====== 3. Higher-order functions ======

Objectives:
  * implement and use **higher-order** functions. A **higher-order** function takes other functions as parameter or returns them
  * implement **curry** and **uncurry** functions, and how they should be properly used (review lecture).

**3.1.** Define the function ''foldWith'' which uses an operation ''op'' to reduce a range of integers to a value. For instance, given that ''op'' is addition (+), the result of folding the range 1 to 3 will be 1+2+3=6. ''foldWith'' should be curried (it will take the operation and return another function which expects the bounds).

<code scala>
def foldWith (op: (Int,Int) => Int)(start: Int, stop: Int): Int = {
  def tail_fold(crt: Int, acc: Int): Int  = ???
  ??
}
</code>

**3.2.** Define the function ''foldConditional'' which extends ''foldWith'' by also adding a predicate ''p: Int => Int''. ''foldConditional'' will reduce only those elements of a range which satisfy the predicate.

<code scala>
def foldConditional(op: (Int,Int) => Int, p: Int => Boolean)(start: Int, stop: Int): Int = ???
</code>

**3.3. [//should be revised//] ** Let $math[count_k(n) = k + 2k + 3k + ... x*k], with $math[ x*k \leq n] be the sum of all multiples of $math[k] within the range 1,n. Write a function ''alldivs'' which computes the sum: $math[count_1(n) + count_2(n) + ... + count_k(n)]. (Hint, use ''foldConditional''). 

<code scala>
def alldivs(n: Int): Int = ???
</code>

**3.4.** Write a function ''foldMap'' which takes values $math[a_1, a_2, \ldots, a_k] from a range and computes $math[f(a_1)\;op\;f(a_2)\;op\;\ldots f(a_k)].
<code scala>
def foldMap(op: (Int,Int) => Int, f: Int => Int)(start: Int, stop: Int): Int = ???
</code>

**3.5.** Write a function which computes $math[1 + 2^2 + 3^2 + \ldots + (n-1)^2 + n^2] using ''foldMap''.
<code scala>
def sumSquares(n: Int): Int = ???
</code>

**3.6.** Write a function ''hasDivisor'' which checks if a range contains a multiple of k. Use ''foldMap'' and choose ''f'' carefully.
<code scala>
def hasDivisor(k: Int, start: Int, stop: Int): Boolean = ???
</code>

**3.7.** We can compute the sum of an area defined by a function within a range a,b (the integral of that function given the range), using the following recursive scheme:
  * if the range is small enough, we treat f as a line (and the area as a trapeze). It's area is $math[(f(a) + f(b))(b-a)/2].
  * otherwise, we compute the mid of the range, we recursively compute the integral from a to mid and from mid to b, and add-up the result.

Implement the function ''integrate'' which computes the integral of a function f given a range:
<code scala>
def integrate(f: Double => Double)(start: Double, stop: Double): Double = ???
</code>