Question 1

Write the following second-order equation as a system of first-order equations.

Accepted Answer

The answer of Write the following second-order equation as a...

Question 2

Find all equilibrium points for the following coupled equations. Identify each equilibrium point as stable or unstable.

Accepted Answer

A)  (0, 0)unstable; (1.33, 1.25)stable; (0.60, 0)unstable; (0, 1.10)unstable 
B)  (0, 0)unstable; (0, 0.20)unstable; (0.50, 0)unstable; (0.60, 0.20)stable 
C)  (0, 0)unstable; (0, 0.20)unstable; (0.60, 0)unstable; (1.33, 1.25)stable 
D)  (0, 0)unstable; (0, 1.25)unstable; (1.33, 0)unstable; (0.60, 1.10)stable 
A)  (0, 0)unstable; (1.33, 1.25)stable; (0.60, 0)unstable; (0, 1.10)unstable 
B)  (0, 0)unstable; (0, 0.20)unstable; (0.50, 0)unstable; (0.60, 0.20)stable 
C)  (0, 0)unstable; (0, 0.20)unstable; (0.60, 0)unstable; (1.33, 1.25)stable 
D)  (0, 0)unstable; (0, 1.25)unstable; (1.33, 0)unstable; (0.60, 1.10)stable D

Question 3

Consider a chemical system containing species A, B, and C; and that A and B can react to make C in a bimolecular reaction with rate constant of k1, and C can decompose to make A and B in a first order reaction with rate constant of k-1. If the instantaneous amounts of A, B, and C are represented as a, b, and c, and the initial amounts are given as A0, B0, and C0, the change in C can be represented with the differential equation   . If A0 = 0 , B0 = 1, C0 = 5, k1 = 0.02 s -1, and k-1 = 0.04 s -1, how much C is present after 15 seconds? [Note: c cannot be larger than C0 plus the smaller of A0 or B0. Nor can it be smaller than 0.]

Accepted Answer

A)  8.36 
B)  3.33 
C)  4.81 
D)  2.73 
A)  8.36 
B)  3.33 
C)  4.81 
D)  2.73 B

Question 4

Find the solution of the differential equation, y' = -3y, satisfying the condition, y(2) = 1.

Accepted Answer

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

Question 5

Match the appropriate slope field with the differential equation   .

Accepted Answer

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

Question 6

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation   ,   .

Accepted Answer

A)  1.38571, 0.58826 
B)  1.26381, 0.54967 
C)  1.61500, 0.13000 
D)  1.42489, 0.58854 
A)  1.38571, 0.58826 
B)  1.26381, 0.54967 
C)  1.61500, 0.13000 
D)  1.42489, 0.58854

Question 7

The Polymerase Chain Reaction (PCR) is used to replicate segments of DNA. It is used to make DNA samples big enough for testing, starting from very small samples collected, for instance, from a crime scene. PCR can double the number of a particular DNA segment every two minutes. If one wants a DNA sample with   copies of a particular segment, how long must the PCR process be carried out to produce them? Assume that there is just one segment in the original sample.

Accepted Answer

A)  133 minutes 
B)  66 minutes 
C)  23 minutes 
D)    minutes 
A)  133 minutes 
B)  66 minutes 
C)  23 minutes 
D)    minutes

Question 8

Write the following second-order equation as a system of first-order equations.

Accepted Answer

The answer of Write the following second-order equation as a...

Question 9

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation   ,   .

Accepted Answer

A)  -3.5735, -4.6954 
B)  -3.9045, -4.7390 
C)  -3.5001, -4.5997 
D)  -3.4735, -4.5954 
A)  -3.5735, -4.6954 
B)  -3.9045, -4.7390 
C)  -3.5001, -4.5997 
D)  -3.4735, -4.5954

Question 10

Calculate how much you would need to invest now in order to fund a year of college twenty years from now, assuming a year of college costs $19,000 now and is inflating at 7%, and your investment will earn 11%. Assume continuous compounding.

Accepted Answer

A)  $8537 
B)  $2105 
C)  $4685 
D)  $42,285 
A)  $8537 
B)  $2105 
C)  $4685 
D)  $42,285

Question 11

Use the following direction field to identify the stability of the equilibrium point (0.57, 0.14).

Accepted Answer

A)  Stable 
B)  Unstable 
A)  Stable 
B)  Unstable

Question 12

$20,000 that was invested in 1990 was worth $147,740 in 2000. What annual interest rate did the investment earn in that 10 year period? Assume continuous compounding.

