CS 61A: Extra Homework 1

Due by 11:59pm on Thursday, 2/12


Download extra01.zip. Inside the archive, you will find a file called extra01.py, along with a copy of the OK autograder.

Submission: When you are done, submit with python3 ok --submit. You may submit more than once before the deadline; only the final submission will be scored.

Using OK

The ok program helps you test your code and track your progress. The first time you run the autograder, you will be asked to log in with your @berkeley.edu account using your web browser. Please do so. Each time you run ok, it will back up your work and progress on our servers. You can run all the doctests with the following command:

python3 ok

To test a specific question, use the -q option with the name of the function:

python3 ok -q <function>

By default, only tests that fail will appear. If you want to see how you did on all tests, you can use the -v option:

python3 ok -v

If you do not want to send your progress to our server or you have any problems logging in, add the --local flag to block all communication:

python3 ok --local

When you are ready to submit, run ok with the --submit option:

python3 ok --submit

Readings: You might find the following references useful:

Table of Contents

This homework is required for the "extra lectures" track of CS 98: Additional Topics on the Structure and Interpretation of Computer Programs.

Question 1

Implement intersect, which takes two functions f and g and their derivatives df and dg. It returns an intersection point x, at which f(x) is equal to g(x).

def intersect(f, df, g, dg):
    """Return where f with derivative df intersects g with derivative dg.

    >>> parabola, line = lambda x: x*x - 2, lambda x: x + 10
    >>> dp, dl = lambda x: 2*x, lambda x: 1
    >>> intersect(parabola, dp, line, dl)
    "*** YOUR CODE HERE ***"

Question 2

Differentiation of polynomials can be performed automatically by applying the product rule and the fact that the derivative of a sum is the sum of the derivatives of the terms.

In the following example, polynomials are expressed as two-argument Python functions. The first argument is the input x. The second argument called derive is True or False. When derive is True, the derivative is returned. When derive is False, the function value is returned.

For example, the quadratic function below returns a quadratic polynomial. The linear term X and constant function K are defined using conditional expressions.

X = lambda x, derive: 1 if derive else x
K = lambda k: lambda x, derive: 0 if derive else k

def quadratic(a, b, c):
    """Return a quadratic polynomial a*x*x + b*x + c.

    >>> q_and_dq = quadratic(1, 6, 8) # x*x + 6*x + 8
    >>> q_and_dq(1.0, False)  # value at 1
    >>> q_and_dq(1.0, True)   # derivative at 1
    >>> q_and_dq(-1.0, False) # value at -1
    >>> q_and_dq(-1.0, True)  # derivative at -1
    A, B, C = K(a), K(b), K(c)
    AXX = mul_fns(A, mul_fns(X, X))
    BX = mul_fns(B, X)
    return add_fns(AXX, add_fns(BX, C))

To complete this implementation and apply Newton's method to polynomials, fill in the bodies of add_fns, mul_fns, and poly_zero below.

def add_fns(f_and_df, g_and_dg):
    """Return the sum of two polynomials."""
    "*** YOUR CODE HERE ***"

def mul_fns(f_and_df, g_and_dg):
    """Return the product of two polynomials."""
    "*** YOUR CODE HERE ***"

def poly_zero(f_and_df):
    """Return a zero of polynomial f_and_df, which returns:
        f(x)  for f_and_df(x, False)
        df(x) for f_and_df(x, True)

    >>> q = quadratic(1, 6, 8)
    >>> round(poly_zero(q), 5) # Round to 5 decimal places
    >>> round(poly_zero(quadratic(-1, -6, -9)), 5)
    "*** YOUR CODE HERE ***"