# Homework 2 Solutions

## Solution Files

You can find solutions for all questions in hw02.py.

# Required Questions

## Getting Started Videos

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Several doctests refer to these functions:

```
from operator import add, mul
square = lambda x: x * x
identity = lambda x: x
triple = lambda x: 3 * x
increment = lambda x: x + 1
```

## Higher-Order Functions

### Q1: Product

Write a function called `product`

that returns the product of the first `n`

terms of a sequence.
Specifically, `product`

takes in an integer `n`

and `term`

, a single-argument function that determines a sequence.
(That is, `term(i)`

gives the `i`

th term of the sequence.)
`product(n, term)`

should return `term(1) * ... * term(n)`

.

```
def product(n, term):
"""Return the product of the first n terms in a sequence.
n: a positive integer
term: a function that takes one argument to produce the term
>>> product(3, identity) # 1 * 2 * 3
6
>>> product(5, identity) # 1 * 2 * 3 * 4 * 5
120
>>> product(3, square) # 1^2 * 2^2 * 3^2
36
>>> product(5, square) # 1^2 * 2^2 * 3^2 * 4^2 * 5^2
14400
>>> product(3, increment) # (1+1) * (2+1) * (3+1)
24
>>> product(3, triple) # 1*3 * 2*3 * 3*3
162
"""
prod, k = 1, 1
while k <= n:
prod, k = term(k) * prod, k + 1
return prod
```

Use Ok to test your code:

`python3 ok -q product`

The `prod`

variable is used to keep track of the product so far. We
start with `prod = 1`

since we will be multiplying, and anything multiplied
by 1 is itself. We then initialize the counter variable `k`

to use in the while
loop to ensures that we get through all values `1`

through `k`

.

### Q2: Accumulate

Let's take a look at how `product`

is an instance of a more
general function called `accumulate`

, which we would like to implement:

```
def accumulate(fuse, start, n, term):
"""Return the result of fusing together the first n terms in a sequence
and start. The terms to be fused are term(1), term(2), ..., term(n).
The function fuse is a two-argument commutative & associative function.
>>> accumulate(add, 0, 5, identity) # 0 + 1 + 2 + 3 + 4 + 5
15
>>> accumulate(add, 11, 5, identity) # 11 + 1 + 2 + 3 + 4 + 5
26
>>> accumulate(add, 11, 0, identity) # 11 (fuse is never used)
11
>>> accumulate(add, 11, 3, square) # 11 + 1^2 + 2^2 + 3^2
25
>>> accumulate(mul, 2, 3, square) # 2 * 1^2 * 2^2 * 3^2
72
>>> # 2 + (1^2 + 1) + (2^2 + 1) + (3^2 + 1)
>>> accumulate(lambda x, y: x + y + 1, 2, 3, square)
19
"""
total, k = start, 1
while k <= n:
total, k = fuse(total, term(k)), k + 1
return total
# Alternative solution
def accumulate_reverse(fuse, start, n, term):
total, k = start, n
while k >= 1:
total, k = fuse(total, term(k)), k - 1
return total
```

`accumulate`

has the following parameters:

`fuse`

: a two-argument function that specifies how the current term is fused with the previously accumulated terms`start`

: value at which to start the accumulation`n`

: a non-negative integer indicating the number of terms to fuse`term`

: a single-argument function;`term(i)`

is the`i`

th term of the sequence

Implement `accumulate`

, which fuses the first `n`

terms of the sequence defined
by `term`

with the `start`

value using the `fuse`

function.

For example, the result of `accumulate(add, 11, 3, square)`

is

```
add(11, add(square(1), add(square(2), square(3)))) =
11 + square(1) + square(2) + square(3) =
11 + 1 + 4 + 9 = 25
```

Assume that

`fuse`

is commutative,`fuse(a, b) == fuse(b, a)`

, and associative,`fuse(fuse(a, b), c) == fuse(a, fuse(b, c))`

.

