(A)  Substitution in expressions.

>  restart;
> a :=  t + ln(t) - sin(t);
> subs(t=5, a);
> b: = subs(t=3, a);
> evalf(b);
>  subs(x=4.17,  x^1.8 - x^0.4 + tan(x));
> evalf(%);
>  c:= evalf( subs(x=1.23, y=9.41, x^2 - y^2 +x*y - 1) );

Exercise:  Define g(x) =  x sin(1/x).   Find the slope of the line joining the points (1, g(1)) and (3, g(3)).

(B)  Solving equations.

>  restart;
>  y := x^4 + 2.5*x^3 - 9.13*x^2 - 13.633*x + 21.25;
>  plot( y, x=-4..4);   # Guess the number of roots of this polynomial
>  fsolve( y = 0);
>  fsolve (y = 0, x=1..3);   # Here we search for solutions over the interval 1 < x < 3.

>  g := t^5 - 5.3*t^4 + 9.9*t^3 - 7.9*t^2 + 2.3*t;
>  plot( g, t = -2..5);
>  fsolve (g=0);
>  fsolve (g=0, t, avoid={t=0, t=1} );  # Here we exclude the two solutions t=0 and t=1.

Sometimes fsolve yields only one of the solutions that we desire.  With a little bit of effort, we can find other solutions.

>  restart;
>  f := (x^4 - x^3 + x^2 - 13)*cos(x);
>  plot(f, x = -10..10);
>  plot(f, x = -2..2);
>  plot(f, x = -1.4..1.6);  #  Try to guess how many roots the function f has.
>  fsolve(f = 0);    #  Is this a surprise?
>  fsolve(f = 0, x = -10..-7);
>  fsolve(f = 0, x = -6..-4);
>  fsolve(f = 0, x= 1..2);
>  fsolve(f = 0, x= 1.4..1.6);

Can you find other roots?

(C)  Graphing.

>  restart;
>  h :=  x*cos(1/x);
>  plot( h, x=0.1 .. 1);
>  plot( h, x=0.1 .. 1, color = BLUE);
>  plot( h, x=0.1 .. 1, color = GREEN, thickness = 3);
>  plot([sqrt(x), 3*log(x)], x=0..300, title = `The Square Root and Log Functions`);
>  # Note that this single quotation mark (often called the backquote) is located on the far left key just under Esc.

>  plot( x^3 - 5, x=0..3, axes = FRAME);
>  plot( x^3 - 5, x=0..3, axes = BOXED);
>  plot( x^3 - 5, x=0..3, axes = NORMAL);

We can even specify individual colors for multiple graphs:

>  plot ( [sqrt(x), log(x), x^0.8],  x = 0.2..1.5, color = [GREEN, VIOLET, BLUE], x=0..2);

Plotting several graphs on the same set of axes requires use of the display command if one of the graphs is that of a parameterized curve.  The display command requires the use of the plots library, invoked by with(plots).

>  with(plots):  # Note the use of the colon to suppress any output.
>  p1 := plot (sin(x), x=0..2*Pi, color = GREEN):
>  p2 := plot (sin(2*x), x=0..2*Pi, color = BLACK):
>  p3 := plot ([t, 0.5*t^0.4, t=0..3], color = BLUE):   #  This is the graph of a parameterized curve.
>  display (p1, p2, p3);

(D)  Controlling the number of digits.

>  Digits := 5;
>  evalf(exp(5);
>  evalf(sin(Pi/5));
>  Digits := 15;
>  evalf(tan(Pi/19);
> Digits := 7;
> fsolve (sin(x)*exp(x) =1);