lab_sols04

Lab sheet 4

1 Parametric plotting

Exercise 1.1

-->	wxplot2d ( [ parametric , cos ( 10 · t ) / ( 1 + t ^ 2 ) , sin ( 10 · t ) / ( 1 + t ^ 2 ) , [ t , − 5 , 5 ] ] , [ same_xy ] ) ;

\[\operatorname{ }\]

\[\operatorname{ }\]

-->	wxplot2d ( [ parametric , sin ( 3 · t ) , sin ( 2 · t ) , [ t , 0 , 2 · %pi ] ] , [ same_xy ] , [ box , false ] , [ axes , false ] ) ;

\[\operatorname{ }\]

-->	wxdraw ( gr2d ( nticks = 100 , proportional_axes = ' xy , xtics = false , ytics = false , axis_top = false , axis_bottom = false , axis_left = false , axis_right = false , color = red , parametric ( sin ( 3 · t ) , sin ( 2 · t ) , t , 0 , 2 · %pi ) ) ) ;

\[\operatorname{ }\]

-->	wxplot2d ( [ parametric , t − sin ( t ) , 1 − cos ( t ) , [ t , 0 , 8 · %pi ] ] , [ same_xy ] , [ box , false ] , [ axes , false ] ) ;

\[\operatorname{ }\]

-->	wxplot2d ( [ parametric , 2 · t / ( 1 + t ^ 2 ) , ( 1 − t ^ 2 ) / ( 1 + t ^ 2 ) , [ t , − 10 , 10 ] ] , [ same_xy ] ) ;

\[\operatorname{ }\]

Exercise 2

-->	x : 4 · t ^ 2 − 1 ; y : 8 · t ^ 3 − 8 · t ;

\[\operatorname{(x) }4 {{t}^{2}}-1\]

\[\operatorname{(y) }8 {{t}^{3}}-8 t\]

-->	pic1 : [ color = red , nticks = 100 , parametric ( x , y , t , − 1 . 5 , 1 . 5 ) ] $ wxdraw ( apply ( gr2d , pic1 ) ) ;

\[\operatorname{ }\]

-->	pic2 : [ color = blue , explicit ( ( X − 3 ) · sqrt ( X + 1 ) , X , − 1 , 8 ) ] $ wxdraw ( apply ( gr2d , append ( pic1 , pic2 ) ) ) ;

\[\operatorname{ }\]

-->	map ( lambda ( [ e ] , subst ( e , [ t , x , y ] ) ) , solve ( y = 0 , t ) ) ;

\[\operatorname{ }[[-1,3,0],[1,3,0],[0,-1,0]]\]

-->	kill ( x , y ) $

2 Implicit plotting

Exercise 2.1

-->	wxdraw2d ( ip_grid = [ 200 , 200 ] , implicit ( y ^ 2 = x ^ 3 − x , x , − 1 . 5 , 3 , y , − 3 , 3 ) ) ;

\[\operatorname{ }\]

-->	a0 : 2 / 3 ^ ( 3 / 2 ) , float ;

\[\operatorname{(a0) }0.3849001794597506\]

-->

wxdraw ( gr2d (
       ip_grid = [ 100 , 100 ] ,
       color = red ,
       implicit ( y ^ 2 = x ^ 3 − x + a0 , x , − 1 . 5 , 2 , y , − 3 , 3 ) ,
       color = blue ,
       implicit ( y ^ 2 = x ^ 3 − x + a0 + 0 . 1 , x , − 1 . 5 , 2 , y , − 3 , 3 ) ,
       color = green ,
       implicit ( y ^ 2 = x ^ 3 − x + a0 − 0 . 1 , x , − 1 . 5 , 2 , y , − 3 , 3 )
   ) ) ;

\[\]\[rat: replaced -0.3849001794597506 by -14326002/37220045 = -0.3849001794597508 \]\[rat: replaced -0.4849001794597506 by -39245734/80935697 = -0.4849001794597506 \]\[rat: replaced -0.2849001794597505 by -4241599/14888018 = -0.2849001794597508\]

\[\operatorname{ }\]

Exercise 2.2

-->	z : x ^ 2 + y ^ 2 + a · ( sin ( 2 · %pi · x ) + cos ( 2 · %pi · y ) ) ;

\[\operatorname{(z) }a\, \left( \cos{\left( 2 \ensuremath{\pi} y\right) }+\sin{\left( 2 \ensuremath{\pi} x\right) }\right) +{{y}^{2}}+{{x}^{2}}\]