Then, implement `summation`

(from lecture) and `product`

as one-line calls to
`accumulate`

.

**Important:** Both `summation_using_accumulate`

and `product_using_accumulate`

should be implemented with a single line of code starting with `return`

.

```
def summation_using_accumulate(n, term):
"""Returns the sum: term(1) + ... + term(n), using accumulate.
>>> summation_using_accumulate(5, square)
55
>>> summation_using_accumulate(5, triple)
45
>>> # This test checks that the body of the function is just a return statement.
>>> import inspect, ast
>>> [type(x).__name__ for x in ast.parse(inspect.getsource(summation_using_accumulate)).body[0].body]
['Expr', 'Return']
"""
return accumulate(add, 0, n, term)
def product_using_accumulate(n, term):
"""Returns the product: term(1) * ... * term(n), using accumulate.
>>> product_using_accumulate(4, square)
576
>>> product_using_accumulate(6, triple)
524880
>>> # This test checks that the body of the function is just a return statement.
>>> import inspect, ast
>>> [type(x).__name__ for x in ast.parse(inspect.getsource(product_using_accumulate)).body[0].body]
['Expr', 'Return']
"""
return accumulate(mul, 1, n, term)
```

Use Ok to test your code:

```
python3 ok -q accumulate
python3 ok -q summation_using_accumulate
python3 ok -q product_using_accumulate
```

We want to abstract the logic of `product`

and `summation`

into `accumulate`

.
The differences between `product`

and `summation`

are:

- How to fuse terms. For
`product`

, we fuse via`*`

(`mul`

). For`summation`

, we fuse via`+`

(`add`

). - The starting value. For
`product`

, we want to start off with 1 since starting with 0 means that our result (via multiplying with the start) will always be 0. For`summation`

, we want to start off with 0.

### Q3: Make Repeater

Implement the function `make_repeater`

that takes a one-argument function `f`

and a positive integer `n`

. It returns a one-argument function, where
`make_repeater(f, n)(x)`

returns the value of `f(f(...f(x)...))`

in which `f`

is
applied `n`

times to `x`

. For example, `make_repeater(square, 3)(5)`

squares 5
three times and returns 390625, just like `square(square(square(5)))`

.

```
def make_repeater(f, n):
"""Returns the function that computes the nth application of f.
>>> add_three = make_repeater(increment, 3)
>>> add_three(5)
8
>>> make_repeater(triple, 5)(1) # 3 * 3 * 3 * 3 * 3 * 1
243
>>> make_repeater(square, 2)(5) # square(square(5))
625
>>> make_repeater(square, 3)(5) # square(square(square(5)))
390625
"""
def repeater(x):
k = 0
while k < n:
x, k = f(x), k + 1
return x
return repeater
```

Use Ok to test your code:

`python3 ok -q make_repeater`

There are many correct ways to implement `make_repeater`

. This solution
repeatedly applies `h`

.

## Check Your Score Locally

You can locally check your score on each question of this assignment by running

`python3 ok --score`

**This does NOT submit the assignment!** When you are satisfied with your score, submit the assignment to Gradescope to receive credit for it.

## Submit

Submit this assignment by uploading any files you've edited **to the appropriate Gradescope assignment.** Lab 00 has detailed instructions.

In addition, all students who are **not** in the mega lab must complete this attendance form. Submit this form each week, whether you attend lab or missed it for a good reason. The attendance form is not required for mega section students.

## Exam Practice

Here are some related questions from past exams for you to try. These are optional. There is no way to submit them.

Note that exams from Spring 2020, Fall 2020, and Spring 2021 gave students access to an interpreter, so the question format may be different than other years. Regardless, the questions below are good problems to try

withoutaccess to an interpreter.

- Fall 2019 MT1 Q3: You Again [Higher-Order Functions]
- Spring 2021 MT1 Q4: Domain on the Range [Higher-Order Functions]
- Fall 2021 MT1 Q1b: tik [Functions and Expressions]