Derivatives & Integrals

Includes Taylor Series

1. Example: a function of one variable. Here

f(x) = x3 - 4x2 + cos x

> f := x -> x^3 - 4*x^2 + cos(x) ;  
    defines f as a function of x (rather than as an expression)

> fp1 := unapply( diff( f(x), x ), x ) ;  
    gives f '(x) and stores it as a function named fp1.

> fp2 := unapply( diff( f(x), x$2 ), x ) ;  

    gives f ''(x) and stores it as a function named fp2.

> fp5 := unapply( diff( f(x), x$5 ), x ) ;  
    gives f (5)(x) and stores it as a function named fp5.

> g := unapply( 4*f(x) + fp1(x)^2, x ) ;  
    combines functions f and fp1 and stores it as a function named g.

> int( f(x), x ) ;  
    gives the integral of f with respect to x

> int( f(x), x = -1..3 ) ;  
    gives the definite integral of f(x) from x = -1 to 3

1. Example: a function of more than one variable. Partial derivatives and partial integration. Here

f(x,z) = x3 sin z - 4 x z2 + e4z cos x

> f := (x,z) -> x^3*sin(z) - 4*x*z^2 + exp(4*z)*cos(x) ;  
    defines f as a function of x and z (rather than as an expression)

> diff( f(x,z), x ) ;      differentiates f with respect to variable x only

> diff( f(x,z), z ) ;      differentiates f with respect to variable z only

> int( f(x,z), z ) ;      integrates f with respect to variable z only

> int( f(x,z), x = -1..3 ) ;      integrates f(x) from x = -1 to 3

1. Example: An improper integral. Here f(x) = 1/x3

> f := x -> 1/x^3;      defines f as a function of x (rather than as an expression)

> int( f(x), x ) ; 

> int( f(x), x = 1..infinity ) ;      result: 1/2

NOTE: In maple, the number infinity is represented by infinity.

1. Example: f(x) = e-x2

> f := exp(-x^2);      defines f as an expression (rather than as a function)

> int( f, x = -infinity..infinity ) ;      result: Pi1/2

> int( f, x ) ;      gives what?

1. Taylor Series: To expand an expression as a Taylor series about x=a, usetaylor:

> f := cos(x);     f is defined as an expression

> ftaylor := taylor( f, x = 0, 7 );  
    constructs the Taylor series of f about x = 0 up to x7 and stores it in ftaylor

> f7 := convert( ftaylor, polynom );  
    stores the 7th degreeTaylor polynomial in f7

> plot( { f, f7 }, x = -5 .. 5 );  
    plots function f and the 7th degreeTaylor polynomial f7 on a common graph

> ftaylor := taylor( f, x = Pi/2, 5 );  
    Taylor series of f about x = Pi/2 up to x5 and stores it in ftaylor

> convert( ftaylor, polynom );  
    returns the 5th degreeTaylor polynomial