[1] Observe that this is a combination whose operator is itself a combination. Exercise 1.4 already demonstrated the ability to form such combinations, but that was only a toy example. Here we begin to see the real need for such combinations—when applying a function that is obtained as the value returned by a higher-order function.
[2] See exercise 1.45 for a further generalization.
[3] Elementary calculus books usually describe Newton's method in terms of the sequence of approximations $x_{n+1}=x_n-g(x_n)/Dg(x_n)$. Having language for talking about processes and using the idea of fixed points simplifies the description of the method.
[4] Newton's method does not always converge to an answer, but it can be shown that in favorable cases each iteration doubles the number-of-digits accuracy of the approximation to the solution. In such cases, Newton's method will converge much more rapidly than the half-interval method.
[5] For finding square roots, Newton's method converges rapidly to the correct solution from any starting point.
[6] The notion of first-class status of programming-language elements is due to the British computer scientist Christopher Strachey (1916–1975).
[7] We'll see examples of this after we introduce data structures in chapter 2.
[8] The major implementation cost of first-class functions is that allowing functions to be returned as values requires reserving storage for a function's free names even while the function is not executing. In the JavaScript implementation we will study in section 4.1, these names are stored in the function 's environment.
1.3.4 Functions as Returned Values