÷ÈÓ°Ö±²¥

True/False- Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Each point in the plane has a unique representation in rectangular coordinates.

(b) True or False: Each point in the plane has a unique representation in polar coordinates.

(c) True or False: If $b≠c$, then the graphs of the polar equations $r=b$and $r=c$are different.

(d) True or False: If $\mathrm{Î\pm }≠\mathrm{β}$, then the graphs of the polar equations $\mathrm{θ}=\mathrm{Î\pm }$and $\mathrm{θ}=\mathrm{β}$are different.

(e) True or False: The graph of $r=\csc \mathrm{θ}$for $-\frac{\mathrm{π}}{2}<\mathrm{θ}<\frac{\mathrm{π}}{2}$is a horizontal line in a polar coordinate system.

(f) True or False: In a polar coordinate system, the coordinates $(r,\mathrm{θ})$and $(r,\mathrm{θ}+\mathrm{π})$represent the same point if and only if $r=0$.

(g) True or False: When $A$and $B$are nonzero constants, the graph of $r=A\sin \mathrm{θ}+B\cos \mathrm{θ}$ is a circle in a polar coordinate system.

(h) True or False: Every function $r=f\left(\mathrm{θ}\right)$in a polar coordinate system can be expressed in terms of rectangular coordinates $x$and $y$.

What is the difference between a cardioid and a limacon?

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Every function $y=f\left(x\right)$can be written in terms of parametric equations.

(b) True or False: Given parametric equations $x=x\left(t\right)$and $y=y\left(t\right)$the parameter can be eliminated to obtain the form y = f (x).

(c) True or False: Every parametric curve passes the vertical line test.

(d) True or False: Every curve in the plane has a unique expression in terms of parametric equations.

(e) True or False: If the functions x = f (t) and y = g(t) are differentiable for every t ∈ R, then the parametric curve defined by x and y is differentiable for every value of t.

(f) True or False: A curve parametrized by x = x(t), y = y(t) has a horizontal tangent line at (x(t0), y(t0)) if y (t) = 0.

(g) True or False: A curve parametrized by x = x(t), y = y(t) has a horizontal tangent line at (x(t0), y(t0)) if x (t) = 0 and y (t) = 0.

(h) True or False: The cycloid curve associated with a circle of radius r is made up of a series of semicircles of radius r.

An epicycloid is another variation of a cycloid in which the point tracing the path is on the circumference of a wheel, but the wheel is rolling without slipping on the outside of another wheel, instead of along a horizontal track. If the radius of the rolling wheel is k and the radius of the fixed wheel is r, find parametric equations for the epicycloid.