Description

1             Patterns

1) Write a function named rectangle that given two integers n and m prints out two identical rectangles of width n and height m, separated by an empty column, using the symbol ’∗’. You may use a loop or recursion. For instance, the call rectangle(4,6) should exactly lead to the following result on IDLE:

**** ****

**** ****

**** ****

**** ****

**** ****

**** ****

2) Write a function named triangle that, given a strictly positive integer n given as argument, prints out a triangle of height n, using the symbol ’∗’. You may use a loop or recursion.

By indexing the height from 0 (top) to n − 1 (bottom), this triangle should be such that: (1) Each row at depth i, i > 0, has two more stars than the number of stars of the previous row at depth i−1. (2) At depth 0, the function prints one star.

For instance, the call triangle(6) should exactly lead to the following result on IDLE:

∗

∗∗∗

∗∗∗∗∗

∗∗∗∗∗∗∗

∗∗∗∗∗∗∗∗∗

∗∗∗∗∗∗∗∗∗∗∗

2             Function on lists

Write a function clean1(aList) that takes a list of integers aList as argument, and returns a new list where multiple occurrences of values have been removed. The function should NOT modify its argument. This means that if we print the list (given as argument) after executing the function, it did not change. For instance, the following statements:

al = [1,2,3,4,4,4,5,1,2,1,5] newlist = clean1(al) print(al) print(newlist)

print the following result (a different order of values is acceptable):

[1,2,3,4,4,4,5,1,2,1,5]

[1,2,3,4,5]

The principle of your algorithm may be the following:

• Create a new empty list tmp.
• Use a “for loop” to fill tmp with each new value found in the argument list.
• Return tmp.

3             Function on lists

Write a function clean2(aList) that takes a list of integers aList as argument, and modifies this list, such as multiple occurrences of values have been removed. The procedure does not return a list: it modifies its argument (reference aList).

For instance, the following statements:

al = [1,2,3,4,4,4,5,1,2,1,5] print(al) clean2(al) print(al)

print the following result (a different order of values is acceptable):

[1,2,3,4,4,4,5,1,2,1,5]

[1,2,3,4,5]

4             Function on lists

Write a function named fct(aList) that takes a list of integers aList as argument and returns a string containing elements of the list that are a power of 2, all on the same line (i.e., without a new line after each printed value). Write a call to this function with argument [1,2,3,4,5,16,255,256,−1,−2,84], embedded in a print statement to see the result on IDLE.

5             Taylor Sinus

Required: To have points, you must use loop(s), NO recursion.

From calculus, we know that every continuously differentiable function can be represented as the sum of an infinite series. A finite subset of the series can be used to approximate the function. In this part, you will use Taylor series to approximate a sine function of an angle x in radians. Write a function taylor(x,n) that approximates the sin(x) by returning the sum of the first n elements of its Taylor series:

For instance, the following statements:

x = math.pi/2 print(math.sin(x)) print(taylor(x,7))

lead to the following result on the IDLE:

1.0

1.0000000006627803

6             Extra credit (5 points)

Implement a function consecutives(aList) that takes a list of integers as argument and return the maximum number of consecutive integers that are all equal, within this list. For instance, the following statements:

l1 = [1,1,1,3,3,4,4,4,4,4,3,1,7,3,3,4,4] l2 = []

l3 = [1,2,1,1,2,1,1] print(consecutives(l1)) print(consecutives(l2)) print(consecutives(l3))

lead to the following result on the IDLE:

5

0

2