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MATH1023: Multivariable Calculus and Modelling
创建日期：2024-08-06 17:54:22
所属分类：数学mathematics
标签： MATH1023
Multivariable Calculus and Modelling

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MATH1023: Multivariable Calculus and Modelling


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see the University of Sydney Canvas Guide, available from the Help section of Canvas.

This assignment is worth 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

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Copyright c? 2023 The University of Sydney 1

A researcher has conducted an experiment concerning the placement of an object, given as y

in centimetres, at time t in hours, after an initial force on 22/02/2023 at 9:00 a.m. disturbed

the object from rest. It is a particularly delicate experiment and only the following data is

gathered successfully.

Date Time y Y, the rate of change of y The rate of change of Y

22/02/2023 11:00 a.m. 0.1 ?0.2 0.5

22/02/2023 3:09 p.m. 10?5 2× 10?6 ?6× 10?5

Even with such limited data, to advance on the project the researcher considers, separately,

the following three possible models for the experiment. Each model consists of a differential

equation.

1. Find the general solution y(t) to model (1). Then give the particular solution that satisfies

y being 3.2 cm from an initial reference point at the very start of the experiment.

2. The solutions y(t) = Ae?2t and y(t) = e?2t(A cos(t) + 5 sin(t)) (where A is an arbitrary

constant) solve models (2) and (3), but the researcher cannot remember which one solves

which.

(a) Find which solution solves which model, clearly showing your working.

(b) Give the particular solution for each model that satisfies y being 3.2 cm from an

initial reference point at the very start of the experiment.

3. Accepting that y is 3.2 cm from an initial reference point at the very start of the experi-

ment, construct a table mirroring the one above, where the values for y, Y and its rate of

change, are the values given by model (2). Then, produce the same table that fits model

(3). Give results to 2 significant figures.

4. Construct a table of the differences in the recorded and predicted y values and the

recorded and predicted rates of change of the y values for both model (2) and (3). Give

results to a single significant figure. Does a particular model appear to fit the data better

and if so, which one and why? Your explanation should be short, simply showing your

understanding. A single sentence of around 10-20 words is all that is required for clarity.

5. Feeling curious, the researcher then considers the following model for the experiment,

accepting y as 3.2 cm from an initial reference point at the very start of the experiment:

dy

dt= 2y + sin(t). (4)

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