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MAT135 Assignment 6

Due 6 November, 2022 at 9pm EST

Submitting the assignment

Due date: Sunday 6 November, 2022 at 9pm

Where to submit: On Crowdmark. You should receive an email with your personal link to the assignment on Crowdmark.

What to do: Write solutions to the problems on blank or lined sheets of paper, or write them digitally. For each question, take a photo, scan, or save a file of your solutions. (You may want to start each question on a new page.)

Uploading: Click on the link in the email from Crowdmark to submit your solutions. Make sure to read the instructions on Crowdmark carefully. You will be asked to upload the same solution file(s) in two places! (Both are on Crowdmark, right next to each other.)

Notes:

1. Deadline: Deadlines to submit assignments are strict. Missing the deadline by even a few seconds will mean that Crowdmark automatically gives you a zero. Please give yourself lots of time to upload your assignment!

2. Upload: A handwritten assignment can be scanned, or you can take a photo of it to submit. It is important that the images (e.g. the scans or photos) are clear and easy to read (not too dark or too bright, etc.) Your submitted files are what will be marked. If the Grader can’t read them, you will not get full marks. Note: It is also OK to write your solutions on e.g. an iPad with a pen, or to type your solutions (e.g using TEX) as long as your solutions are clear, easy to read and follow, and are all your own work.

3. Format: 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. You will be graded on your answer, as well as your explanation of your solution.

4. Grading and feedback: The assignment is out of 10 marks: −→ 8 marks for problem solutions and mathematical answers.

−→ 2 marks for the writing and organization of your solutions, and the clarity of your explanations.

Marks will depend on a combination of your answers, and your explanations . The graders may leave feedback for you on your assignment work. Please log in to Crowdmark to view the feedback. (We will announce when it is ready.)

5. Purpose: The purpose of the written assignments is to give you some practice in writing solutions to mathematical problems, without any time 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.

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

Your solutions are all you own work, explained in your own words.

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

1. Romina is an oceanographer who recently obtained ocean temperature and ocean density data from the Labrador Sea using an argo float . Her colleague, Nico, however, goes scuba-diving to obtain data from only two depths. Nico measures a water temperature of 7◦ C at a depth of 50 meters, and 3.6◦ C at a depth of 125 meters. This research group is collaborating with our MAT135 course to help them analyze their data.

(a) (2 marks) Determine a linear mathematical model for the temperature, T (in degrees Celsius), as a function of depth, z (in meters) by finding an equation of the secant line that goes between Nico’s two data points. (b) (1 marks) Nico asks us to predict the ocean temperature at the surface using his data. What value would

you provide to Nico?

(c) (3 marks) Romina delivers her data to a group of MAT135 students. The temperature, T (in degrees Celsius) and density, ρ (in kg/m3 ) profiles are shown in the graphs below. Notice that Romina has used depth on the vertical axis. The horizontal axes are temperature (in the first graph) and density (in the second graph). Using Depth on the vertical axis is the standard in oceanography since it mimics how we visualize the ocean (depth being vertical). We still think of temperature and density as functions of depth, here. Rotating your paper 90 degrees counter clockwise may help you visualize.

Note: There is a nearly vertical part of both graphs from depth 0 meters, to depth approximately 50 meters.

(This is on purpose.)

Consider the derivative dρ/dz . Sketch a graph that shows dρ/dz on the vertical axis, and z on the horizontal axis. You should draw a rough sketch and exact values are NOT expected. However, try to capture the approximate locations of key features in the dρ/dz graph.

Again, rotating the graphs above 90 degrees counter clockwise may help you visualize.

(d) (4 marks) Oceanographers define layers of the ocean by patterns in the density. There are three layers to the Labrador Sea - the mixed layer (near the surface), pycnocline layer (below the mixed layer), and the deep layer (below the pycnocline layer).

The derivative dρ/dz is nearly zero (or small) in the mixed layer and also in the deep layer. Use your derivative graph to find the boundaries between the layers (approximate answers are enough). Draw horizontal dotted

lines on Romina’s density plot at the approximate locations of the two boundaries. Then write a sentence

or two describing the derivative dρ/dz in the pycnocline layer, and how the density changes in each of the three layers.

Hurrah! You are now an amateur oceanographer!

Important: 2 marks are also given based on the quality of your writing and your explanations. See the Quercus page on Written Assignments for some tips and information on mathematical writing.