Exercises

After completing the tutorial attempt the following exercises.

If you are not sure how to do something, have a look at the “How To” section.

1. Use the class created in Tutorial to find the roots of the following quadratics:

   1. $f(x)=-4x^2+x+6$

   2. $g(x)=3x^2-6$

   3. $h(x)=f(x)+g(x)$

2. Write a class for a Linear expression and use it to find the roots of the following expressions:

   1. $f(x)=2x+6$

   2. $g(x)=3x-6$

   3. $h(x)=f(x)+g(x)$

3. If rain drops were to fall randomly on a square of side length $2r$ the probability of the drops landing in an inscribed circle of radius $r$ would be given by:

   $$P=\frac{\text{Area of circle}}{\text{Area of square}}=\frac{\pi r^2}{4r^2}=\frac{\pi }{4}$$

   Thus, if we can approximate $P$ then we can approximate $\pi$ as $4P$. In this question we will write code to approximate $P$ using the random library.

   First create the following class:

   class Drop:
       """
       A class used to represent a random rain drop falling on a square of
       length r.
       """
   
       def __init__(self, r=1):
           self.x = (0.5 - random.random()) * 2 * r
           self.y = (0.5 - random.random()) * 2 * r
           self.in_circle = (self.y) ** 2 + (self.x) ** 2 <= r ** 2
   

   Note that the above uses the following equation for a circle centred at $(0,0)$ of radius $r$:

   $$x^2+y^2\le r^2$$

   To approximate $P$ create $N=1000$ instances of Drops and count the number of those that are in the circle. Use this to approximate $\pi$.

4. In a similar fashion to question 3, approximate the integral ${\int }_0^{1}1-x^{2}dx$. Recall that the integral corresponds to the area under a curve.