Add exercises/exercise-02.md

exercises/exercise-02.md

+# Exercise Session 2
+
+This week we will work on playing with functions as values.
+
+## Question 1.
+
+Define the function `flip`. It takes a function and returns the same function, but with the arguments flipped.
+
+```scala
+def flip(f: (Int, Double) => Int): (Double, Int) => Int = ???
+```
+
+## Question 2.1
+
+Define the identity function for integers, which, given an `Int`, returns it.
+
+```scala
+val id: Int => Int = ???
+```
+
+## Question 2.2
+
+Define the compose function, that, given 2 functions `f`, `g`, returns a function that composes them, i.e., `f ∘ g`.
+
+```scala
+def compose(f: Int => Int, g: Int => Int): Int => Int = ???
+```
+
+What does `compose(id, f)(k)` evaluate to, for some function `f` and integer `k`?
+
+## Question 2.3
+
+Define the function `repeated`, which takes a function and repeatedly applies it `n` times (`n ≥ 0`).
+
+```scala
+def repeated(f: Int => Int, n: Int): Int => Int = ???
+```
+
+_Hint_: What values should be returned by `repeated(x => x + 1, 0)` and `repeated(x => x + 1, 3)`?
+
+## Question 3.1
+
+Define the function `curry2`, that curries a two arguments function. That is, `curry2(f) = g` such that `f(x, y) == (g(x))(y)`
+
+```scala
+def curry2(f: (Double, Int) => Boolean): Double => (Int => Boolean) = ???
+```
+
+_Hint_: what should `curry2((x, y) => x < y)(1.0)` return?
+
+
+## Question 3.2
+
+Define the function `uncurry2`. It takes a curried function, and creates a two-argument function.
+
+```scala
+def uncurry2(f: Double => Int => Boolean): (Double, Int) => Boolean = ???
+```
+
+## Question 4.
+
+Write a function `fixedPoint` with the following signature:
+
+```scala
+def fixedPoint(f: Int => Int): Int => Int
+```
+
+The function takes a function `f` and returns a function that maps an integer into the fixed point of f that is obtained by iterating `f` some finite number of times starting from the initial value.
+
+A value `x` is a fixed point of `f` if `f(x) == x`.
+
+For each of the following expressions, indicate whether it terminates, and if so, what is the value returned:
+
+- `fixedPoint(x => x/2)(4)`
+- `fixedPoint(id)(123456)`
+- `fixedPoint(x => x + 1)(0)`
+- `fixedPoint(x => if (x % 10 == 0) x else x + 1)(35)`
+- `fixedPoint((x: Int) => x / 2 + 5)(20)`
+
+## Question 5.1
+
+Write the `sum` function with the following signature:
+
+```scala
+def sum(a: Int, b: Int)(f: Int => Int): Int = ???
+```
+
+Which returns the sum of `f(i)` where `i` ranges from `a` to `b`.
+
+_Bonus point_: Can your implementation be tail recursive ?
+
+## Question 5.2
+
+Write the `quadratic` function with the following signature:
+
+```scala
+def quadratic(c: Int): Int => Int = ???
+```
+
+Which returns a function that takes an integer `x` as argument and returns `(x - c)²`.
+
+## Question 5.3
+
+Using the above functions, define the function `quad3Integrate` which, given two integers `a` and `b`, computes the sum of `(i - 3)²`  where `i` ranges from `a` to `b - 1`.
+
+```scala
+def quad3Integrate(a: Int, b: Int): Int = ???
+val quad3Integrate: (Int, Int) => Int = ???
+```