-->

wxdraw (
gr2d ( ip_grid = [ 200 , 200 ] , proportional_axes = xy , implicit ( subst ( a = 1 , z ) = 100 , x , − 15 , 15 , y , − 15 , 15 ) ) ,
gr2d ( ip_grid = [ 200 , 200 ] , proportional_axes = xy , implicit ( subst ( a = 5 , z ) = 100 , x , − 15 , 15 , y , − 15 , 15 ) ) ,
gr2d ( ip_grid = [ 200 , 200 ] , proportional_axes = xy , implicit ( subst ( a = 20 , z ) = 100 , x , − 15 , 15 , y , − 15 , 15 ) ) ,
columns = 3
) ;

\[\operatorname{ }\]

3 Plotting lists of points

Exercise 4.1

-->	vals : [ 10 , 40 , 20 , 30 , 50 , 10 , 20 ] ;

\[\operatorname{(vals) }[10,40,20,30,50,10,20]\]

-->	makelist ( [ i , vals [ i ] ] , i , 1 , length ( vals ) ) ;

\[\operatorname{ }[[1,10],[2,40],[3,20],[4,30],[5,50],[6,10],[7,20]]\]

-->	wxdraw ( gr2d ( xrange = [ 0 , 8 ] , yrange = [ 0 , 60 ] , points ( makelist ( [ i , vals [ i ] ] , i , 1 , length ( vals ) ) ) ) ) ;

\[\operatorname{ }\]

-->	wxdraw ( gr2d ( xrange = [ 0 , 8 ] , yrange = [ 0 , 60 ] , points_joined = true , point_type = filled_circle , points ( makelist ( [ i , vals [ i ] ] , i , 1 , length ( vals ) ) ) ) ) ;

\[\operatorname{ }\]

Exercise 3.2

-->	N : 1000 $

-->	primes : [ 2 ] $ for i from 2 thru N do ( primes : cons ( next_prime ( primes [ 1 ] ) , primes ) ) $ primes : reverse ( primes ) $

-->	wxdraw ( gr2d ( points ( makelist ( [ i , primes [ i ] ] , i , 1 , N ) ) , color = red , explicit ( x · ( log ( log ( x ) ) + log ( x ) − 1 ) , x , 1 , N ) ) ) ;

\[\operatorname{ }\]

-->	kill ( N ) $

Exercise 3.3

-->

20 ! ;

\[\operatorname{ }2432902008176640000\]

-->	1 . 0 · 20 ! ;

\[\operatorname{ }2.43290200817664 {{10}^{18}}\]

-->	f ( x ) : = sqrt ( 2 · %pi ) · x ^ ( x + 1 / 2 ) · exp ( − x ) ;

\[\operatorname{ }\operatorname{f}(x):=\sqrt{2 \ensuremath{\pi} } {{x}^{x+\frac{1}{2}}} \operatorname{exp}\left( -x\right) \]

-->	wxdraw ( gr2d ( points ( makelist ( [ i , log ( i ! ) ] , i , 1 , 20 ) ) , color = red , explicit ( log ( f ( x ) ) , x , 1 , 20 ) ) ) ;

\[\operatorname{ }\]

-->	kill ( f ) $

Exercise 3.4

-->	q ( c , x ) : = c · x · ( 1 − x ) ;

\[\operatorname{ }\operatorname{q}\left( c,x\right) :=c x\, \left( 1-x\right) \]

-->	f ( n , c ) : = block ( [ ] , if n = 0 then return ( 0 . 5 ) else return ( q ( c , f ( n − 1 , c ) ) ) ) ;

\[\operatorname{ }\operatorname{f}\left( n,c\right) :=\operatorname{block}\left( [],\operatorname{if} n=0 \operatorname{then} \operatorname{return}\left( 0.5\right) \operatorname{else} \operatorname{return}\left( \operatorname{q}\left( c,\operatorname{f}\left( n-1,c\right) \right) \right) \right) \]

-->	F ( n , c ) : = block ( l : [ [ 0 , 0 . 5 ] ] , for i from 1 thru n do ( l : cons ( [ i , q ( c , l [ 1 ] [ 2 ] ) ] , l ) ) , return ( reverse ( l ) ) ) ;

\[\operatorname{ }\operatorname{F}\left( n,c\right) :=\operatorname{block}\left( l:[[0,0.5]],\operatorname{for} i \operatorname{thru} n \operatorname{do} l:\operatorname{cons}\left( [i,\operatorname{q}\left( c,{{\left( {l_1}\right) }_2}\right) ],l\right) ,\operatorname{return}\left( \operatorname{reverse}(l)\right) \right) \]

-->	wxdraw ( gr2d ( points ( F ( 500 , 3 . 44 ) ) ) ) ;

\[\operatorname{ }\]

Created with wxMaxima.

The source of this Maxima session can be downloaded here.