Contents

Tutorial 2

Question 1

Use for loops and print(end="") to write functions which print the following patterns:

  1. print_square(n) where n is the number of stars along each edge.

*****
*   *
*   *
*   *
*****

  1. print_rhombus(n) where n is the number of stars along each edge.

    *****
   *   *
  *   *
 *   *
*****

  1. print_numbers(n) where n is the number at the centre.

1       1
 2     2
  3   3
   4 4
    5
   4 4
  3   3
 2     2
1       1

Question 2

An integer \(n\) is a prime number if it is divisible only by 1 and \(n\).

  1. Write a function is_divisible(n, m) which returns True if n is divisible by m, and otherwise returns False.

  2. Write a function is_prime(n) which returns False if n is divisible by any integer between 2 and n-1, and otherwise returns True.

  3. Write a function number_of_primes(n) which returns the number of prime numbers less than or equal to n [NB 1 is not a prime number].

Check the correctness of your functions by writing two tests for each.

Question 3

A solid of revolution is a three-dimensional figure contstructed by rotating a curve about a straight line. We can estimate the volume of a solid of revolution by dividing it into a sequence of stacked discs and summing the volume of each.

A sphere of radius \(R\) is formed by rotating the curve \(y = \sqrt{R^2 - x^2}\) around the x-axis between \(-R\) and \(R\).

a

Use the following steps to estimate the volume of a sphere of radius 1.

  1. Write a function vol_disc(R, x, dx) which returns the volume of the disc centred at position x with thickness dx.

  2. Estimate the volume of a sphere of radius 1 by dividing the figure into 10 discs equally spaced between -1 and 1 [use a value of 3.14159 for \(\pi\)].

  3. Write a function sphere_vol(R, n) which returns the estimate of the volume of a sphere of radius R calculated by dividing it into n discs.

  4. The estimate should get more accurate as we increase n. We can estimate the accuracy by calculating the difference between sphere_vol(R, n) and sphere_vol(R, n-1). For R = 1, how large does n need to be so that difference between consecutive estimates is less than \(10^{-4}\)?

Extension Questions (Optional)

These questions are open-ended and designed to allow to you challenge yourself beyond the material we have studied.

  1. Investgate the Prime Number Theorem.

  2. Write a function volume_of_revolution(func, x_min, x_max, n) which calculates the volume of the surface of revolution of the curve given by function func. func(x) should be a function of a single variable which returns the y-value of the curve given the x value. You’ll have to learn how to pass a function as an argument to another function. Test your function by calculating the volume of a sphere, a paraboloid and another curve of your choice.