The University of Sydney School of Mathematics and Statistics Assignment 2 MATH1062: Mathematics 1B Semester 2, 2024 Lecturers: Tiangang Cui, Jack Freestone, Brad Roberts, and Zhou Zhang This individual assignment is due by 11:59pm Sunday 13 Oct 2024, via Can- vas. Late assignments will receive a penalty of 5% per day until the closing date. Your answers must be compiled in two separate documents, and uploaded in Canvas to two different submission boxes, as outlined in the submission instructions below. Both documents should include your SID. Please make sure you review your submis- sions carefully. What you see is exactly how the marker will see your assignment. Submissions can be overwritten until the due date. To ensure compliance with our anonymous marking obligations, please do not under any circumstances include your name in any area of your assignment; only your SID should be present. The School of Mathematics and Statistics encourages some collaboration between students when working on problems, but students must write up and submit their own version of the solutions. If you have technical difficulties with your submission, see the University of Sydney Canvas Guide, available from the Help section of Canvas. 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: Copyright © 2024 The University of Sydney 1 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/59770/assignments/553029. Solutions to Part B must be prepared as a single knitted html file and submitted to https: //canvas.sydney.edu.au/courses/59770/assignments/553030. Part A: calculus questions Please justify your answers by showing all relevant works. Correct answers without adequate justification will not receive full marks. 1. (a) Find the particular solution for the following differential equation for the unknown function y(x), d2y dx2 + dy dx = 0, y(0) = y′(0) = 1. (b) Consider the following differential equation d2y dx2 − 2dy dx + y = ex for the unknown function y(x) by following the steps below. (i) For z(x) = e−xy(x), show that e−x ( dy(x) dx − y(x) ) = dz(x) dx . (ii) Still for z(x) = e−xy(x), using (i) or not, show that e−x ( d2y dx2 − 2dy dx + y ) = d2z(x) dx2 . (iii) Assuming (ii), the original differential equation is transformed to d2z(x) dx2 = 1. (A) Obtain the general solution for this new differential equation for the unknown function z(x). (B) Using z(x) = e−xy(x), obtain the general solution for the original differential equation for the unknown function y(x). 2. In R3, there is the following sphere centred at the origin, S = {(x, y, z) | x2 + y2 + z2 = 3}. We consider the point (1, 1,−1) on S in the following. (a) Consider the part of the sphere S near (1, 1,−1) as the graph of z = f(x, y). (i) What is the natural domain of the function f(x, y)? (ii) What is the equation of the level curve containing the point (x, y) = (1, 1)? 2 (b) Calculate the equation for the tangent plane P of the sphere S at the point (1, 1,−1). (c) Show that the line L, x(t) = 1 + t, y(t) = 1 + t, z(t) = −1 + 2t, t ∈ R contains the point (1, 1,−1), intersects with the z-axis and doesn’t intersect with the y-axis. (d) Show that the line L in (c) sits on the plane P in (b). Part B: statistics questions Solutions must be prepared as a single html file and submitted to https://canvas.sydney. edu.au/courses/59770/assignments/553030. You should use the following files to complete the assignment. • A data file AutumnCleaned.csv at https://canvas.sydney.edu.au/courses/59770/ files/39409879. This data file is needed to knit the worksheet and complete your assignment questions. • An R Markdown worksheet Assignment2Worksheet.Rmd at https://canvas.sydney. edu.au/courses/59770/files/38324259. Questions are provided in this R Markdown worksheet. 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 knitted html file. The generated html file should contain all your work, including code. 3 51作业君版权所有