#
# 5.1   Defining functions
#
#
#
# MAPLE SESSION 1
#
>   f:= x -> x^2 - 3*x + 5;
#
#
# MAPLE SESSION 2
#
>   f(2);
#
#
# MAPLE SESSION 3
#
>   g := (x,y) -> x*y/(1+x^2+y^2);
#
#
# MAPLE SESSION 4
#
>   g(sin(t),cos(t));
>   simplify(%);
#
#
# MAPLE SESSION 5
#
>   q := Z^5+3*Z^4-12*Z^3-35*Z^2
>   +42*Z+119:
>   h := unapply(q,Z);
#
#
# MAPLE SESSION 6
#
>   H := x -> evalf( h(x), 4):
>   X := [seq(evalf(-4+i/10,4),i=0..10)];
>   Y := map(H,X);
#
# 5.2   Composition of functions
#
#
#
# MAPLE SESSION 7
#
>   (sin@cos)(x);
>   f := x -> x^2:
>   g := x -> sqrt(1-x):
>   (f@g)(x);
>   (g@f)(x);
#
# 5.3   Summation and product
#
#
#
# MAPLE SESSION 8
#
>   f := 'f':
>   Sum(f(i),i=1..n);
>   Sum(i^2,i=1..10);
>   sum(i^2,i=1..10);
#
#
# MAPLE SESSION 9
#
>   Sum(i^2,i=1..10);
>   value(%);
>   sum(i^2,i=1..n);
>   factor(%);
#
#
# MAPLE SESSION 10
#
>   f := 'f': q := 'q':
>   Product(f(i),i=1..n);
>   Product(1-q^i,i=1..5);
>   value(%);
>   expand(%);
#
#
# MAPLE SESSION 11
#
>   i:=2;
>  sum(i^3,i=1..5);
#
#
# MAPLE SESSION 12
#
>  sum('i'^3,'i'=1..5);
#
# 5.4   Limits
#
#
#
# MAPLE SESSION 13
#
>   Limit((x^2-4)/(x-2),x=2); value(%);
#
#
# MAPLE SESSION 14
#
>   f:=(x^2-4)/(x^2-5*x+6);
>   Limit(f,x=3,right); value(%);         
>   Limit(f,x=infinity); value(%);         
#
# 5.5   Differentiation
#
#
#
# MAPLE SESSION 15
#
>   f := sqrt(1 - x^2);
>   diff(f,x);
>   g := z -> z^2*exp(z) + sin(log(z)):
>   diff(g(x),x);
#
#
# MAPLE SESSION 16
#
>   z := exp(x*y)*(1+sqrt(x^2+3*y^2-x));
>   diff(z,x);
>   normal(diff(z,x,y)-diff(z,y,x));
#
#
# MAPLE SESSION 17
#
>   g := z -> z^2*exp(z) + sin(z);
>   D(g);
#
# 5.6   Extrema
#
#
#
# MAPLE SESSION 18
#
>    maximize(sin(x));
>    maximize(sin(x)+cos(x));
>    maximize(x^2-5*x+1,x=0..3);
>    maximize(sin(x),x=0..1);
>    maximize(sin(x)+cos(x),x=0..1);
>    maximize(sin(x)+cos(x),x=0..1/2);
#
#
# MAPLE SESSION 19
#
>   minimize(x^2+y^2);
>   minimize(x^2+y^2,x);
#
#
# MAPLE SESSION 20
#
>   readlib(extrema):
#
#
# MAPLE SESSION 21
#
>   f := 2*x^2 + y + y^2;
>   g := x^2 + y^2 - 1;
>   extrema(f,{g},{x,y},'s');
>   s;
>   simplify(subs(s[1],f));
>   simplify(subs(s[2],f));
>   simplify(subs(s[3],f));
#
# 5.7   Integration
#
#
#
# MAPLE SESSION 22
#
>   Int(x^2/sqrt(1-x^3),x);
>   value(%);
>   Int(1/x/sqrt(x^2 - 1),x=1..2/sqrt(3));
>   value(%);
#
#
# MAPLE SESSION 23
#
>   Int(sqrt(1+x^6),x=0..1);
>   value(%);
>   evalf(%);
#
# 5.7.1   Techniques of integration
# 5.7.1.1   Substitution
#
#
#
# MAPLE SESSION 24
#
>    with(student): 
>    G:=Int(x^4/sqrt(1-x^10),x);
>    changevar(u=x^5,G,u);
>    G2 := value(%);
>    subs(u=x^5,G2);
>    diff(%,x);
