Math 210 Sec. 201 Assignment 1 - due January 26, 1996

1. Consider the equation

x^{4} − 11x^{3} + 54x^{2} − 108x + 55 = 0  (1)

(a) Find the exact solutions of (1). Note: solve doesn’t usually solve fourth-degree equations explicitly. You need to set an “environment variable” with the command

>  EnvExplicit:= true;

Do this before you try solving the equation, otherwise solve will just remember its previous result.

(b) Use evalf to numerically evaluate these solutions, to 14 significant digits.

(c) Use fsolve to numerically compute the real solutions to 14 significant digits.

(d) Which answers are more accurate? Check the accuracy by substitution. Discuss why this might happen.

2. The following equation occurs in the theory of crystal growth.

x^{3} + e^{1/x} = c  (2)

(a) Let f(x) be the left side of (2), and use Maple to compute f′(x) and f′′(x). Note that E, not e, is used for the base of natural logarithms (but it’s best to use exp).

(b) Show that for any given value of c, there are at most 2 positive solutions of (2). (A “theoretical” answer is intended here — a graph may help guide your intuition, but doesn’t prove anything)

(c) Find numerically the smallest value of c for which (2) has positive solutions. Explain your reasoning.

(d) Find numerically the positive solutions of (2) when c = 4.

(e) Discuss the accuracy of the solutions in (d).

3. (a) Using Maple, show that the function g(x) = 1.5 cos x has a fixed point that is a repeller. (b) By experimenting, guess the typical behaviour of the iteration x_{n} = g(x_{n−1}).

(c) Prove using Maple that your guess is correct.

(d) What happens for g(x) = 2 cos x? Try some experiments, and draw cobweb diagrams.

(e) Find a constant c such that the iteration g(x) = c cos x has a cycle of period 3 that is an attractor.