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Jupyter notebook Week 1/Homework 1- Basic python notebook.ipynb

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Kernel: Python 3

#Earth Systems Modeling ##Lab 1: Introduction to python and iPython notebook

Problem 1

Write a function to solve the definite integral ∫0100(aebx+c)dx\int_0^{100} (ae^{bx} + c) dx See the following documentation: Scipy documentation.

Your function must be set up to run with input aa, bb, and cc, which are passed into the function as arguments. Present your function with a description in your blog. Provide the code. [10 pts]

from scipy.integrate import quad as I
from numpy import exp as e

def function_problem1(a, b, c, x=100.0):
    """
    compute the integral:
    f(Ae^(Bx) + C) over x from 0 to x.
    """

    expression = lambda x: a * e(b * x) + c
    y, err = I(expression, a=0, b=x)

    return {'area': y, 'error': err}

# test
A, B, C = 5, -10, 3
output = function_problem1(A, B, C, x=50)
print("the area is ", output['area'], " with +/- ", output['error'])
the area is 150.5 with +/- 4.558312174712722e-10

Problem 2

Write a function create a histogram using a random number generator. The histogram should represent a normal distribution of 1x1041 x 10^4 values. Make sure your distribution has an average of avgavg, and a standard deviation of bb. Plot these values as a histogram with nn number of bins distribution. The arguments of your function should be avgavg, bb and nn. See the following documentation: Random Numbers from a Normal Distribution | Mathworks and Part 3: Histograms | Bespoke Blog. In your blog, briefly describe your calculation, provide your code and a plot of your histogram. Your plot should include a figure caption describing what the plot depicts. [10 pts]

%matplotlib inline

import numpy
import matplotlib.pyplot as plt
import pandas as pd

def function_problem2(avg, b, n):
    """
    for a given mean (avg)
    and stdev (b) return a histogram
    with 1*10^4 samples
    """
    # seed with the grand number
    numpy.random.seed(42) 
    dist = numpy.random.normal(loc=avg, scale=b, size=10000)

    # plot histogram
    fig = plt.figure()
    plt.subplot(111)
    plt.hist(dist, bins=n)
    TXT ="CAPTION:\nThis chart shows a Random Normal Distribution with a mean \n \
of {0} and a standard deviation of {1} allocated into {2} bins.".format(avg, b, n)
    fig.text(.1,-.1,TXT)
    plt.title("Random Normal Distribution: mean({0}) stdev({1})".format(avg, b))
    plt.xlabel("Value")
    plt.ylabel("Frequency")
    plt.show()
    
# test
function_problem2(500, 1, 20)
Image in a Jupyter notebook

Problem 3: Challenge Problem for Extra Credit

Find the volume of a sphere in 4-dimensional euclidean space (a 3-sphere) with a radius of 5cm. (Hint: you will use a random number generator.) Describe your solution to the challenge problem in your blog. [5 pts]

# problem 3.py

# estimate the volume of a sphere
# given a radius

import numpy as np

def function_problem3(radius=5):
    """
    creates a cube
    fills cube with lots of points
    calculated distance between center of cube and points
    estimates volume of sphere as percentage of points in sphere.
    """
    
    volume_of_cube = (2 * radius) ** 3.0
    origin = (radius, radius, radius)
    
    # apply the grand number
    # and get a large set of (x, y, z)
    # coordinates
    np.random.seed(42)
    coords = np.random.uniform(low=0, high= 2*radius, size=(1000000, 3))

    # calculate distance to origin
    distances = np.sqrt(np.sum((coords - radius)**2, axis=1)).flatten()
    
    # how many points are inside vs outside of sphere
    in_sphere, out_sphere = len(distances[distances < radius]), len(distances[distances > radius])
    pcnt = in_sphere / (in_sphere + out_sphere)

    # guess volume
    volume_of_sphere = volume_of_cube * pcnt

    print("volume of sphere: ", volume_of_sphere, "cm^3")


# test
function_problem3(5)
volume of sphere: 523.899 cm^3