[SOLVED] MAT135 Differential Calculus – Fall 2024 Assignment 9 SQL

MAT135 – Differential Calculus – Fall 2024

Written Assignment

Clarified MAT135 Assignment 9 (Written Component)

Due 1 December 2024, at 9pm

More details

0. What’s this new version? Students expressed some confusion with Q1(a) and Q1(b), and in retrospect Q1(a) is worded in a way that really does not seem to need a theorem. This version clarifies those things. All changes are in red.

1. Deadline: The deadline to submit this assignment is strict, to the second. Assignments that are even a few seconds late will normally receive a grade of 0. Technical issues are not a valid reason to be late, so you are taking a risk if you leave uploading your solutions to the last minute. Please give yourself lots of time to upload your assignment!

2. Purpose and feedback: The purpose of the written assignments is to give you some practice in thinking about and writing solutions to mathematical problems, without anytime pressure. You will receive feedback on your writing and on your solutions. You are encouraged to take this opportunity to carefully write your solutions and think about how to best present your reasoning behind them.

3. Writing solutions: Explain all your work, show your steps as well as your reasoning. You should write in words what you are doing and why.  The person reading your solution should easily be able to understand what you have done, because you explained it, in words. Submissions with little or no written explanations will not receive full marks.

4. Uploading: A handwritten assignment can be photographed or scanned.  It is important that the images (the scans or photos) are clear and easy to read (not too dark, too bright, too blurry, etc.). Your submitted files are what will be marked, so if the grader(s) can’t read them or can’t make sense of what you’ve written, you will not get full marks.

You may also write your solutions on a tablet, or type your solutions (LaTeX only please) as long as your solutions are clear, easy to read and follow, and are all your own work.

Make sure you upload solutions to each problem to the correct place. If you upload solutions to the wrong problems, or uploadallyour solutions to the same problem, it may result in getting0on all problemsuploaded to the wrong places.

5. Grading: The assignment is out of 10 marks, with the marks for each question indicated beside it.

Marks for each solution will depend on your answer, as well as the quality of your explanation of those answers.

6. Academic Integrity: The solutions that you submit to this assignment must be all your own work. By submitting this assignment you declare that:

(a)  Your solutions are all your own work, explained in your own words.

(b)  You have not copied any part of the assignment solutions from anyone or anywhere.

(c)  You have not let anyone else copy any part of your solutions to the assignment.

(d)  You have not used Generative AI (e.g., ChatGPT) for any part of the assignment.

Background

Several months have passed since the aliens arrived. The world’s leading scientists are jealous that the aliens continue to seek the help of MAT135 students at UTM, even after 10% of the class tried to kill Estra on Written Assignment 6.

As incredible as their time here with us has been, the aliens have begun making preparations to leave Earth and head back to their home planet. Despite all their incredible technological advancements, they still have some basic problems that first-year calculus students (and only first-year calculus students) can help them solve.

Problems

1.  (3 points) Estra, who you’ve come to learn is the alien ship’s navigator, is planning their route and is concerned about their fuel levels for their journey home. The alien ship uses a mysterious fuel called orsh, a substance unfamiliar to humanity.

Estra estimates that the regular functions of their ship while it sits at UTM (lights, communications, hydroponic fruit gar- dens, and so on) consume orsh at a rate between 4000 and 5000 litres per day.  The ship’s fuel gauge says there are currently 300000 litres of orsh on board, but Estra tells you that the fuel gauge’s reading can be off by as much as 3000 litres (they’ve been meaning to get it fixed, but it’s so hard to find a good mechanic…).

You may assume that the amount of orsh in the ship’s tank is a differentiable function of time.

(a)  (2 points) Assuming Estra’s estimates are correct, and accounting for the potential error from the orsh gauge, what

is the range of possible amounts of orsh that might be left in the ship’s tank after two weeks of normal operation?   State your final answer as an interval (like [a,b]). Explain your answer, and justify the use of any theorems you use.

Clarification: Due to the wording of the information about orsh usage given above (it’s given as range of litres used per day, rather than a direct statement about a derivative) it is possible to interpret this question in a way that can be solved without any calculus theorems. Our original intention with this question was to mirror examples from LEC 27 (it’s virtually the same as those examples in all but its numbers.) We will accept answers of either form.

