MAT9004

Assignment 1 School of Mathematics

Monash University

Present your solution so that the sequence of logical steps is clear, with succinct justfication for each step where it is not obvious. Note that marks will be deducted for solutions that do not show all the important logical steps and reasoning in reaching a conclusion. Answers should consist of complete sentences with a mixture of English words and mathematical symbols. Marks can also be deducted if solutions are much more convoluted than required. Handwriting must be legible and all sketches must be neat.

It is your responsibility to verify that what you upload is a complete and legible scan of your work. Assignments received after the due date will receive a late penalty, so we encourage you not to plan to submit last minute. Extensions are only given in exceptional cases, please consult Moodle and the special consideration guidelines.

This assignment has 60 marks and is worth 30% of the final unit mark.

Question 1.1 (15 marks)

Answer the following and explain in full detail why your answer is correct.

a) Is the function f : [0,8) ? [0,8) with f(x) = 4x4 + 12x2 + 9 bijective?

b) Let E = {0,±3,±6,...} be the set of integers that are divisible by 3 and let T = {0,±5,±10,...} be the set of integers that are divisible by 5. Is the function f : E ? T with f(x) = 5x/3 + 10 bijective?

c) Is the function f : [-2,2] ? [-4,4] with f(x) = x3 - 2x bijective?

d) Is there a bijective function f : [-1,1] ? [-50,50]?

e) If a convex function f : R?R satisfies f(10) = -4 and f(20) = 30, what’s the smallest possible value for f(7)?

Question 1.2 (8 marks)

A city experiences outbreaks of two variants of a virus. The psi variant outbreak begins with 4 people infected and grows exponentially so that every week the number of people infected is 1.5 times the number the week before. The omega variant outbreak begins exactly 2 weeks after the psi variant outbreak with 3 people infected and grows exponentially so that every week the number of people infected doubles. This continues for 8 weeks after the first psi infection. (For the purposes of our model we allow the number of people infected to take non-integer values.)

a) How many people have each of the variants 8 weeks after the first psi variant infection?

b) The city institutes a lockdown exactly 8 weeks after the first psi infection. During the lockdown the number of omega cases decreases exponentially so that every week the number of people infected is times the number the week before. How long must the lockdown last before the omega variant is

eliminated (assume this happens when the number of infected people reaches 1).

c) At some point before the lockdown the number of people infected by each variant was equal. Exactly how long from the first psi infection did this occur?

Question 1.3 (9 marks)

A food company creates a type of mushroom with two species called moosh and room. To create enough mushroom, they add b kg of moosh and z kg of room, which then start to grow continuously such that every three days the number of moosh doubles and every four days the number of room triples.

a) When starting with b = 4, how many kg of moosh are in the mixture after 6 days?

b) When starting with z = 6, how many kg of room are in the mixture after 8 days?

c) The company starts with b = 4 and z = 3. To make sure that the final product is perfect (and customers don’t get sick), it is required that there are the same amount of room as moosh. If the mushroom production starts Monday 6:00am, at what day and time is this achieved?

Question 1.4 (12 marks)

Determine the following derivatives and explain in detail what rule you have used in each of your steps when computing f0(x).

Note: The aim of this exercise is to demonstrate that you can apply these rules correctly; in later assignments it will not be required to give the same level of justification.

a) Compute f0(x) for f(x) = (2x2 + 3ex + 5)10.

b) Compute .

c) Compute f0(x) for f(x) = 102x2+x+10.

Question 1.5 (8 marks)

You run a coffee shop. At present you sell coffees for $4.00 and average 34 sales per hour. For every $1 increase (decrease) in your price you will average 20 fewer (more) sales per hour. For example, at a price of $4.50 you will average 24 sales per hour. Assume you cannot drop the price below the point where you average 40 sales per hour (you will be too rushed) and cannot raise the price above the point where you average 0 sales per hour.

a) Find the function f(x) that outputs your average hourly revenue when you charge x dollars for a coffee. Be sure to give a suitable domain for f.

b) By finding the global maximum of f, determine what coffee price maximises your average hourly revenue and what this maximum average hourly revenue is.

c) Now suppose that you have the option of hiring an assistant barista. Doing this will allow you to drop prices below the point where you average 40 sales per hour (but not below $0, of course). Now what coffee price maximises your average hourly revenue and what is this maximum average hourly revenue?

Question 1.6 (8 marks)

A pizza company is experimenting with a new yeast, called KABOOM. By adding x grams of KABOOM to each kilogram of their dough, the rise of the dough can be increased by a factor of f(x), where f(x) = 2x3 - 9x2 + 12x + 3. However, if they add more than 2.4gram per kilogram, then the dough becomes inedible.

a) Determine the stationary points of f(x) on (-8,8).

b) Classify each of the stationary points determined in a): demonstrate this in two ways, firstly, by using the second derivative of f, secondly, by looking at the signs of f0(x).

c) What’s the optimal value for x such that the dough remains edible and has the best rise?

