Description
1 Archimedean spiral (30 points)
Using Pyplot and Numpy, state a function arc that draws a green Archimedean
spiral according to the following parameterized equation:
• x(t) = t.cos(t)
• y(t) = t.sin(t)
t should vary in [0, 5π] (step 0.01) and the function should call plt.plot with
arguments x and y, as well as plt.show() function at the end. Call the function.
2 Heart (30 points)
Using Pyplot and Numpy, state a function heart that draws in dashed red the
following parameterized equation:
• x(t) = 16sin3
(t)
• y(t) = 13cos(t) − 5cos(2t) − 2.5cos(3t) − cos(4t)
t should vary in [0, 2π] (step 0.01) and the function should call plt.plot with
arguments x and y, as well as plt.show() function at the end. Call the function.
To see the result you should first close the window of part 1 (Archimedean spiral).
3 Graphs (30 points)
We wish to use graph theory in order to solve the following problem. A company
should carry different chemical products P1, P2, . . ., Pk from the factory to a
city. For security reasons, some products should not be carried in the same truck:
∀i, 0 < i < k, Pi
is not compatible with Pi+1. Moreover Pk is not compatible with P1.
1) Write a comment: how can we state an undirected graph that visually
represents this problem: 1. What are the vertices, what are the edges? 2. What is
the specific property of this graph?
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2) Write an algorithm that, given an integer k > 1, returns the minimum
number of trucks necessary to carry all the products.
4 Algorithm (10 points)
In ancient times, Egyptians did not use multiplication tables. Instead, they transformed any calculation into multiplication by two, and additions. The principle of
Egyptian multiplication is the following: (x ∗ y) = 2 ∗ (x ∗ (y/2)) if y is even, and
(x ∗ y) = x + (x ∗ (y − 1)) if y is odd. For instance,
(7 ∗ 10) = (2 ∗ (7 ∗ 5)) = (2 ∗ (7 + (7 ∗ 4))) = (2 ∗ (7 + (2 ∗ (7 ∗ 2)))).
Write a recursive Python function that, given two strictly positive integers x and
y (the arguments of the function), returns the result of x ∗ y obtained using the
Egyptian method. Of course, directly multiplying x by y is allowed in your code if
and only if x ≤ 2 or y ≤ 2.
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