You will create an executable Python program for each task and you can finally collect them in a .py file and upload. Also provide a printout showing the results of Exercises 1, 2 and 3.
Task 1 The weierstrass.py weierstrass.py contains a function w needed for this task, so download it and place it in the same folder as the file you are about to submit. Here you need to program (Python):
1. Plot the function w on the interval [-2, 2]. Try different values for num in np.linspace for how fine division of the x-axis in np.linspace you need to show enough details.
Then create a function called wderivert (x, h) that uses a method that calculates the derivative in each of the points you defined in 1. for a given h = 0.001.
3. Plot the derivative. What can you say about the function w based on what you see.
4. Try different values of num under 30 to find your favorite graph of the derivative of w and leave this value as the final value for the division of the x-axis for all the plots.
Task 2
In this exercise we will use numerical integration on the function
We will use Simpson's rule, which approximates the function as a series of quadratic functions that go through the endpoints and midpoints of the subintervals. The figure shows Simpson's rule for the function similar to ours at the interval [2, 8].
You can read more about Simpson's rule (Simpson's rule in English) ) her. The Norwegian Wikipedia article also has some tips on how the method can be implemented, and can be read here. her. (Lenker til eit ekstern område.) Note the difference between dividing into n subintervals (plus their midpoint) and dividing into 2n subintervals (where some of the divisions are the midpoints). The latter is the method we want.
1- Divide the interval [2, 8] into 2 equal sub-intervals and use Simpsons rule to calculate the integral.
2- Repeat step 1, but use 5, 10, and 100 subintervals. What happens to the value of the integral? Can you guess what the true integral is? Based on the image above, what is the minimum number of sub-intervals you expect to provide a good approach? 3- Use Simpson's rule on
at the same interval. What do you find when you try different number of sub-intervals? Can Simpson's formula take odd number intervals? Can you explain this?
Task 3
Write a Python function (not a script) that calculates the area of a circle sector. The function should take a radius, called r, and an angle given in radians, called theta (theta is the Greek letter that is also written ?) as arguments, and begins like this (a so-called docstring):
def sirkelsektor(r,theta):
---
Parameters
---------- r : TYPE
DESCRIPTION.
theta : TYPE DESCRIPTION.
Returns
------- TYPE
DESCRIPTION.
---
Complete this function and test that it gives the expected answer
for r = 1, ? = 0, p, 2pr = 1, ? = 0, p, 2pr = 1, ? = 0, p, 2p. What happens if someone uses this feature but specifies the angle in degrees instead of radians? Do we get an error message?