CS99 Lecture6
- Functions
- Chapter 5 (forthcoming)
- week of 9/2

Mathematical Functions
- definition 
  correspondance between domain and range
- y = f(x) = something with x in it
  + y is dependent variable
  + x is independent variable
  + f(x) takes x as input and spits out a value
- what about domain and range?
  + essentially ignore this concept for now
  + will focus on input/output concept

Maple Functions
- perform task (think, "commands")

- syntax:
  + expr(exprseq)
    usually, expr is name
    exprseq is expr,expr,...

  ex) sqrt(4)
  ex) plot(x,x=0..10)  

- INPUT arguments (parameters): exprseq
  Maple evaluates each expr before PASSING into code

- OUTPUT expression
  Maple 
  + evaluates entire function using inputs
  + produces output

  ex) Ans := sqrt(4);
     
        Ans := 2

      Maple replaces function with OUTPUT!

  ex) x := 3;
      Ans := sqrt(x^2);
          Ans := 3

- Nesting
  ex) A:=Pi/6: B:=sin(A): sqrt(B);

      Active  Exercise: How to make nested?
      sqrt(sin(Pi/6))

          (1/2) sqrt(2)

- Powers 
     2
  sin (x)  means  sin(x)^2

- Incorrect notation
  > f(x) := m*x + b will NOT work!
  correct: 
  > restart;
  > f := x -> m*x + b;
  > f(2);

     Active Exercise: answer?
          m + b  

Variety of Functions
- see ?polynom, ?trig

  note: must use radians for angles!
    convert(angle,radians)
    or use rads/2*Pi = degs/360

Powers and Roots
- exponentiation: ** and ^
- roots: expr^(expr)
         root(expr,n)
         surd(expr,n) -> finds more real roots!

  ex) > simplify((-1)^(1/3));
      > root(-1,3) 
   
      1/2 + 1/2 I sqrt(3)

      > surd(-1,3)

Logarithms
 
  x                    
 b  = y  means log x   means Maple: log[b](y)
                  b

 log[b](y) -> ln(y)/ln(b)

 ln? Natual log -> log base e
 e? exp(1) = 2.71828...

 Maple: ln(x), exp(x)