# 1.40

## Exercise 1.40: Define a procedure cubic that can be used together with the newtons-method procedure in expressions of the form

(newtons-method (cubic a b c) 1)


## to approximate zeros of the cubic $x^3 + a x^2 + bx +c$ .

### fixed-point

(define tolerance 0.00001)

(define (fixed-point f first-guess)
(define (close-enough? v1 v2)
(< (abs (- v1 v2))
tolerance))
(define (try guess)
(let ((next (f guess)))
(if (close-enough? guess next)
next
(try next))))
(try first-guess))


### deriv

(define dx 0.00001)

(define (deriv g)
(lambda (x)
(/ (- (g (+ x dx)) (g x))
dx)))


### newton-transform

(define (newton-transform g)
(lambda (x)
(- x (/ (g x)
((deriv g) x)))))


### newtons-method

(define (newtons-method g guess)
(fixed-point (newton-transform g)
guess))


### Now let's define cubic

$f(x)=x^3 + a x^2 + bx +c$

(define (cubic a b c) (lambda (x) (+ (* x x x) (* a (* x x)) (* b x) c)))
((cubic 1 1 1) 1)

((cubic 1 1 1) -1)


### Finally let's verify the result

(newtons-method (cubic 1 1 1) 1)