- May 15, 2020

MATH1023: 题意：

给定x,t的积分关系，根据题目里不同的问题解出两个x(t)，然后在不同条件下比较这两个函数

解析：

首先给出了dx与dt的关系，可以通过这个算式得出积分后的x与t的关系，随后通过给定的条件可以解出x(t)；求出两种题设下的x(t)后，再比较二者 涉及知识点：

不定积分，根据条件解待定系数

The University of SydneySchool of Mathematics and StatisticsAssignment 1MATH1023: Multivariable Calculus and Modelling Semester 2, 2019Web Page: http://sydney.edu.au/science/maths/u/UG/JM/MATH1023/Lecturers: Eduardo Altmann and Leo TzouThis individual assignment is due by 11:59pm Thursday 29 August 2019, viaCanvas. Late assignments will receive a penalty of 5% per day until the closing date.A single PDF copy of your answers must be uploaded in the Learning ManagementSystem (Canvas) at https://canvas.sydney.edu.au/courses/17310. Please submit only one PDF document (scan or convert other formats). It should include yourSID, your tutorial time, day, room and Tutor’s name. Please note: Canvas does NOTsend an email digital receipt. We strongly recommend downloading your submission to check it. What you see is exactly how the marker will see your assignment.Submissions can be overwritten until the due date. To ensure compliance with ouranonymous marking obligations, please do not under any circumstances include yourname in any area of your assignment; only your SID should be present. The Schoolof Mathematics and Statistics encourages some collaboration between students whenworking on problems, but students must write up and submit their own version of thesolutions. If you have technical difficulties with your submission, see the Universityof Sydney Canvas Guide, available from the Help section of Canvas.This assignment is worth 2.5% of your final assessment for this course. Your answers should bewell written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite anyresources used and show all working. Present your arguments clearly using words of explanationand diagrams where relevant. After all, mathematics is about communicating your ideas. Thisis a worthwhile skill which takes time and effort to master. The marker will give you feedbackand allocate an overall letter grade and mark to your assignment using the following criteria:Mark Grade Criterion5 4 3 2 1 0 A B C D E F Outstanding and scholarly work, answering all parts correctly, with clearaccurate explanations and all relevant diagrams and working. There areat most only minor or trivial errors or omissions.Very good work, making excellent progress, but with one or two substantialerrors, misunderstandings or omissions throughout the assignment.Good work, making good progress, but making more than two distinctsubstantial errors, misunderstandings or omissions throughout the assignment.A reasonable attempt, but making more than three distinct substantialerrors, misunderstandings or omissions throughout the assignment.Some attempt, with limited progress made.No credit awarded.Copyright c 2019 The University of Sydney 1Let x(t) 2 [0; 1] be the fraction of maximum capacity of a live-music venue at time t (in hours)after the door opens. The rate at which people go into the venue is modeled bydxdt = h(x)(1 – x); (1)where h(x) is a function of x only.1. Consider the case in which people with a ticket but outside the venue go into it at aconstant rate h = 1=2 and thusdxdt =1 2(1 – x):(a) Find the general solution x(t).(b) The initial crowd waiting at the door for the venue to open is k 2 [0; 1] of themaximum capacity (i.e. x(0) = k). How full is the venue at t?2. Suppose people also decides whether to go into the venue depending on if the place lookspopular. This corresponds to h(x) = 3 2x and thusdxdt =3 2x(1 – x):(a) Find the general solution x(t).(b) What should be the initial crowd x(0) if the band wants to start playing at t = 2hours with 80% capacity?3. Consider the two models, A and B, both starting at 10% full capacity. Model A isgoverned by the process of question (1) and model B is governed by the process describedin question (2).Start this question by writing down the respective particular solutions xA(t) and xB(t).(a) Which of the two models will first reach 50% of full capacity?(b) Which of the two models will first reach 99% of full capacity?(c) Plot the curves xA(t) and xB(t). Both curves should be consistent with:(i) your answers to the two previous items;(ii) the rate of change at t = 0 (i.e., dxdt at t = 0);(iii) the values of x in the limit t ! 1.2