Introduction

In the first Welcome to Maple project you learned how to enter expressions, assign variables, define functions and plot functions in Maple. In this project you will learn how to use some other important tools of Maple including the solve and  fsolve  commands.

The solve  command

The solve  command in Maple is used to solve an equation algebraically. Suppose we want to solve the equation . Of course this equation can be rewritten as a single quadratic equation and we could apply the quadratic formula. Maple will do this for us automatically with the solve  command. The solve command requires the equation and the variable we wish to solve for.

>	solve(3*x+3 = x^2-4*x+1,x);

The equation we just solved had only one variable. Maple can deal with equations involving more than one variable providing the equation can be solved algebraically. For example consider the equation . Suppose we wanted to solve this equation for x.

>	solve(1/(x+k) = k/(x+2*k),x);

We could also solve this equation for k in terms of x

>	solve(1/(x+k) = k/(x+2*k),k);

The first argument of a solve command is normally an equation . If you enter an expression without an equals sign, Maple assumes you are setting the expression equal to zero.

>	solve(x^2-4,x);

>	solve(x^2+4,x);

The solve  command can also work with equations involving functions that have been defined in Maple

>	f:=x->x^2+7*x-5; solve(f(x)=3,x);

>	solve(sin(x)=sin(2*x),x);

The solve command attempts to solve an equation algebraically . There are many equations that cannot be solved algebraically. An example of such an equation is .

>	solve(cos(x)=x,x);

The output indicates that Maple cannot solve this equation.

Maple can be used to solve systems of equations. The system is entered as a set  of equations in curly braces  and the variables are also entered as a set  of variables.

>	solve({3*x+4*y=2,x=-2*y+3},{x,y});

Sometimes Maple stops short of completely solving a single equation or a system of equations.

>	solve({x^2+y^2=1,y=x^2},{x,y});

The following command forces Maple to try as hard as possible to completely solve an equation or system of equations. Note that the command starts with an underscore.

>	_EnvExplicit:=true;

>	solve({x^2+y^2=1,y=x^2},{x,y});

Project - Part 1

1) Use the solve  command to find the point ( x , y ) where the lines  and  intersect.

2) Determine the points of intersection of the circle of radius 1 centered at the origin and the circle of radius 4 centered at the point (0,4).

3) Let k  denote the number of letters in your last name. Determine where the graphs of  and  intersect. Make a plot of these functions. Get a good view so that the points of intersection are clearly indicated.

The fsolve  Command

Most equations encountered in the real world cannot be solved algebraically, and one must turn to obtaining good numerical approximations to solutions of such equations. Consider the equation . A plot of the functions  and  is shown below.

Clearly these graphs intersect at a point ( x , y ) with x  between 0 and 1. To get a good approximation of this value of x  we use the fsolve  command. It is very similar to the solve  command except that it tells Maple to solve the equation numerically .

>	fsolve(cos(x)=x,x);

As another example consider the equation . A plot of the functions  and  is shown below. Note that there are apparently two solutions between x  = 0 and x  = 3. Basic knowledge of the graphs of these two functions indicates that these are the only solutions.

>	fsolve(exp(x)=5*x,x);

Note that when the fsolve  command is applied, only one solution is returned. An important feature of the fsolve  is that one can specify an interval [ a , b ] on which Maple is to search for a numerical solution. This information is inserted as a third component in the fsolve  command. It is entered as a range a .. b .

>	fsolve(exp(x)=5*x,x,2..3);

An alternative way of specifying the range is

>	fsolve(exp(x)=5*x,x=2..3);

The fsolve  command together with the plot  command provide a powerful tool for getting numerical  solutions of equations. In most cases the plot tool will allow you to determine an x - interval that contains a single solution. Then the fsolve  command can be used to get a good numerical approximation to the solution.

Project - Part 2

Define the functions  and  where k  is the number of letters in your last name. Plot the graphs of these functions in the same window and show that they have two points of intersection. Use the fsolve  command to determine the points of intersection. The two points of intersection determine a line in the plane. Determine an equation of this line. Express the line as a function of x . Plot the graphs of the functions f(x), g(x) and your line in the same window. Adjust the view of the window so that you have a good plot.