### Exercises

#### Exercise 1.35

Show that the golden ratio (the section Tree Recursion) is a fixed point of the transformation , and use this fact to compute by means of the `fixed-point/2`

function.

#### Exercise 1.36

Modify `fixed-point/2`

so that it prints the sequence of approximations it generates, using `io:format`

as shown in exercise 1.22. Then find a solution to by finding a fixed point of . (Use Erlang's `math:log`

function, which computes natural logarithms.) Compare the number of steps this takes with and without average damping. (Note that you cannot start `fixed-point/2`

with a guess of 1, as this would cause division by .)

#### Exercise 1.37

a. An infinite *continued fraction* is an expression of the form

As an example, one can show that the infinite continued fraction expansion with the and the all equal to 1 produces , where is the golden ratio (described in the section Tree Recursion]). One way to approximate an infinite continued fraction is to truncate the expansion after a given number of terms. Such a truncation -- a so-called *k-term finite continued fraction* -- has the form

Suppose that `n`

and `d`

are functions of one argument (the term index ) that return the and of the terms of the continued fraction. Define a function `cont-frac/3`

such that evaluating `(cont-frac n d k)`

computes the value of the -term finite continued fraction. Check your function by approximating using

```
(cont-frac (lambda (i) 1.0)
(lambda (i) 1.0)
k)
```

for successive values of `k`

. How large must you make `k`

in order to get an approximation that is accurate to 4 decimal places?

b. If your `cont-frac/3`

function generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

#### Exercise 1.38

In 1737, the Swiss mathematician Leonhard Euler published a memoir *De Fractionibus Continuis*, which included a continued fraction expansion for , where is the base of the natural logarithms. In this fraction, the are all 1, and the are successively . Write a program that uses your `cont-frac/3`

function from exercise 1.37 to approximate , based on Euler's expansion.

#### Exercise 1.39

A continued fraction representation of the tangent function was published in 1770 by the German mathematician J.H. Lambert:

where is in radians. Define a function `(tan-cf x k)`

that computes an approximation to the tangent function based on Lambert's formula. `k`

specifies the number of terms to compute, as in exercise 1.37.