Mobius 4 assignment PDF

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Mobius 4 assignment...

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2/16/2020

University of Ottawa -

Assignment Worksheet

Online Homework System

2/16/20 - 2:55:48 PM EST

Name:

____________________________

Class:

MAT 1332 - Winter 2020 - All sections

Class #:

____________________________

Section #:

____________________________

Instructor: Benoit Dionne

Assignment: Assignment 4

Question 1: (1 point) For which values of

does the function

satisfy the differential equation

List all the values in the textbox below, separating multiple entries by a semi-colon.

Question 2: (1 point) Find the solution to the differential equation

with the initial condition Answer:

__________

(If needed: Recall that in MapleTA, we write

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for

.)

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Question 3: (1 point) A bacterial culture starts with

bacteria and grows at a rate proportional to its numbers. It has already

(i) Find an expression for the number of bacteria after __________ bacteria. Give an exact formula (do not round the exponents).

(ii) Calculate the number of bacteria after Answer: ____________ bacteria. Round the answer to the nearest integer.

bacteria after

hours.

hours.

hours.

(iii) What is the growth rate of the population after Answer: ____________ bacteria per hour. Round the answer to the nearest integer.

hours?

(iv) How long will it take for the number of bacteria to reach Answer: ____________ hours. Round the answer to the nearest integer.

units?

Question 4: (1 point) A bacterial culture starts with a certain number of bacteria and expands at a rate proportional to the number of bacteria. a) If is the number of bacteria at time measured in hours, write a differential equation describing the behaviour of will involve a constant representing the proportional relation, and the variable .

. Your answer

__________ b) Solve the differential equation above to find the number of bacteria as a function of the time . Your solution should depend on initial number of bacteria.

and the

__________ c) Given that after

hours, the number of bacteria has tripled, find the exact value of .

__________ d) With the value of

you found in (c), what can be said about the following ratios

____________ ,

____________ and in general ____________ e) Still in the specific case discussed in (c), how long does it take for the number of bacteria to be

times bigger?

Answer: ____________

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Question 5: (1 point) A pork roast is removed from an oven at . Its initial temperature is . The roast is left on the kitchen counter until , at which point it is put into the refrigerator; at that time its temperature is . When the roast is taken out of the refrigerator, at , its final temperature is . The room temperature is and the refrigerator is kept at a temperature of . In order to answer the following questions, let us assume that the temperature of the roast follows Newton’s Law of Cooling while on the counter as well as in the fridge, but for different constants.

(i) What is the temperature of the roast at Answer: ____________ C

? Give the answer with

precision.

(ii) What is the temperature of the roast at Answer: ____________ C

? Give the answer with

precision.

Question 6: (1 point) Some biologists decided to seed a lake with trout. They estimated the carrying capacity of the lake to be year they noticed that the population of trout had tripled.

(i) Assuming the population follows the logistic model, calculate the relative growth rate ____________ . Round your answer to decimal places.

(ii) After how many years did the population in the lake reach Answer: ____________ years. Round the answer to decimal places.

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trout. By the end of

for an unconstrained environment.

trout?

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Question 7: (1 point) Solve the initial-value problem shown below:

Give an exact formula for

.

__________

Question 8: (1 point) Solve the initial-value problem shown below:

Give an exact formula for __________

valid for

(such that

).

Question 9: (1 point)

A patch of moss initially occupies an area of hours, the patch has grown to

. The size of the patch of moss grows at a rate inversely proportional to its size. After

.

a) If the size of the patch of moss at time is

, the differential equation describing the growth of the patch of moss is

__________ , where your solution will depend on the constant of proportionality

that will be determined later.

b) Find the general solution of the differential equation given in (a). __________ Your solution will depend on

and the arbitrary constant of integration

.

c) Use the information given in the question to determine the exact values of

and

for thin problem. Then, write the particular solution.

__________ d) What is the size of the patch after

hours? Give your answer with a precision of at least three decimal places.

Answer: ____________ e) How long does it take for the size of the patch of moss to reach places.

? Give your answer with a precision of at least three decimal

Answer: ____________

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Question 10: (1 point) We want to solve the differential equation

a) This differential equation is separable and can be writen.

where __________ and __________ b) To compute the integral

, you need to use the substitution

__________ With this substitution, we get

, where

__________ We find that __________ and hence __________

,

where we have added the constant of integration c) To compute the integral

for you; so you don't need to add one in your answer.

, you need to use integration by parts with

__________ and __________ With this choice, we get

where __________ and __________ We then find that __________ where we have added the constant of integration d)Solve

for

satisfying the initial condition

for you; so you don't need to add one in your answer.

to find the general solution of the given differential equation. Then, give the particular solution

.

__________

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