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# Functions in Maple - PowerPoint PPT Presentation

95503 統資軟體課程講義. Functions in Maple. 教授：蔡桂宏 博士 學生：柯建豪 學號 ： 95356026. 1.1 Introduction. Understanding functions in Maple is also a good starting point for our discussions about Maple programming.

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95503統資軟體課程講義

### Functions in Maple

• Understanding functions in Maple is also a good starting point for our discussions about Maple programming.

• There are two distinct ways to represent mathematical functions in Maple. That is Maple expression ( := ) and a Maple function ( -> ).

• the formula has as its domain the set of all real numbers, its codomain is the set of all positive real numbers.

• the formula has as its domain the set of all positive real numbers, its codomain is the set of all positive real numbers.

• the formula has as its domain the set of all negative real numbers, its codomain is the set of all positive real numbers.

• the function f is not invertible.

• The inverse of the function g is .

• The inverse of the function h is .

• So f, g, and h all have the same formula (i.e., rule) but they are not the same function. The domain and codomain are important parts of the definition of a mathematical function.

• These two ways of representing mathematical functions are not equivalent. And it is subtle and non-obvious.

• A Maple function is something defined using arrow notation ( ->).

• A Maple expression is something defined using ( := ).

• x -> x^2;

• x -> a*x^2;

• (x,a) -> a*x^2;

• .

• Maple will treat all unassigned names (i.e., all unknowns) as variables.

• The mathematical function .

• g := x -> x^2 – 1;g(2);

• g(x) := x^2-1;g(2);

• g := cos + ln;

• k := x -> cos(x) + ln(x);

• g(Pi); k(Pi);

• h := cos + (x -> 3*x-1);

• h(z);

• f := x -> (1 + x^2)/x^3;

• g := (1 + (x -> x^2))/(x -> x^3);

• h := (1 + (z -> z^2))/(y -> y^3);

• f(1); g(1); h(1);

• m := x -> (1 + exp(x))/x^3;

• n := (1 + exp)/(x -> x^3);

• m(1); n(1);

• f := x^2;

• g := x -> 2 * x^3 * f;

• g(x); g(2);

• f := x -> x^2;

• g := 2 * (x -> x^3) * f;

• g(x); g(2);

1.4 Expressions vs. functions: Some puzzles

• The following examples are meant to show that there are still a lot of subtle things to learn about variables and functions and how Maple handles them.

• f1 := x^2+1;

• f2 := y^2+1;

• f3 := f1 + f2;

• f3 is a function of two variables.

• g1 := x -> x^2+1;

• g2 := y -> y^2+1;

• g3 := g1 + g2;g3(x);

• g3 is not a function of two variables.

• x:='x': a:=1: b:=2: c:=3:

• a*x^2+b*x+c;

• f := unapply( a*x^2+b*x+c, x );

• g := x -> a*x^2+b*x+c;

• f(x); g(x);

• D(f); D(g);

• p := x^2 + sin(x) + 1; p(2);

• p := unapply(p,x); p(2);

• plot( x^2, x=-10..10 );

• plot( x->x^2, -10..10 );

• plot( x^2, -10..10 );

• plot( x->x^2, x=-10..10 );

• x := 5;

• plot( x^2, x=-10..10 );

• plot( x->x^2, -10..10 );

• f := x^2;

• f := x*f;

• f;

• g := x -> x^2;

• g := x -> x*g(x);

• g(x);

• x^2;

• f := %;

• plot( f, x=-3..3 );

• x^2;

• g := x -> %;

• plot( g, -3..3 );

1.5 Working with expressions and Maple functions (review)

• g := x -> x^2-3*x-10;

• g(x);

• g;

• print(g);

• eval(g);

• op(g);

• f := x^2 - 3*x-10;

• g := x -> x^2-3*x-10;

• subs( x=1, f );

• eval( f, x=1 );

• g(1);

• f(1);

• subs( x=1, g );

• eval( f, x=1 ); subs( x=1, f ); say at 1.

• Think of reading eval( f, x=1) as " evaluate f at x=1" and think of reading subs (x=1,f) as " substitute x=1 into f".

• factor( f );

• factor( g(x) );

• factor( f (x) );

• factor( g );

• diff( f, x ); say at 1.

• D( g );

• diff command needed a reference to x in it but the D command did not.

• D( f );

• diff( g, x );

Let us do an example of combining two mathematical functions f and g by composing them to make a new function .

For the expression:

• f := x^2 + 3*x;

• g := x + 1;

• h := subs( x=g, f );

• subs(x=1, h);

For the function: f and g by composing them to make a new function .

Let us do an example of representing a mathematical function of two variables. Here is an expression in two variables.

• f := (x^2+y^2)/(x+x*y);

• g := (x,y) -> (x^2+y^2)/(x+x*y);

• subs( x=1, y=2, f );

• eval( f, {x=1, y=2} );

• g(1,2);

• simplify( f );

• simplify( g(x,y) );

• diff( f, x );

• simplify( % );

• diff( f, y );

• simplify( % );

• D[1](g);

• simplify( %(x,y) );

• D[2](g);

• simplify( %(x,y) );

1.6 Anonymous functions and expression. expressions (review)

• Here we define an anonymous function and then evaluate, differentiate, and integrate it.

• (z -> z^2*tan(z))(Pi/4);

• x -> x^3 + 2*x;

• %(2);

• D( %% );

• int( (%%%)(x), x );

• plot(w^3+1, w=-1..1); and naming a function are two distinct steps.

• plot( ((x,y)->x^3-y^3)(w,-1), w = -1..1 );

• plot( w->(((x,y)->x^3-y^3)(w,-1)), -1..1 );

1.7. Functions that return a and naming a function are two distinct steps. function (optional)

• f := a -> ( y->a*y );

• f(3);

• f(3)(4);

• f := (x,y) -> 3*x^2+5*y^2; and naming a function are two distinct steps.

• f(x,3);

• fx3 := x -> f(x,3);

• fx3(x);

• slice_f_with_y_fixed := c -> ( x->f(x,c) );

• fx3 := slice_f_with_y_fixed(3);

• fx3(x);

The End ! and naming a function are two distinct steps. Thank you for your listening !