>    restart;
with( plots ): 

Warning, the name changecoords has been redefined

Here is the Maple code to create the 8 plots for #45 (Varberg, Purcell, Rigdon, 8th edition).  Animation provides a more informative view of these curves.  The animatecurve commands for each of these curves is provided.  Be forewarned that some of these plots are large and could crash Maple on some computers.  Please save your work before running the animations.

The following commands create the 8 plots - static and animated - for #55.

>    R45a := cos(theta/2);      DOMa := theta=0..4*Pi;
R45b := sec(3*theta);      DOMb := theta=-2*Pi/3..2*Pi/3;
R45c := 2-3*sin(5*theta);  DOMc := theta=0..2*Pi;
R45d := 1-2*sin(5*theta);  DOMd := theta=0..2*Pi;
R45e := cos(theta/4);      DOMe := theta=0..8*Pi;
R45f := theta*cos(theta);  DOMf := theta=-25..25;
R45g := 1/theta^(3/2);     DOMg := theta=0.5..20;
R45h := 2*cos(3*theta);    DOMh := theta=0..2*Pi;

R45a := cos(1/2*theta)

DOMa := theta = 0 .. 4*Pi

R45b := sec(3*theta)

DOMb := theta = -2/3*Pi .. 2/3*Pi

R45c := 2-3*sin(5*theta)

DOMc := theta = 0 .. 2*Pi

R45d := 1-2*sin(5*theta)

DOMd := theta = 0 .. 2*Pi

R45e := cos(1/4*theta)

DOMe := theta = 0 .. 8*Pi

R45f := theta*cos(theta)

DOMf := theta = -25 .. 25

R45g := 1/(theta^(3/2))

DOMg := theta = .5 .. 20

R45h := 2*cos(3*theta)

DOMh := theta = 0 .. 2*Pi

>    polarplot( R45a, DOMa, title="I" );
polarplot( R45b, DOMb, view=[-10..10,-10..10], title="II" );
polarplot( R45c, DOMc, title="III" );
polarplot( R45d, DOMd, title="IV" );
polarplot( R45e, DOMe, title="V" );
polarplot( R45f, DOMf, title="VI" );
polarplot( R45g, DOMg, title="VII" );
polarplot( R45h, DOMh, title="VIII" );

[Maple Plot]

[Maple Plot]

[Maple Plot]

[Maple Plot]

[Maple Plot]

[Maple Plot]

[Maple Plot]

[Maple Plot]

>   

The following animations provide a better understand of how these curves are formed.

>    animatecurve( [R45a,theta, DOMa], coords=polar, numpoints=200, title="I" );

[Maple Plot]

>    animatecurve( [R45b,theta, DOMb], view=[-10..10,-10..10], coords=polar, numpoints=200, title="II" );

[Maple Plot]

>    animatecurve( [R45c,theta, DOMc], coords=polar, numpoints=200, title="III" );

[Maple Plot]

>    animatecurve( [R45d,theta, DOMd], coords=polar, numpoints=200, title="IV" );

[Maple Plot]

>    animatecurve( [R45e,theta, DOMe], coords=polar, numpoints=200, title="V" );

[Maple Plot]

>    animatecurve( [R45f,theta, DOMf], coords=polar, numpoints=200, title="VI" );

[Maple Plot]

>    animatecurve( [R45g,theta, DOMg], coords=polar, numpoints=200, title="VII" );

[Maple Plot]

>    animatecurve( [R45h,theta, DOMh], coords=polar, numpoints=200, title="VIII" );

[Maple Plot]

>   

>