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Functions in Maple 95356026 95503 Slide 2 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 ( -> ). Slide 3 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. Slide 4 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. Slide 5 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; Slide 6 . 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); Slide 7 g := cos + ln; k := x -> cos(x) + ln(x); g(Pi); k(Pi); h := cos + (x -> 3*x-1); h(z); Slide 8 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); Slide 9 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); Slide 10 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. Slide 11 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. Slide 12 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); Slide 13 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 ); Slide 14 x := 5; plot( x^2, x=-10..10 ); plot( x->x^2, -10..10 ); Slide 15 Puzzle 4 f := x^2; f := x*f; f; g := x -> x^2; g := x -> x*g(x); g(x); Slide 16 Puzzle 5 x^2; f := %; plot( f, x=-3..3 ); x^2; g := x -> %; plot( g, -3..3 ); Slide 17 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); Slide 18 plot( a*x^2, -5..5 ); plot( x->a*x^2, -5..5 ); plot3d( (x,a)->a*x^2, -5..5, -10..10 ); Slide 19 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 ); Slide 20 eval( f, x=1 ); subs( x=1, f ); 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 ); Slide 21 diff( f, x ); D( g ); diff command needed a reference to x in it but the D command did not. D( f ); diff( g, x ); Slide 22 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); Slide 23 For the function: f := x -> x^2 + 3*x; g := x -> x + 1; h := f@g; h(x); h(1); Slide 24 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) ); Slide 25 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) ); Slide 26 1.6 Anonymous functions and 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 ); Slide 27 These next two commands show again that defining a function and naming a function are two distinct steps. z -> z/sqrt(1-z); f := %; Slide 28 f := ((x,y) -> x^2) + ((x,y) -> y^3); f(u,v); f(2,3); g := (x -> x^2) + (y -> y^3); g(u,v); g(2,3); Slide 29 plot(w^3+1, w=-1..1); plot( ((x,y)->x^3-y^3)(w,-1), w = -1..1 ); plot( w->(((x,y)->x^3-y^3)(w,-1)), -1..1 ); Slide 30 1.7. Functions that return a function (optional) f := a -> ( y->a*y ); f(3); f(3)(4); Slide 31 f := (x,y) -> 3*x^2+5*y^2; 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); Slide 32 The End ! Thank you for your listening !