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The University of Sydney
Math1003 Integral Calculus and Modelling
Semester 2
Exercises for Week 7
2011
Assumed Knowledge
Proportionality and inverse proportionality. Integration techniques.
Taylor series expansion.
Objectives
(6a)
Given a verbal description of a simple model, to be able to express it as a mathematical
equation.
(6b
) To be able to recognise an ordinary diFerential equation.
(6c)
To be able to sketch the solution curves for a ±rstorder diFerential equation from its
direction ±eld.
Preparatory Questions
1.
(
i
) The diFerential equation
dy
dx
=
f
(
x
) has a direction ±eld given by the diagram
below.
On the direction ±eld draw the graphs of two solutions of
dy/dx
=
f
(
x
), where one
solution
y
=
g
(
x
) passes through the point (0
,
1) and the other solution
y
=
h
(
x
)
satis±es the equation
h
(1) = 0.
(
ii
) Do the graphs of
y
=
g
(
x
) and
y
=
h
(
x
) intersect? If not, why not?
1
2
3
−
1
1
2
3
−
1
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View Full DocumentPractice Questions
2.
Heat tends to fow From hot bodies to cold bodies. Newton observed that the rate
at which temperature rises or Falls within a body is proportional to the temperature
di±erence between the body and its surroundings.
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