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## PowerPoint Slideshow about 'Functions in Maple' - zaltana-torres

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1.1 Introduction

- 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 ( -> ).

1.2 Functions in Mathematics

- 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.

1.3 Functions in Maple

- 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.

Puzzle 1

- 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.

Puzzle 2

- 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);

Puzzle 3

- 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 );

Puzzle 4

- f := x^2;
- f := x*f;
- f;
- g := x -> x^2;
- g := x -> x*g(x);
- g(x);

Puzzle 5

- 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);

- plot( a*x^2, -5..5 );
- plot( x->a*x^2, -5..5 );
- plot3d( (x,a)->a*x^2, -5..5, -10..10 );

We wanted to evaluate our mathematical function at a point, say at 1.

- 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 .

- f := x -> x^2 + 3*x;
- g := x -> x + 1;
- h := [email protected];
- h(x);
- h(1);

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) );

Here is how we compute partial derivatives of the expression.

- 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 );

- These next two commands show again that defining a function and naming a function are two distinct steps.
- z -> z/sqrt(1-z);
- f := %;

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

- 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 !

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