Assignment 1
MATH1023: Multivariable Calculus and Modelling Semester 1, 2023
Lecturer: Clio Cresswell
This individual assignment is due by 11:59pm Wednesday 15 March 2023, via
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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)
Using the Direction Field Grapher available on the MATH1023 Canvas site, give the
predicted value of this model for 8 p.m. 23/02/2023.