*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.

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:

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

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)
4.0
"""
"*** YOUR CODE HERE ***"
```

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
15.0
>>> q_and_dq(1.0, True) # derivative at 1
8.0
>>> q_and_dq(-1.0, False) # value at -1
3.0
>>> q_and_dq(-1.0, True) # derivative at -1
4.0
"""
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
-2.0
>>> round(poly_zero(quadratic(-1, -6, -9)), 5)
-3.0
"""
"*** YOUR CODE HERE ***"
```