Sunday's Exercises

Exercise 1:

Write a procedure that when given an a,b,c, computes the (more)
positive root of the quadratic polynomial ax^2+bx+c.  If the root is
complex signal an error: (error "Complex roots!")

(quad-roots 1 -3 2)
;Value: 2
(quad-roots 1 2 1)
;Value: -1
(quad-roots 1 1 1)
;Error: Complex roots!

Exercise 2:

Write a procedure that uses Euclid's algorithm to compute the GCD of
two numbers.  Euclid's algorithm (according to Knuth it's the oldest
known algorithm) goes as follows: if r is the remainder of a divided
by b, then the common divisors of a and b are the same as those of b
and r.  Additionally, the gcd of a number and 0 is the number.

(gcd 206 40)
;Value: 2

Exercise 3:

We can approximate sin x using a recursive process:
if x is close to zero (< .1 for example), sin x = x
if x is greater reduce using trig: sin x = 3 sin (x/3) - 4 (sin (x/3))^3
  (the end reads: minus sin cubed of x over 3)
Suggested plan:
A) write cube
B) write a procedure that computes sin x given sin (x/3)
C) write a procedure that combines the two.

(sine 0)
;Value: 0
(sine 3.14159)
;Value: -0.000785519012433511
(sine (/ 3.14159 2.0))
;Value: 0.9999999999991198