Problem 3
Write a program that generates all odd numbers from 1 to n. Set \(n\) in the beginning of the program and use a while loop to compute the numbers. (Make sure that if \(n\) is an even number, the largest generated odd number is n-1.) Name of program file: odd.py.
Problem 6
Write a program that prints a table with \(t\) values in the first column and the corresponding \(y(t)=v_{0} t-0.5 g t^{2}\) values in the second column. Use \(n\) uniformly spaced \(t\) values throughout the interval \(\left[0,2 v_{0} / g\right] .\) Set \(v_{0}=1, g=9.81\), and \(n=11\). Name of program file: ball_table1.py. \(\diamond\)
Problem 8
Set a variable primes to a list containing the numbers \(2,3,5,7\) 11 , and \(13 .\) Write out each list element in a for loop. Assign 17 to a variable \(\mathrm{p}\) and add \(\mathrm{p}\) to the end of the list. Print out the whole new list. Name of program file: primes.py.
Problem 9
You are given the following program: \(\mathrm{a}=[1,3,5,7,11]\) \(\mathrm{b}=[13,17]\) \(c=a+b\) print \(\mathrm{c}\) \(\mathrm{b}[0]=-1\) \(\mathrm{~d}=[\mathrm{e}+1\) for e in a] print d \(\mathrm{d}\). append \((\mathrm{b}[0]+1)\) \(\mathrm{d}\). append \((\mathrm{b}[-1]+1)\) print \(\mathrm{d}[-2:]\) Explain what is printed by each print statement.
Problem 10
We want to generate \(x\) coordinates between 1 and 2 with spacing 0.01. The coordinates are given by the formula \(x_{i}=1+i h\), where \(h=0.01\) and \(i\) runs over integers \(0,1, \ldots, 100\). Compute the \(x_{i}\) values and store them in a list. Use a for loop, and append each new \(x_{i}\) value to a list, which is empty initially. Name of program file: coor1.py.
Problem 16
interest rate amount = initial_amount years = 0 while amount <= 1.5*init… # Consider the following program for computing with interest rates: initial_amount = 100 p = 5.5 # interest rate amount = initial_amount years = 0 while amount <= 1.5*initial_amount: amount = amount + p/100*amount years = years + 1 print years (a)Explain with words what type of mathematical problem that is solved by this program. Compare this computerized solution with the technique your high school math teacher would prefer. (b)Use a pocket calculator (or use an interactive Python shell as substitute) and work through the program by hand. Write down the value of amount and years in each pass of the loop. (c) Change the value of \(p\) to 5 . Why will the loop now run forever? (See Exercise \(2.12\) for how to stop the program if you try to run it.) Make the program more robust against such errors. (d)Make use of the operator += wherever possible in the program. Insert the text for the answers to (a) and (b) in a multi-line string in the program file. Name of program file: interest_rate_loop.py.
Problem 23
Rewrite the generation of the nested list q, $$ \mathrm{q}=[r * * 2 \text { for } \mathrm{r} \text { in }[10 * * \mathrm{i} \text { for } \mathrm{i} \text { in } \mathrm{range}(5)]] $$ by using standard for loops instead of list comprehensions. Name of program file: listcomp2for.py.
Problem 25
Explain the outcome of each of the following boolean expressions: C = 41 C == 40 C != 40 and C < 41 C != 40 or C < 41 not C == 40 not C > 40 C <= 41 not False True and False False or True False or False or False True and True and False False == 0 True == 0 True == 1 Note: It makes sense to compare True and False to the integers 0 and 1 , but not other integers (e.g., True \(==12\) is False although the integer 12 evaluates to True in a boolean context, as in bool(12) or if 12).
Problem 26
Maybe you have tried to hit the square root key on a calculator multiple times and then squared the number again an equal number of times. These set of inverse mathematical operations should of course bring you back to the starting value for the computations, but this does not always happen. To avoid tedious pressing of calculator keys we can let a computer automate the process. Here is an appropriate program: from math import sqrt for n in range(1, 60): r = 2.0 for i in range(n): r = sqrt(r) for i in range(n): r = r**2 print ’%d times sqrt and **2: %.16f’ % (n, r) Explain with words what the program does. Then run the program. Round-off errors are here completely destroying the calculations when \(\mathrm{n}\) is large enough! Investigate the case when we come back to 1 instead of 2 by fixing the \(\mathrm{n}\) value and printing out \(\mathrm{r}\) in both for loops over i. Can you now explain why we come back to 1 and not 2 ? Name of program file: repeated_sqrt.py.
Problem 27
Type in the following code and run it: eps = 1.0 while 1.0 != 1.0 + eps: print ’...............’, eps eps = eps/2.0 print ’final eps:’, eps Explain with words what the code is doing, line by line. Then examine the output. How can it be that the "equation" \(1 \neq 1+\) eps is not true? Or in other words, that a number of approximately size \(10^{-16}\) (the final eps value when the loop terminates) gives the same result as if eps \(^{9}\) were zero? Name of program file: machine_zero.py. If somebody shows you this interactive session >>> 0.5 + 1.45E-22 0.5 and claims that Python cannot add numbers correctly, what is your answer? \(\diamond\)