Accepted Answer

A)  119.03% 
B)  12.02% 
C)  99.03% 
D)  20.00% 
A)  119.03% 
B)  12.02% 
C)  99.03% 
D)  20.00%

Question 13

Identify the equilibrium solutions for   , for   , and determine if they are stable or unstable.

Accepted Answer

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

Question 14

Write the following third-order equation as a system of equations.

Accepted Answer

The answer of Write the following third-order equation as a...

Question 15

Use the following direction field to identify the stability of the equilibrium point (0.50, 0.84).

Accepted Answer

A)  Stable 
B)  Unstable 
A)  Stable 
B)  Unstable

Question 16

A bank offers to sell a bank note that will reach a maturity value of $15,000 in 9 years. How much should you pay for it now if you wish to receive an 8% return on your investment?

Accepted Answer

A)  $1543.21 
B)  $7301.28 
C)  $13,800.00 
D)  $1533.33 
A)  $1543.21 
B)  $7301.28 
C)  $13,800.00 
D)  $1533.33

Question 17

Find the solution of the differential equation, y' = -2y, satisfying the initial condition, y(-5) = 5.

Accepted Answer

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

Question 18

The differential equation is separable. Find the general solution in an explicit form.

Accepted Answer

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

Question 19

Use the direction field below to sketch a solution curve and estimate the initial value y(0) for the differential equation   , such that the solution curve passes through the point (-1,-2).

Accepted Answer

The answer of Use the direction field below to sketch...

Question 20

Find the solution of the differential equation, y' = 3y, satisfying the initial condition, y(0) = 1.

Accepted Answer

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

Write the following second-order equation as a system of first-order equations.

Find all equilibrium points for the following coupled equations. Identify each equilibrium point as stable or unstable.

Find the solution of the differential equation, y' = -3y, satisfying the condition, y(2) = 1.

Match the appropriate slope field with the differential equation .

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation , .

Write the following second-order equation as a system of first-order equations.

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation , .

Calculate how much you would need to invest now in order to fund a year of college twenty years from now, assuming a year of college costs $19,000 now and is inflating at 7%, and your investment will earn 11%. Assume continuous compounding.

Use the following direction field to identify the stability of the equilibrium point (0.57, 0.14).

$20,000 that was invested in 1990 was worth $147,740 in 2000. What annual interest rate did the investment earn in that 10 year period? Assume continuous compounding.

Identify the equilibrium solutions for , for , and determine if they are stable or unstable.

Write the following third-order equation as a system of equations.

Use the following direction field to identify the stability of the equilibrium point (0.50, 0.84).

A bank offers to sell a bank note that will reach a maturity value of $15,000 in 9 years. How much should you pay for it now if you wish to receive an 8% return on your investment?

Find the solution of the differential equation, y' = -2y, satisfying the initial condition, y(-5) = 5.

The differential equation is separable. Find the general solution in an explicit form.

Use the direction field below to sketch a solution curve and estimate the initial value y(0) for the differential equation , such that the solution curve passes through the point (-1,-2).

Find the solution of the differential equation, y' = 3y, satisfying the initial condition, y(0) = 1.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

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Exam 9: First-Order Differential Equations

Write the following second-order equation as a system of first-order equations.

Find all equilibrium points for the following coupled equations. Identify each equilibrium point as stable or unstable.

Find the solution of the differential equation, y' = -3y, satisfying the condition, y(2) = 1.

Match the appropriate slope field with the differential equation .

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation , .

Write the following second-order equation as a system of first-order equations.

Use Euler's method with h = 0.1 to approximate y(1.0) and y(2.0) for the differential equation , .

Calculate how much you would need to invest now in order to fund a year of college twenty years from now, assuming a year of college costs $19,000 now and is inflating at 7%, and your investment will earn 11%. Assume continuous compounding.

Use the following direction field to identify the stability of the equilibrium point (0.57, 0.14).

$20,000 that was invested in 1990 was worth $147,740 in 2000. What annual interest rate did the investment earn in that 10 year period? Assume continuous compounding.

Identify the equilibrium solutions for , for , and determine if they are stable or unstable.

Write the following third-order equation as a system of equations.

Use the following direction field to identify the stability of the equilibrium point (0.50, 0.84).

A bank offers to sell a bank note that will reach a maturity value of $15,000 in 9 years. How much should you pay for it now if you wish to receive an 8% return on your investment?

Find the solution of the differential equation, y' = -2y, satisfying the initial condition, y(-5) = 5.

The differential equation is separable. Find the general solution in an explicit form.

Use the direction field below to sketch a solution curve and estimate the initial value y(0) for the differential equation , such that the solution curve passes through the point (-1,-2).

Find the solution of the differential equation, y' = 3y, satisfying the initial condition, y(0) = 1.

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters