FiltersDone

Question type

Essay

Multiple Choice

Short Answer

True False

Matching

Exam 8: First-Order Differential Equations

Show Answer

Share

---

All types

Filters

Study FlashcardsPractice ExamLearn

Question 31

Essay

Review LaterAdd Note

Review Later

An object falling freely through the atmosphere will accelerate due to the force of gravity at 9.86 m/s² [Acceleration is the derivative of velocity with respect to time.] The atmosphere, however, will commonly exert a retarding force proportional to the object's velocity squared. The proportionality constant will depend largely on the object's shape. Write and solve the differential equation, then identify a free-falling object's terminal velocity (the limiting velocity) in terms of its drag proportionality constant, k. [To solve explicitly, you may use the initial condition v(0) = 0.]

Correct Answer

verified

Taking the limit a...

View Answer

Show Answer

Question 32

Multiple Choice

Review LaterAdd Note

Review Later

Find the solution of the differential equation, y' = -y, satisfying the condition, y(2) = -4.

A)
B)
C)
D)

Correct Answer

verified

Show Answer

Question 33

Multiple Choice

Review LaterAdd Note

Find the solution of the differential equation, y' = 5y, satisfying the initial condition, y(0) = -4.

An object's cooling constant, k, is often taken to be proportional to the its surface area. An iron ball cools from 200^oC to 100^oC in 50 seconds in a fast flowing stream of 30^oC water. Use Newton's Law of Cooling to estimate how cool a similar volume of hot iron would become in 50 seconds if it were in the shape of a rod with surface area 4 times larger than the ball's.

Construct a direction field for the differential equation.

Is the following differential equation separable or not?

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

A volume discount on a certain item is expressed as a differential equation in which the derivative of the price per item with respect to the number purchased is proportional to the difference between the price and some base price, below which the price can never go. If the price for just one item is $60, and the base price is $30, and the price per item when buying 10 items is $54, what is the price per item when buying 100 items?

The number of bacteria in a culture increases exponentially with a growth constant of 0.2 hour^-1. How long will it take for the population to increase from 4000 to 20,000 ?

Find all equilibrium points for the following system of equations.

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' = -y, satisfying the initial condition, y(0) = -5.

Find the solution of the differential equation, y' = -y, satisfying the initial condition, y(4) = 3.

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.

A treasury note that will be worth $100,000 in 25 years currently sells for $48,432. What constant interest rate does that correspond to?

You select an insurance policy for your company that agrees to pay you the depreciated value of a piece of equipment if it is destroyed in an accident, but you have to choose one of two depreciation schedules when you begin the policy. One depreciation schedule is linear and depreciates the value to zero over 10 years. The other depreciation schedule is exponential and depreciates the value at a constant rate of 14% per year. An accident destroys the equipment after 6 years of ownership. Calculate the insurance company's obligation for each plan if the piece of equipment was valued initially at $100,000. Which depreciation schedule would have paid the most? (The cost of the policy is the same, no matter which depreciation schedule you choose.)

A bank offers to sell a bank note that will reach a maturity value of $12,000 in 12 years. How much should you pay for it now if you wish to receive an 8% return on your investment?

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

At the cafeteria, two identical glasses of juice were poured at the same time and put on the counter waiting for customers to take them. Their temperatures were 33^oF when they were poured, and the cafeteria was a stable 72^oF. One glass was 40^oF when it was taken after 40 seconds. What was the temperature of the other when it was taken after 200 seconds?

Solve the following initial value problem explicitly.