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- Exercises
- Exercise 0: Preparation
- Exercise 1: Product
- Exercise 2: Squaring Lists
- Exercise 3: Lengths of Lists
- Exercise 4: How Much Did You Skip?
- Exercise 5: No Skipping Allowed
- Exercise 6: Tallying Symbols
- Exercise 7: Counting Odd Numbers
- Exercise 8: Extracting Odds
- Exercise 9: Closest to Zero
- Exercise 10: Finding the Gaps
- Exercise 11: Are They All Valid?
- Exercise 12: Is it there?
- Exercise 13: Types
- Exercise 14: How Many Values?

- Notes

Scan through the reading on recrusion.

Define and test a Scheme procedure `product`

that takes a list
of numbers as its argument and returns the result of multiplying them all
together. Warning: `(product null)`

should *not* be 0.
It should be the identity for multiplication, just as ```
(sum
null)
```

is the identity for addition. Explain why.

Define and test a Scheme procedure `square-each-element`

that
takes a list of numbers as its argument and returns a list of their
squares.

>(square-each-element (list -7 3 12 0 4/5))(49 9 144 0 16/25)

Hint: For the base case, consider what the procedure should return when given a null list; for the other case, separate the car and the cdr of the given list and consider how to operate on them so as to construct the desired result.

Define and test a Scheme procedure `lengths`

that takes a list
of lists as its argument and returns a list of their lengths:

>(lengths (list (list 'alpha 'beta 'gamma) (list 'delta) null (list 'epsilon 'zeta 'eta 'theta 'iota 'kappa)))(3 1 0 6)

Define and test a Scheme procedure named `tally-skips`

that
takes one argument, a list, and determines how many times the symbol
`skip`

occurs in the list.

Define and test a Scheme procedure named `filter-out-skips`

that
takes a list of symbols as its argument and returns a list that does not
contain the symbol `skip`

, but is otherwise identical to the
given list. (Use the predicate `eq?`

to test whether two
symbols are alike.)

>(filter-out-skips (list 'hop 'skip 'jump 'skip 'and 'skip 'again))(hop jump and again)

The example illustrates the intended effect of the procedure. By itself,
however, it's not an adequate test of your procedure. It would be a good
idea to test the case in which the given list is empty, a case it which it
contains only `skip`

s, and one in which it contains only symbols
other than `skip`

. You might also test different positions
opf `skip`

: at the front, at the end, and in the middle.

I recommend that you test the procedures you create very thoroughly. In most cases, testing does not reveal any errors in your procedures; but finding and correcting the errors that testing exposes is one of the most productive and rewarding uses of a programmer's time.

Define and test a Scheme procedure named `tally-occurrences`

that takes two arguments, a symbol and a list of symbols, and determines
how many times the given symbol occurs in the given list.

**Hint:** Use direct recursion. Here are the questions that you must
resolve: What is the base case? What value should the procedure return in
that case? How can you simplify the problem in order to recursively invoke
the procedure being defined? What do you need to do with the value of the
recursive procedure call in order to obtain the final result?

>(tally-occurrences 'apple (list 'pear 'apple 'cranberry 'banana 'apple))2 >(tally-occurrences 'apple (list 'oak 'elm 'maple 'spruce 'pine))0

Write a Scheme procedure, `num-odds`

, that returns the
number of odd numbers in a list.

If you'd like to be extra careful, make sure that your procedure works correctly even if some of the values in the list are not numbers.

Write a Scheme procedure, `odds`

that, given a list,
produces another list that contains only the odd numbers in the
first list.

Write a Scheme procedure, `closest-to-zero`

that, given
a list of numbers (including both negative and positive nubmers),
returns the value closest to zero in the list.

Define and test a Scheme procedure named `gaps`

that takes
a non-empty list of real numbers as its argument and returns a list of
the disparities between numbers that are adjacent on the given list.

>(gaps (list 30 16 21 9 42))(14 5 12 33)

Note that gaps always returns a list one element shorter than the one it is given.

Hint: What is the base case?

Define and test a Scheme predicate `all-in-range?`

that takes a
list as argument and determines whether all of its elements are in the
range from 0 to 100, inclusive.

Define and test a Scheme predicate `element?`

that takes two
arguments, a symbol and a list, and determines whether the given symbol is
an element of the given list.

Define and test a Scheme predicate, `types`

, that takes a
list as an argument and returns a list of the types of the individual
elements.

For example,

>(types '(34 a (1 2 3)))(number symbol list) >(types '())() >(types '(a b c))(symbol symbol symbol)

You may want to use your `type`

procedure from a previous
lab.

**This problem is for extra credit.**

Some of you have criticized the build-in `length`

method
for counting only the number of values in the top-level list. Write
a procedure, `count-values`

that counts the total number
of non-list values in a list or its sublists.

For example,

>(count-values '(a b c))3 >(count-values '())0 >(count-values '((1 2 3) (a b) ((2))))6 >(count-values '(() () () ()))0

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