(b)  (1point)Estra believes thejourney to theirhomeplanetwill require no lessthan200000 litres oforsh. Again, assuming their estimate for the ship’s orsh consumption is correct (and still accounting for the gauge’s error), how many more

full days can the ship remain at UTM before it no longer has enough orsh to make the journey home?    State your final answer as an integer. Explain your answer, and justify the use of any theorems you use.

Clarification: Everything mentioned for part (a) also applies here. In addition, we’d like to clarify that this question is asking for how many full days the aliens can stay at UTM starting from now (when the gauge reads 300000). Many students mistakenly thought we were asking for how many days after the two weeks mentioned in part (a).

2.  (3 points) The aliens want to stay longer than your estimate from Q1(b) allows, so their chief scientist Genly has come up with a way of converting water into orsh. He says that by applying a great deal of pressure to water, the aliens can obtain many litres of orsh from a single litre of water, with the exact amount varying as a function of the pressure.

Genly’s first version of this process produces orsh according to the following function, where G(P) is the number of litres of orsh produced by applying a pressure P > 0 (measured in kilopascals, kP) to a litre of water:

G(P) = 10+(P−10)2/5P2

The aliens can apply very large amounts of pressure to the water, but need your help to determine what to do.

(a) (Warm–up! Do not submit this part to Crowdmark.) Check that G(10) = 50. What are the units for this quantity? Compute the value of G for some other values of P. What do you think is happening for large values of P?

(b)  (1 point) Show with a limit that increasing the pressure P to very, very large levels doesn’t produce more orsh per litre of water.

(c)  (2 points) Find the ideal pressure Pi  at which this process is most efficient (i.e,.  at which it produces the largest quantity of orsh per litre of water). How much orsh is produced (per litre of water) at the pressure Pi?

Clearly state your two answers at the beginning of your solution, and show your work below.  Don’t forget to justify why your answer is where this maximum occurs.

3.  (2 points) Genly’s first version of the process was good, but he is confident he can do better.

He spendsanother day refining his ideas, and comes up with a new, adjustable version of his process that produces Gb(P) litres of orsh per litre of water, where

Gb(P) = 10+(P−b)2/5P2,

where b is a positive constant that Genly can manipulate with his laboratory equipment. We’ve used the name Gb above so it’s clear that different values of b make for different functions.

(a)  (1 point) For a (general) value of b, find the ideal pressure Pb at which this process is most efficient (i.e., at which it produces the most litres of orsh per litre of water).

Clearly state your answer at the beginning of your solution, and show your work below. Don’t forget to justify why your answer is actually where this maximum occurs.

Hints: Think about whether your answer should depend on b. You’ve already found P10!

(b)  (1 point) For a (general) value of b, let O(b) be the maximum output value of Gb. Find a formula for O(b), and show that as b increases, O(b) also increases.  In other words, the maximum efficiency of this process increases as b increases.

Clearly state your formula at the beginning of your solution, and show your work below.

4.  (2 points) A rival alien scientist named Faxe has come up with their own conversion process for turning water into orsh.

Their process is described by the function

F(P) = 1+(P−2)2/P3+10.

(a)  (1 point) We already know that Genly’s processes have a high peakefficiency at a low pressure.

If the aliens are able to apply very high pressures (much higher than the values of Pb you were thinking about in Q2 and Q3), which process would be produce more orsh per litre of water?

State your answer (“Genly” or “Faxe”) at the top of your answer and, in at most two sentences and some short computations, justify your choice below.

(b)  (1 point) Another alien named Tibe works in Faxe’s laboratory and has experimented with Faxe’s process at high pressures.

Tibe says that for very high pressures, the increase in orsh production per litre of water from Faxe’s process is directly proportional to the increase in pressure (i.e., Tibe believes that F is a line for large values of P). Faxe hears this and thinks this is at best a good approximation for what happens at high pressures. The two of them come to the MAT135 students, who have just learned about different types of asymptotes in class, for some clarification.

In at most a few sentences and some computations, explain which one of Faxe or Tibe is more correct about F, and why.