##########################
### Lecture Slide Examples
##########################
interface(echo=2);
interface(screenwidth=60);
### Exact calculations
12315/35;
(22431)*(832748)*(387281);
sqrt(27);
evalf(%,10);
8^(2/3);
simplify(%);

### Order of operations
(-27/64)^2/3;
(-27/64)^(2/3);

### The surd function
simplify((-27/64)^(2/3));
surd((-27/64),3);
surd((-27/64),3)^2;

### Transforming expressions
factor(12*x^2+27*x*y-84*y^2);
expand((x+y)^2*(3*x-y)^3);
simplify(cos(x)^5+sin(x)^4+2*cos(x)^2);
normal((x^2-y^2)/(x-y)^3);
normal((x^2-y^2)/(x-y)^3,`expanded`);
convert(1/((x-3)*(x-1)),parfrac);
convert(sin(x),exp);

### Defining expressions
exp1:=x^2;
exp1:=x^3;
exp:=x^2;
f:=(x^3+2*x^2)/(x^3+x^2-4*x-4);
subs(x=4,f);
eval(f,x=4);

### Expressions
x:=2;
exp1:=x^2;
exp1:='x^3';
exp1;
unassign('exp1');
exp1;
unassign('x');

### Functions
f:=x->x/(x^2+1);
f(3);
f(3+h);
n1:=simplify((f(3+h)-f(3))/h);
subs(h=0,n1);
map(f,[0,1,2,3]);
[seq(f(n),n=0..3)];

### Solving equations
solve(x^3+x^2+x+1=0);
solve(y=(x-5)^3/8,x);
solve(y=(x-5)^3/8,x)[1];
sys:={3*x-y=4,x+y=2};
sols:=solve(sys);
subs(sols,sys);

### Limits
limit(sin(x)/x,x=0);
limit((1+a/x)^x,x=infinity);

### Differentiation
diff(x^4+4/3*x^3-3*x^2,x);
diff(x^4+4/3*x^3-3*x^2,x$2);

### Integration
int(1/x^2*exp(1/x),x);
int(sin(x)/x,x);
int(exp(-x^2)*ln(x),x);
int(1/sqrt(2*Pi)*exp(-(1/2)*x^2),x=-infinity..a);
int(1/sqrt(2*Pi)*exp(-(1/2)*x^2),x=-infinity..1.96);

### Procedures
fib:= proc(n::nonnegint) 
  if n<2 then 
    n 
  else 
    fib(n-1)+fib(n-2) 
  end if 
end proc:

seq(fib(n),n=0..10);

### Producing computer code
polyeqn:=x^3-a*x-1;
sols:=solve(polyeqn,x);
sol1:=sols[1];
with(CodeGeneration);
C(sol1,optimize=true,declare=[a::float]);

### latex example
expr:=int(exp(-x^2)*ln(x),x);
latex(expr);