#
#
# MAPLE SESSION 25
#
>    G:=Int((3*x^2+1)/sqrt((1-x-x^3)*(1+x+x^3)),x);
>    value(G);
#
#
# MAPLE SESSION 26
#
>    radsimp(changevar(u=x+x^3,G,u));
#
# 5.7.1.2   Integration by parts
#
#
#
# MAPLE SESSION 27
#
>   with(student):
>   Int(x*cos(3*x),x);
>   intparts(%,x);
>   value(%);
#
# 5.7.1.3   Partial fractions
#
#
#
# MAPLE SESSION 28
#
>   rat := (4*x^4+9*x^3+12*x^2+9*x+4)
>   /(x + 1)/(x^2 + x + 1)^2:
>   convert(rat,parfrac,x);
>   int(%,x);
#
# 5.8   Taylor and series expansions
#
#
#
# MAPLE SESSION 29
#
>   y := 1/sqrt(1-x);
>   taylor(y,x=0,5);
#
#
# MAPLE SESSION 30
#
>   J := product(1-x^'i','i'=1..50):
>   taylor(J^3,x=0,20);
>   coeff(%,x,15);
#
# 5.9   The  student package
#
#
#
# MAPLE SESSION 31
#
>    with(student): 
>    p := x^2 + 6*x + 13;
>    completesquare(p);
>    q := x^2 + 10*x + 2*y^2 + 12*y + 12;
>    completesquare(q);
>    completesquare(q,x);
>    completesquare(%,y);
#
#
# MAPLE SESSION 32
#
>    with(student): 
>    distance(-3,5);
>    distance([1,2],[-3,4]);
#
#
# MAPLE SESSION 33
#
>    with(student): 
>    equate(x,y);
>    equate([x+y,x-y],[3,-1]);
>    solve(%);
#
#
# MAPLE SESSION 34
#
>    with(student): 
>    F := Int(sin(x^3*y),x=0..2*Pi);
>    integrand(F);
#
#
# MAPLE SESSION 35
#
>    with(student): 
>    intercept(y=5*x-3);
>    intercept(y=x^2+3*x-20,y=2*x^2+x-23);
#
#
# MAPLE SESSION 36
#
>    with(student): 
>    leftbox(sin(x^2), x=0..Pi, 6, shading=green);
#
#
# MAPLE SESSION 37
#
>    with(student): 
>    leftsum(sin(x^2), x=0..Pi, 6);
>    value(%);
>    evalf(%);
>    evalf(int(sin(x^2),x=0..Pi));
#
#
# MAPLE SESSION 38
#
>    with(student): 
>    y := x^2 + x - 3;
>    f := makeproc(y,x);
>    f(x);
#
#
# MAPLE SESSION 39
#
>    with(student): 
>    f := makeproc([-1,1],[3,7]);
>    f(-1), f(3);
#
#
# MAPLE SESSION 40
#
>    with(student): 
>    f := makeproc([2,5],'slope'=3);
>    f(2);
>    diff(f(x),x);
#
#
# MAPLE SESSION 41
#
>    with(student):
>    midpoint([2,3],[5,7]);
#
#
# MAPLE SESSION 42
#
>    with(student): 
>    showtangent(sin(x),x=Pi/4);
#
#
# MAPLE SESSION 43
#
>    with(student): 
>    simpson(1/sqrt(1+x^4),x=0..1,12): 
>    value(%): 
>    app := evalf(%);
>    xval := evalf(int(1/sqrt(1+x^4),x=0..1));
>    abs(app-xval);
#
#
# MAPLE SESSION 44
#
>    with(student): 
>    slope(y=2*x-5);
>    slope(2*y+12=3*x);
>    slope(2*y+12=3*x,y(x));
>    slope([12,5],[3,7]);
#
#
# MAPLE SESSION 45
#
>    with(student): 
>    z3 := Sum(1/n^3,n=1..infinity);
>    summand(z3);
#
#
# MAPLE SESSION 46
#
>    with(student): 
>    trapezoid(1/sqrt(1+x^4),x=0..1,12): 
>    value(%): 
>    app := evalf(%);
>    xval := evalf(int(1/sqrt(1+x^4),x=0..1));
>    abs(app-xval);