work number 1

27.
restart;
R:=3;
x:=t->R*(t-sin(t));
y:=t->R*(1-cos(t));
plot([x(t),y(t), t=0..6*Pi], color = black, scaling = constrained);

28.
restart;
f:=x->trunc((((1+sqrt(5))/2)^x-((1-sqrt(5))/2)^x)/sqrt(5));
A:=[seq([x,f(x)], x=0..10)];
plot(A, style = point, color = black);

restart;
f:=x-> f(x-2)+f(x-1);
f(0):=0;
f(1):=1;
A:=[seq([x,f(x)], x=0..10)];
plot(A, style = point, color = black);

29.
restart;
y:=x-> -(x^2-4*x)/(x-2)^2;
a:=t-> 2+0*t;
b:=t-> 0+t;
A:=[seq([a(t),b(t)],t=-4..8)];
plot([A,y(x)], x=-4..8, y=-4..8, color = black);

30.
restart;
with(plots):
r:=(s,t)->4+s*cos(t/2);
f:=(s,t)->t;
h:=(s,t)->s*sin(t/2);
plot3d([r(s,t),f(s,t),h(s,t)],s=-Pi..Pi,t=0..2*Pi,color=white, coords = cylindrical);

31.
restart;
with(plots):
a:=1;
implicitplot(r^2=2*a^2*cos(2*h), r=-2..2, h=0..2*Pi, color = black, coords=polar, numpoints = 2000, scaling = constrained);

restart;
with(plots);
c := 1;
implicitplot((x^2+y^2)^2 = 2*c^2*(x^2-y^2), x = -2 .. 2, y = -2 .. 2, scaling = constrained, gridrefine = 2, color = blue, numpoints = 2000)

32.
restart;
with(plots);
k := 1;
spiral := polarplot(k*phi, phi = 0 .. 12*Pi, axes = none, color = black):
ball := proc (phi, ro)
 pointplot([[phi, ro]], color = red, coords = polar, symbol = circle, symbolsize = 30, axes = none)
end proc:
anim := animate(ball, [phi, k*phi], phi = 0 .. 12*Pi, scaling = constrained, frames = 150);
spiral := polarplot(k*phi, phi = 0 .. 12*Pi, axes = none, color = black);
display(anim, spiral);

33.
restart;
with(plots):
with(plottools):
a :=2:
b := 1:
N := 23:
Ellipse := plot([a*cos(t), b*sin(t), t=0..2*Pi], color = blue):
Ani := [seq(rotate(Ellipse, ((k-1))*Pi/N), k=1..N+1)]:
display(Ani, insequence = true, scaling = constrained, axes = none);

34.
restart;
S:={x^2+2*y^2=17, x^2-2*x*y=-3};
sol:=solve(S,{x,y}, explicit);
assign(sol[1]);
[x,y];

35.
restart;
urav:=x^3-(2*a-1)*x^2+x+(2*a+1)=0;
sol:=solve(urav,x,explicit);
urav2:=sol[1] + sol[2] + sol[3] = sol[1]^2 + sol[2]^2 + sol[3]^2;
sol2:=solve(urav2, a, explicit);

36.
restart;
restart;
result := 0:
num_of_roots := 0:
eqn := (x-1)^2 - (x^2-2*x)^3 = 1;
sol := solve(eqn, x, explicit = true);
sol := [sol];
for i from 1 to nops(sol) do
  if trunc(sol[i]) > 0
    then result := result + sol[i];
    num_of_roots := num_of_roots + 1;
  end if:
end do:
result;
num_of_roots;

restart;
result := 0;
num_of_roots := 0;
eqn := (x-1)^2-(x^2-2*x)^3 = 1;
sol := solve({eqn, x > 0}, x, explicit = true);
sol := [sol];
sol := map(op, sol);
sol := map(rhs, sol);
for i to nops(sol) do
 result := result+sol[i];
 num_of_roots := num_of_roots+1
end do;
result;
num_of_roots;

37.
restart;
with(plots):
eqn1 := x^2 - 2*x*y + 2*y^2 = 6:
eqn2 := x^2 - 2*y^2 + 8*x = -6:
eqns := {eqn1, eqn2}:
sol := solve(eqns, {x, y}, explicit = true, real);
first := implicitplot(eqn1, x=-4..4, y=-4..4, color = blue, thickness = 2, gridrefine = 3, numpoints = 5000):
second := implicitplot(eqn2, x=-4..4, y=-4..4, color = red, thickness = 2, gridrefine = 3, numpoints = 5000):
display(first, second);