Section 2. Symbolics: Equations and Assignments

Solving equations is the bread and butter of mathematics. Maple does it in a natural way. It is always a good idea to assign a name to the equation and the solution.

0. Pull down the Help menu, choose New Users/Full Tour, then click on the third item: Algebraic Calculations and work through the examples in that worksheet.

Open a new Maple worksheet and do the following.

1. Make the following entry.

restart; x := y - z;  y = 3: z := 4;

Then enter x and explain the output.

2. Make the following entry.

restart; x := y - z;  y := 3: z = 4;

Then enter x and explain the output.

3. Make the following entry

restart; x := y - z;  y := 3: z := 4;

Then enter x and explain the output.

Now enter the following

restart;  x := z:  z := y - w:  w := 5:

Write on a pad of paper what you think Maple has stored for x, y, z, and w. Then predict the output for the entry

x + z;

Execute this entry to check your guess. Then do a restart by clicking on the restart button on the button bar (just to the right of the bug).

4. Enter the equation

x^3-x^2+x-1 = 0

with the name eqn.

Solve the equation and name the solutions solns with the entry

solns := solve( eqn, {x});

Check the third solution with the entry

eval( eqn, solns[3]);

Explain why the output confirms the validity of the solution.

Check all three solutions with the entry

eval(eqn,solns[k]) $ k=1..3;

Read the error message carefully and fix the problem (premature evaluation) by enclosing the eval procedure in single quotes to delay evaluation as shown below . Do not retype the entry, just add the single quotes and press the [return] key.

'eval(eqn,solns[k])' $ k=1..3;

5. Use fsolve to obtain an approximate real solution to the equation

x^3-Float(9, -1)*x^2+x-1 = 0

Name the solution soln.

Hint. Name the equation eqn and simply enter soln := fsolve(eqn,{x}).

Check the approximate solution using eval(eqn,soln).

6.* Plot the expression

x^3-x^2+Float(5, -2)*x+cos(x)-Float(7, -1)

with the entry

y := x^3 - x^2 + 0.05*x + cos(x) - 0.7;   plot( y, x=-2..2, -2..2);

Enter fsolve(y=0,x) to see which zero fsolve finds.

Using the graph as a guide, obtain an approximation to the largest positive zero using fsolve(y=0,{x},a..b).

7. Find the first three positive solutions to the equation

cos(x) = x*tan(x)

Hint. Define the function y = cos(x) -  x tan(x) with the entry

y := cos(x) - x*tan(x);

Plot y using appropriate domain and range values (and discont=true), then use fsolve with specific interval settings obtained from the graph.

8. Let y be the function of x defined in Exercise 7. Enter

solve(y=0,{x});  evalf(%);

and comment on the output.

9. Graph the function

y = cos(x^2)-sin(2*x)

over the interval from x = 0 to x = 2. Define yp as its derivative and find the zero of yp near x = 1.5. Name it xmin. Then find the minimum value of y over this interval using eval(y,x=xmin). Name it ymin.

Use the following entry to plot the graph and the low point.

plot( [y, [ [xmin,ymin] ]  ], x=0..2, style=[line,point]);

10.* Find the area of the region between the graph in Exercise 9 and the x-axis.

Hint: Numerically integrate the absolute value of y from x = 0 to x = 2 via the entry

evalf( Int( abs(y), x=0..2) );