Assignment 1 School of Mathematics

Monash University

Present your solution so that the sequence of logical steps is clear, with succinct justfication for each step where it is not obvious. Note that marks will be deducted for solutions that do not show all the important logical steps and reasoning in reaching a conclusion. Answers should consist of complete sentences with a mixture of English words and mathematical symbols. Marks can also be deducted if solutions are much more convoluted than required. Handwriting must be legible and all sketches must be neat.

It is your responsibility to verify that what you upload is a complete and legible scan of your work. Assignments received after the due date will receive a late penalty, so we encourage you not to plan to submit last minute. Extensions are only given in exceptional cases, please consult Moodle and the special consideration guidelines.

This assignment has 60 marks and is worth 30% of the final unit mark.

Question 1.1 (15 marks)

Answer the following and explain in full detail why your answer is correct.

a) Is the function f : [0,8) ? [0,8) with f(x) = 4x4 + 12x2 + 9 bijective?

b) Let E = {0,±3,±6,...} be the set of integers that are divisible by 3 and let T = {0,±5,±10,...} be the set of integers that are divisible by 5. Is the function f : E ? T with f(x) = 5x/3 + 10 bijective?

c) Is the function f : [-2,2] ? [-4,4] with f(x) = x3 - 2x bijective?

d) Is there a bijective function f : [-1,1] ? [-50,50]?

e) If a convex function f : R?R satisfies f(10) = -4 and f(20) = 30, what’s the smallest possible value for f(7)?

Question 1.2 (8 marks)

A city experiences outbreaks of two variants of a virus. The psi variant outbreak begins with 4 people infected and grows exponentially so that every week the number of people infected is 1.5 times the number the week before. The omega variant outbreak begins exactly 2 weeks after the psi variant outbreak with 3 people infected and grows exponentially so that every week the number of people infected doubles. This continues for 8 weeks after the first psi infection. (For the purposes of our model we allow the number of people infected to take non-integer values.)

a) How many people have each of the variants 8 weeks after the first psi variant infection?

b) The city institutes a lockdown exactly 8 weeks after the first psi infection. During the lockdown the number of omega cases decreases exponentially so that every week the number of people infected is times the number the week before. How long must the lockdown last before the omega variant is

eliminated (assume this happens when the number of infected people reaches 1).

c) At some point before the lockdown the number of people infected by each variant was equal. Exactly how long from the first psi infection did this occur?

Question 1.3 (9 marks)

A food company creates a type of mushroom with two species called moosh and room. To create enough mushroom, they add b kg of moosh and z kg of room, which then start to grow continuously such that every three days the number of moosh doubles and every four days the number of room triples.

a) When starting with b = 4, how many kg of moosh are in the mixture after 6 days?

b) When starting with z = 6, how many kg of room are in the mixture after 8 days?

c) The company starts with b = 4 and z = 3. To make sure that the final product is perfect (and customers don’t get sick), it is required that there are the same amount of room as moosh. If the mushroom production starts Monday 6:00am, at what day and time is this achieved?

Question 1.4 (12 marks)

Determine the following derivatives and explain in detail what rule you have used in each of your steps when computing f0(x).

Note: The aim of this exercise is to demonstrate that you can apply these rules correctly; in later assignments it will not be required to give the same level of justification.

a) Compute f0(x) for f(x) = (2x2 + 3ex + 5)10.

b) Compute .

c) Compute f0(x) for f(x) = 102x2+x+10.

Question 1.5 (8 marks)

You run a coffee shop. At present you sell coffees for $4.00 and average 34 sales per hour. For every $1 increase (decrease) in your price you will average 20 fewer (more) sales per hour. For example, at a price of $4.50 you will average 24 sales per hour. Assume you cannot drop the price below the point where you average 40 sales per hour (you will be too rushed) and cannot raise the price above the point where you average 0 sales per hour.

a) Find the function f(x) that outputs your average hourly revenue when you charge x dollars for a coffee. Be sure to give a suitable domain for f.

b) By finding the global maximum of f, determine what coffee price maximises your average hourly revenue and what this maximum average hourly revenue is.

c) Now suppose that you have the option of hiring an assistant barista. Doing this will allow you to drop prices below the point where you average 40 sales per hour (but not below $0, of course). Now what coffee price maximises your average hourly revenue and what is this maximum average hourly revenue?

Question 1.6 (8 marks)

A pizza company is experimenting with a new yeast, called KABOOM. By adding x grams of KABOOM to each kilogram of their dough, the rise of the dough can be increased by a factor of f(x), where f(x) = 2x3 - 9x2 + 12x + 3. However, if they add more than 2.4gram per kilogram, then the dough becomes inedible.

a) Determine the stationary points of f(x) on (-8,8).

b) Classify each of the stationary points determined in a): demonstrate this in two ways, firstly, by using the second derivative of f, secondly, by looking at the signs of f0(x).

c) What’s the optimal value for x such that the dough remains edible and has the best rise?

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