# Initialize Otter
import otter
grader = otter.Notebook("hw2-math.ipynb")

🏡📝 Homework 2 - Using Python to Solve Math Problems#

This assignment will explore using python’s capabilities for mathematical calculations.

Question 1: Interest rate calculation

Let \(r\) be a bank’s interest rate in percent per year. An inital amount of money \(P\), also called a principal, will mature to an amount of

\[ P\left(1 + \frac{r}{100}\right)^{n} \]

after \(n\) years have passed.

Write python code to do the following:

  • Define variables: P = 1000, r = 0.95, and n = 5

  • Calculate the interest rate based on the equation above

  • Store the final value in a variable called v

Note: Your code should be indented under what’s called the function definition: def question_1():. This helps the autograder run all the code you write by calling a single name. (Functions will be taught in future weeks.).

Note: Python cares about Tabs make sure the spacing is correct. If you get an Indention Error it likely means your spacing is not correct.

Your solution should look like this:

def question ():
    line1 = line1_code # line 1 of code
    line2 = line2_code # line 2 of code

    return x, y, z # variables to return seperated by commas

Your code replaces the prompt: ...

# define a function called question_1 to be used for grading
def question_1():

    ...

    # The following line outputs these values from the function so that they can be accessed by the grader
    return P, r, n, v

grader.check("q1-interest")

Question 2: Musical note frequencies

On a piano, each key has a fundamental frequency \(f\). Each higher key (black or white) has a fundamental frequency of

\[ f * 2^{(\frac{n}{12})} \]

where \(n\) is the distance away from the starting key. The key \(A_4\), near the center of the piano keyboard, has a fundamental frequency of 440 Hz. Your task is to calculate the fundamental frequency of \(D_5\), which is 5 keys higher.

Write python code to do the following:

  • Define variables: f = 440 and n = 5

  • Calculate the fundamental frequency of \(D_5\) based on the equation above

  • Store the frequency of \(D_5\) in a variable called d

Again, your code should be indented under the function definition: def question_2():.

Your code replaces the prompt: ...

# define a function called question_2 to be used for grading
def question_2():

    ...

    # output these values from the function so that they can be accessed by the grader
    return f, n, d

grader.check("q2-music")

Question 3: Using Python as a calculator

We have seen how Python can be used to evaluate mathematical expressions. This problem provides practice incorporating various mathematical functions and constants.

Write python code to do the following:

  • Define variables: a = 3, b = -6, c = 4, and x = 2

  • For each of the quantities \(E_0\) through \(E_7\), construct a one-line Python expression that computes the value and assigns it to a variable. (You should perform your computations using the variable names \(a\), \(b\), and \(c\) instead of their numerical values). In Python, we will use variable names E0 through E7 to store the values \(E_0\) through \(E_7\). Definitions of the variables E0 through E7 have been started for you in the template.

\(\text{i. } E_0 = \sqrt{a^2+b^2+c^2}\)

\(\text{ii. } E_1 = \ln(3x-a)\)

\(\text{iii. } E_2 = \log_{10} \left(3 \left|b\right|+\frac{c}{5} \right)\)

\(\text{iv. } E_3 = \left( ax+\frac{ab}{c} \right)^{1/3}\)

\(\text{v. } E_4 = \frac{x^2+1}{(ax-1)\left|b-e^x\right|}\)

\(\text{vi. } E_5 = \left( cos \left( \frac{\sqrt{a}}{3}\pi \right) \right)^2+\cos \left(\left( \frac{\sqrt{a}}{3}\pi \right)^2\right)\)

\(\text{vii. } E_6 = \arccos(\cos(x))\)

\(\text{viii. } E_7 = \frac{a+2c}{\sin \left( \frac{b+2c}{\sqrt{a^2+b^2+c^2}} \right) }\)

You should use the NumPy library for the required mathematical functions. You can discover the necessary functions using the Numpy Documentation

We have used the standard NumPy convention import numpy as np, thus, for example the cos function is used by typing np.cos(<value>)

Again, your code should be indented under the function definition: def question_3():.

Your code replaces the prompt: ...

# import the np library to be used
import numpy as np

# define a function called question_3 to be used for grading
def question_3():

    a = ...
    b = ...
    c = ...
    x = ...

    E0 = ...
    E1 = ...
    E2 = ...
    E3 = ...
    E4 = ...
    E5 = ...
    E6 = ...
    E7 = ...

    # output these values from the function so that they can be accessed by the grader
    return a, b, c, x, E0, E1, E2, E3, E4, E5, E6, E7

grader.check("q3-calculator")