**(b)** Use Newton's method with Digits = 80 and x[0] = 2 to solve the
equation 1 + sin(2 x) = x. How many digits are correct in each
iteration until x[n+1]=x[n] (to the given accuracy)?

**(c)** Newton's method converges much more slowly when f' is 0 at the
solution. Why? Consider f(x) = (x-1)^3 and start with x[0] = 2.
Find a formula for x[n]. How does the number of correct digits change
with each iteration? How many are needed to get the root correct to
10 decimal places?

**(d)** For an example with even slower convergence, try

f(x) = { x^(1/x) for x > 0 { 0 for x <= 0with x[0] = 1/2. When is x[n] < 1/10?

**2. (a)** Find a polynomial for which Newton's method started with
x[0] = 0 has a stable 3-cycle.

**(b)** Find the largest interval containing 0 for which you can be
sure that Newton's method, starting with x[0] in this interval, is
attracted to the 3-cycle.

**3.** Consider the equations

x^3 + 3 x y^3 - y = 0 x^2 y^2 - 2 x^3 - 3 y^3 = 4

`implicitplot`

to plot the curves for -5 <= x <= 5
and -5 <= y <= 5. Where do the intersections appear to be?
**(b)** Use `fsolve`

to find numerical values for the intersections
you found in (a).

**(c)** With the help of `resultant`

, find all real solutions of
the equations.
One of the solutions may be a surprise. Why?

**(d)** Use `solve`

to find all real solutions of the equations.
Compare the results to those of (c). A higher setting of Digits
is needed to obtain accurate numerical results this way. Why?

**(e)** Use Newton's method to find a solution to the equations.
Start at a point whose coordinates are small integers.

**(f)** Find a point (x[0],y[0]) with |x[0]| < 1 and |y[0]| < 1 such
that if Newton's method is started at this point, the next point
(x[1], y[1]) will be at (or very close to) the solution of the equations
that is farthest from the origin.

** 4. ** Complete the *Optimization on a Restricted Domain*
example in Lesson 14 by finding

**(a)**The critical points of f(x,y) in the region g1(x,y) < 0, g2(x,y) < 0.**(b)**The critical points of the restriction of f(x,y) to g2(x,y) = 0 with g1(x,y) < 0.**(c)**The intersections of the curves g1(x,y)=0 and g2(x,y) = 0.