Solutions to Assignment 2
MATH1062: Mathematics 1B
Semester 1, 2024
This assignment is worth 5% + 5% = 10% of your final assessment for this course. Your answers should be neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any resources used and show all working. Present your arguments clearly using words of explanation and diagrams where relevant. The marker will give you feedback and allocate an overall mark to your assignment using the following criteria:
Submission instructions
Solutions to calculus questions (Part A) must be prepared in written form, and uploaded as a single pdf file to https://canvas.sydney.edu.au/courses/57267/assignments/520094.
Solutions to Part B must be prepared as a single html file and submitted tohttps://canvas. sydney.edu.au/courses/57267/assignments/520095. You should use the following files to complete Part B.
. A data file CanterburyMarch2023.csv athttps://canvas.sydney.edu.au/courses/ 57267/files/36646049.
. An R Markdown worksheet Assignment2Worksheet.Rmd athttps://canvas.sydney. edu.au/courses/57267/files/36646052.
You need to write your solutions as either embedded R code or text answers in the provided worksheet, and then generate the html file using Knit in R Studio.
Part A: calculus questions
1. (a) Given the function f : R2 一 R where f(x, y) = x2 - y2 ,
(i) Sketch the level curve f(x, y) = 1.
(ii) Show that the level curve f(x, y) = 1 can be parameterised by
where t ∈ R.
(b) The surface S ⊆ R3 is defined as the zero-set of the following equation:
a1 x2 + a2y2 + a3 z2 + a4 xy + a5 xz + a6yz + a7 x + a8y + a9 z + a10 = 0
where a1 , . . . , a10 e R are constants. In each of the parts below give an example of an equation, of the above form, that produces S with the desired properties. Show all working.
(i) The plane z = k intersects the surface S nontrivially when k ≥ 4 and k ≤ -4, but does not intersect the surface when -4 < k < 4.
(ii) The plane z = k intersects the surface S nontrivially when k = 5, but does not intersect the surface for any other k e R.
2. Consider the surface defined by z = f(x, y) where f : R2 一 R is given by
f(x, y) = x3 - x2y - xy2 + y3 .
(a) Determine the equation of the tangent plane to the surface when x = 1 andy = -1.
(b) Determine all points (a, b) e R2 where the tangent plane to the surface at the point (a, b, f(a, b)) has the equation z = k for k e R.
Part B: statistics questions
Part B questions are provided in the R Markdown worksheet Assignment2Worksheet.Rmd at https://canvas.sydney.edu.au/courses/57267/files/36646052. You need to write your solutions as either embedded R code or text answers in the provided worksheet, and then generate the html file using Knit in R Studio. We can only mark the html file. The generated html file should contain all your work, including code.
You also need to download the data file CanterburyMarch2023.csv athttps://canvas. sydney.edu.au/courses/57267/files/36646049. This data file is needed to knit the work- sheet and complete your assignment questions.