logistic differential equation and initial condition with the graph of its solution shown below. ​ ​

Use integration to find a general solution of the differential equation . ​

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. ​ ​ ​

Find the general solution of the differential equation . ​

Solve the differential equation. ​ ​

A 300-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as . ​

Find the orthogonal trajectories of the family . ​ ​

Find the function passing through the point with the first derivative .

A conservation organization releases 20 wolves into a preserve. After 2 years, there are 35 wolves in the preserve. The preserve has a carrying capacity of 125. Determine the time it takes for the population to reach 80.

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Each of the following graphs is from a logistic function . Which one has the smallest value of b?

Determine whether the function is homogeneous and determine its degree if it is.

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.

Use Euler's Method to make a table of values for the approximate solution of the following differential equation with specified initial value. Use 5 steps of size 0.05. ,

The initial investment in a savings account in which interest is compounded continuously is $803. If the time required to double the amount is years, what is the annual rate? Round your answer to two decimal places.

Match the logistic differential equation and initial condition with the graph of its solution shown below.

Use integration to find a general solution of the differential equation .

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.

Find the general solution of the differential equation .

Solve the differential equation.

A 300-gallon tank is full of a solution containing 55 pounds of concentrate. Starting at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as .

Find the orthogonal trajectories of the family .