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 \(r\):

   \[ x^2+y^2≤r^2 \]

   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 \(\int_{0}^11-x^2\;dx\). Recall that the integral corresponds to the area under a curve.