Download Feedback Exercises 5 with solutions

MTH4105 Introduction to
Mathematical Computing
Feedback Exercises 5: Plotting,
strings and drawing
School of Mathematical
Sciences
© 2016 by Dr Francis J. Wright
Answer the questions below in a copy of the answer template document available from
QMplus. Your coursework submissions provide evidence that you are engaging with this
module. Failure to submit coursework may result in you being deregistered.
When answering the questions, use standard typography for any mathematical notation that
you include. Use hyperlinks for any web references and spell check the document before
submitting it. Upload you completed answer document to QMplus by the deadline, which
will normally be about a week after each question sheet is released on QMplus. The precise
deadline and submission mechanism are specified in QMplus. You will be provided with
feedback on your answers and I will release example solutions after the submission
deadline.
Question 1
1 /2
1 /3
1 /4
Plot the graphs of x , x
, x
, x
as functions of x together on the same
axes for x % 2 with a legend that clearly identifies the plots.
My answer
> plot
x,
x
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x
1 /2
,
x
1 /3
,
x
1 /4
, x =K2 ..2, legend = x ,
x
1 /2
,
x
1 /3
,
1 /4
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2
1.5
1
0.5
K2
K1
0
1
2
x
x
x
x 1/3
x 1/4
Question 2
A Pythagorean triple can be defined as a sequence of natural numbers a, b, c with
2
2
2
a ! b ! c such that a Cb = c , and a triangle with sides of these lengths is therefore
right-angled. Four small Pythagorean triples are (3, 4, 5), (5, 12, 13), (7, 24, 25) and (8,
15, 17). Plot the corresponding right-angled triangles in different colours superimposed
on one plot, all with the right angle at the origin, the side of length a going vertically up
and the side of length b going horizontally to the right. Include a suitable title and legend
and ensure that the plot is geometrically correct. It is probably easiest to construct each
plot separately and then display them together.
My answer
This is probably the easiest approach:
> P d plot 0, 0 , 0, 3 , 4, 0 , 0, 0 , legend = "(3, 4, 5)", colour = red ,
plot 0, 0 , 0, 5 , 12, 0 , 0, 0 , legend = "(5, 12, 13)", colour = blue ,
plot 0, 0 , 0, 7 , 24, 0 , 0, 0 , legend = "(7, 24, 25)", colour = green ,
plot 0, 0 , 0, 8 , 15, 0 , 0, 0 , legend = "(8, 15, 17)", colour = violet :
> plots:-display P, scaling = constrained, title
= "Four small right-angled triangles based on Pythagorean triples"
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Four small right-angled triangles based on Pythagorean triples
8
6
4
2
0
0
5
10
(3, 4, 5)
(8, 15, 17)
15
(5, 12, 13)
20
(7, 24, 25)
Here is an alternative using only the plot function:
> plot
0, 0 , 0, 3 , 4, 0 , 0, 0 ,
0, 0 , 0, 5 , 12, 0 , 0, 0 ,
0, 0 , 0, 7 , 24, 0 , 0, 0 ,
0, 0 , 0, 8 , 15, 0 , 0, 0 ,
legend = "(3, 4, 5)", "(5, 12, 13)", "(7, 24, 25)", "(8, 15, 17)" ,
scaling = constrained, title
= "Four small right-angled triangles based on Pythagorean triples"
Four small right-angled triangles based on Pythagorean triples
8
6
4
2
0
0
5
10
(3, 4, 5)
(8, 15, 17)
(5, 12, 13)
15
20
(7, 24, 25)
Question 3
2
Plot the rotated parabola x = y for 0 % x % 9,K3 % y % 3 as (what looks like) a single
blue curve in two ways:
a) as a single curve defined implicitly by using the implicitplot function in the plots
package;
b) as the graphs of the two functions corresponding to y =G x combined on a single
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b)
plot.
In both cases, include a suitable title and label the axes with the variables x and y. What
differences do you notice between the two plots?
My answer
2
> plots:-implicitplot x = y , x = 0 ..9, y =K3 ..3, colour = blue, title
2
= typeset "The parabola ", x = y
2
The parabola x = y
3
2
1
0
y
1
2
3
4
5
6
7
8
9
x
K1
K2
K3
x ,K x , x = 0 ..9, 'y', colour = blue, title = typeset "The parabola ", x
> plot
=y
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2
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The parabola x = y2
3
2
y
1
0
1
2
3
4
5
6
7
8
9
x
K1
K2
K3
The two plots differ in that Maple places the vertical axis label y differently and the
implicit plot is less smooth, especially near the origin where the curvature is highest.
[Remark: Generally, implicit plots tend to be less smooth than plots of graphs of
functions because of the underlying algorithms used.]
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