% Problem 5

clear
close all

% We begin by graphing x^4 and 2^x on the interval [-10, 10].

ezplot('x^4', [-10, 10])
hold on
ezplot('2^x', [-10, 10])
hold off
title 'Figure 5.1'
pause
print -deps figA5-1
% This graph appears in Figure 5.1.

% There appear to be points of intersection between -2 and 2, but none between
% 2 and 10 nor between -10 and -2. Let's plot on [-2, 2] to get a clearer 
% picture.

ezplot('x^4', [-2, 2])
hold on
ezplot('2^x', [-2, 2])
hold off
title 'Figure 5.2'
pause
print -deps figA5-2
% This graph appears in Figure 5.2.

% We see that there is one point of intersection at about -0.9, and another 
% near 1.2. Are there other points of intersection? To the left of 0, 2^x is 
% always less than 1, whereas x^4 goes to infinity as x goes to -infinity. On 
% the other hand, both x^4 and 2^x go to infinity as x goes to infinity, so 
% the graphs may intersect again to the right of 0. Let's check.

ezplot('x^4', [10, 20])
hold on
ezplot('2^x', [10, 20])
hold off
title 'Figure 5.3'
pause
print -deps figA5-3
% This graph appears in Figure 5.3.

% We see that they do cross again, near x = 16. Using calculus you can show that 
% the graphs never cross again (by taking logarithms, for example), so we have 
% found all the points of intersection. Low let's use the fzero command to
% find these points of intersection numerically. 

r1 = fzero('x^4 - 2^x', -0.9)

r1 =

   -0.8613

r2 = fzero('x^4 - 2^x', 1.2)

r2 =

    1.2396

r3 = fzero('x^4 - 2^x', 16)

r3 =

    16


% Let's see if the r1, r2, and r3 satisfy the equation.

r1^4 - 2^r1

ans =

  -2.2204e-16

r2^4 - 2^r2

ans =

   8.8818e-16

r3^4 - 2^r3

ans =

     0


% We see that r1 and r2 very nearly satisfy the equation (as we expect for 
% very accurate numerical solutions), and that r3 = 16 satisfies the equation
% exactly. It is easily seen that 16 is an exact solution.

% Finally, let's use the solve command.

solve('x^4-2^x = 0')
 
ans =
 
[    -4*lambertw(-1/4*log(2))/log(2)]
[ -4*lambertw(-1,-1/4*log(2))/log(2)]
[  -4*lambertw(-1/4*i*log(2))/log(2)]
[     -4*lambertw(1/4*log(2))/log(2)]
[   -4*lambertw(1/4*i*log(2))/log(2)]
 

% We have obtained a vector of 5 solutions, stated in terms of the logarithm
% function and the Lambert W function, a function used in advanced mathematics.
% We can get numerical values with the double command.

double(ans)

ans =

   1.2396          
  16.0000          
  -0.1609 + 0.9591i
  -0.8613          
  -0.1609 - 0.9591i


% So, we have two complex solutions, plus the three real solutions found above.

echo off
diary off