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 Sydney

School of Mathematics and Statistics

Assignment 1

MATH1023: Multivariable Calculus and Modelling Semester 2, 2019

Web Page: http://sydney.edu.au/science/maths/u/UG/JM/MATH1023/

Lecturers: Eduardo Altmann and Leo Tzou

This individual assignment is due by 11:59pm Thursday 29 August 2019, via

Canvas. 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 Management

System (Canvas) at https://canvas.sydney.edu.au/courses/17310. Please submit only one PDF document (scan or convert other formats). It should include your

SID, your tutorial time, day, room and Tutor’s name. Please note: Canvas does NOT

send 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 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 2.5% of your final assessment for this course. Your answers should be

well written, 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. After all, mathematics is about communicating your ideas. This

is a worthwhile skill which takes time and effort to master. The marker will give you feedback

and allocate an overall letter grade and mark to your assignment using the following criteria:

Mark | Grade | Criterion |

5 4 3 2 1 0 | A B C D E F | Outstanding and scholarly work, answering all parts correctly, with clear accurate explanations and all relevant diagrams and working. There are at most only minor or trivial errors or omissions. Very good work, making excellent progress, but with one or two substantial errors, misunderstandings or omissions throughout the assignment. Good work, making good progress, but making more than two distinct substantial errors, misunderstandings or omissions throughout the assign ment. A reasonable attempt, but making more than three distinct substantial errors, misunderstandings or omissions throughout the assignment. Some attempt, with limited progress made. No credit awarded. |

Copyright c 2019 The University of Sydney 1

Let 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 by

dx

dt = 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 a

constant rate h = 1=2 and thus

dx

dt =

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 the

maximum 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 looks

popular. This corresponds to h(x) = 3 2x and thus

dx

dt =

3 2

x(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 = 2

hours with 80% capacity?

3. Consider the two models, A and B, both starting at 10% full capacity. Model A is

governed by the process of question (1) and model B is governed by the process described

in 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., dx

dt at t = 0);

(iii) the values of x in the limit t ! 1.

2